# Solvable models in the KPZ class: approach through periodic and free
boundary Schur measures

Takashi Imamura  T. Imamura, Department of Mathematics and Informatics, Chiba
University, Chiba, 263-8522, Japan<EMAIL_ADDRESS>,  Matteo
Mucciconi  M. Mucciconi, Department of Mathematics, University of Warwick,
Coventry, CV4 7HP, United Kingdom<EMAIL_ADDRESS>and  Tomohiro
Sasamoto  T. Sasamoto, Department of Physics, Tokyo Institute of Technology,
Tokyo, 152-8551, Japan<EMAIL_ADDRESS>
###### Abstract.

We explore probabilistic consequences of correspondences between $q$-Whittaker
measures and periodic and free boundary Schur measures established by the
authors in the recent paper [arXiv:2106.11922]. The result is a comprehensive
theory of solvability of stochastic models in the KPZ class where exact
formulas descend from mapping to explicit determinantal and pfaffian point
processes. We discover new variants of known results as determinantal formulas
for the current distribution of the ASEP on the line and new results such as
Fredholm pfaffian formulas for the distribution of the point-to-point
partition function of the Log Gamma polymer model in half space. In the latter
case, scaling limits and asymptotic analysis allow to establish Baik-Rains
phase transition for height function of the KPZ equation on the half line at
the origin.

###### Contents

  1. 1 Introduction
  2. 2 $q$-Whittaker Measure and its half space variant
  3. 3 Models in the KPZ class and $q$-Whittaker measures
  4. 4 Periodic and free boundary Schur measures
  5. 5 Fredholm determinant and pfaffian formulas for KPZ models
  6. 6 Asymptotic results
  7. 7 KPZ equation
  8. A Fredholm determinant and Fredholm pfaffian
  9. B Estimates on $q$-Pochhammer symbols

##  1\. Introduction

###  1.1. One dimensional KPZ equation and stochastic heat equation

The Kardar-Parisi-Zhang (KPZ) equation [KPZ86] is the default model of growth
of random surfaces and in one dimension it reads

(1.1)
$\partial_{t}\mathcal{H}=\frac{1}{2}\partial_{x}^{2}\mathcal{H}+\frac{1}{2}\left(\partial_{x}\mathcal{H}\right)^{2}+\dot{\mathscr{W}},$

where $\mathcal{H}=\mathcal{H}(x,t),x\in\mathbb{R},t>0,$ is the height
function and $\dot{\mathscr{W}}$ is the space-time gaussian white noise with
covariance
$\mathbb{E}\left(\dot{\mathscr{W}}(x,t)\dot{\mathscr{W}}(y,s)\right)=\delta_{x-y}\delta_{t-s}$.
This equation is notoriously ill posed due to the non linear term
$\left(\partial_{x}\mathcal{H}\right)^{2}$ and making sense of the solution of
(1.1) has been one of the recent major accomplishments in the theory of
stochastic analysis [BG97, Hai14, GIP15]. In this paper we consider the Cole-
Hopf solutions of the KPZ equation defined as

(1.2) $\mathcal{H}(x,t)=\log\mathscr{Z}(x,t),$

where $\mathscr{Z}$ is a _mild solution_ of the stochastic heat equation with
multiplicative noise (abbreviated as “mSHE” in the text)

(1.3)
$\partial_{t}\mathscr{Z}=\frac{1}{2}\partial_{x}^{2}\mathscr{Z}+\mathscr{Z}\dot{\mathscr{W}}.$

The correct prescription to interpret the noise term in the right hand side is
discussed [BC95], whereas for an introduction to mild solutions see [BC95,
Qua13] and references therein. Throughout the text we will mostly discuss the
solution of the mSHE $\mathscr{Z}$ rather than $\mathcal{H}$, as this allows
to phrase our statements rigorously. In all cases the use of the Cole-Hopf
transformation (1.2) will be justified.

Interest in the KPZ equation has, in the last decades, been motivated also by
the fact that certain geometries and initial conditions allow to write (and
sometimes prove) exact formulas for the distribution of the height function
$\mathcal{H}$ [ACQ11, CLDR10, Dot10, SS10, LDC12, GLD12, IS13, BCFV15, BBC16,
BBCW18, KLD20, BKLD20, DNKLDT20, BLD21]. In this paper we deal with two such
choices of solvable geometries and initial data, which we introduce next.

###  1.2. The KPZ equation in full space with narrow wedge initial conditions

The proper formulation of the problem of this case is given in terms of the
mSHE which reads

(1.4)
$\begin{cases}\partial_{t}\mathscr{Z}=\frac{1}{2}\partial_{x}^{2}\mathscr{Z}+\mathscr{Z}\dot{\mathscr{W}},\qquad
x\in\mathbb{R},t\in\mathbb{R}_{+},\\\
\mathscr{Z}(x,0)=\delta_{0}(x),\end{cases}$

where $\delta_{0}$ is the Dirac delta function. Heuristically, for the
function $\mathcal{H}$, the initial data in (1.4) represents a situation where
the random KPZ growth is started from a wedge
$\mathcal{H}(x,0)``="-|x|/\varepsilon$, with $\varepsilon$ infinitesimally
small, which justifies the name. Under delta initial conditions the function
$\mathscr{Z}$ can be interpreted as the point-to-point partition function of
the _continuum directed random polymer_ [ACQ11, AKQ14], which is a model of a
polymer in 1+1 dimension propagating through a space-time in uncorrelated
random environment. This is evident expressing the solution of (1.4) by the
Feyman-Kac formula

(1.5)
$\mathscr{Z}(x,t)=\mathbb{E}\left[:\exp:\left\\{\int_{0}^{t}\dot{\mathscr{W}}(\mathscr{B}(s),s)\mathop{}\\!\mathrm{d}s\right\\}\bigg{|}\mathscr{B}(0)=0,\,\mathscr{B}(t)=x\right],$

where $:\exp:$ indicates the Wick ordered exponential and the expectation is
taken only over the brownian bridge $\mathscr{B}$. In this way the integral in
(1.5) has the natural interpretation of the random energy of the polymer
$\mathscr{B}$.

The first exact solution of the mSHE (1.4) became available in 2010 after the
breakthrough result of four different groups [ACQ11, Dot10, CLDR10, SS10].
Famously, this solution expresses the Laplace transform of the probability
distribution of $\mathscr{Z}$ as a Fredholm determinant.

######  Theorem 1.1.

Consider $\mathscr{Z}(x,t)$ the solution of the stochastic heat equation
(1.4). Then, we have

(1.6)
$\mathbb{E}\left[e^{-e^{-s+\log\mathscr{Z}(0,t)+t/24}}\right]=\det(1-\mathscr{K}_{\mathrm{mSHE}})_{\mathbb{L}^{2}(s,+\infty)},$

where $\mathscr{K}_{\mathrm{mSHE}}$ is the linear operator with integral
kernel

(1.7)
$\mathscr{K}_{\mathrm{mSHE}}(x,y)=\int_{\mathrm{i}\mathbb{R}-d}\frac{\mathop{}\\!\mathrm{d}Z}{2\pi\mathrm{i}}\int_{\mathrm{i}\mathbb{R}+d}\frac{\mathop{}\\!\mathrm{d}W}{2\pi\mathrm{i}}e^{-\frac{t}{2}\left(\frac{Z^{3}}{3}-\frac{W^{3}}{3}\right)+Zx-
Wy}\frac{\pi}{\sin[\pi(W-Z)]}$

and for the integration contours we choose $d\in(0,1/2)$.

######  Remark 1.2.

The kernel $\mathscr{K}_{\mathrm{mSHE}}$ can be written as a “positive
temperature variant” of the Airy kernel,

(1.8) $\mathscr{K}_{\mathrm{mSHE}}(x,y)=\int_{\mathbb{R}}\frac{{\rm
Ai}((x+\xi)/\gamma_{t}){\rm
Ai}((y+\xi)/\gamma_{t})}{1+e^{-\xi}}\mathop{}\\!\mathrm{d}\xi,$

with $\gamma_{t}=(t/2)^{1/3}$.

From a physical standpoint the explicit solution of (1.4) is relevant as it
has allowed to rigorously establish the large time asymptotic limit

(1.9)
$\mathcal{H}(0,t)\approx-\frac{t}{24}+\frac{t^{1/3}}{2^{1/3}}\chi_{\mathrm{GUE}},$

where $\chi_{\mathrm{GUE}}$ follows the GUE Tracy-Widom distrubution[TW94].
Historically, the anomalous scaling exponent $1/3$ had been predicted in
[KPZ86, HH85], whereas more precise conjectures about the universal
fluctuations for a large class of models (KPZ universality class) were
formulated [PS02] after the works [BDJ99, Joh00]. For the case of the KPZ
equation (1.9) confirms the predictions from the KPZ scaling theory.

In this paper we will present a derivation of the solution of theorem 1.1
which is conceptually different from ideas used in literature since 2010. The
advantage of our techniques, compared to pre-existing methods, is that they
explain the emergence of Fredholm determinants, by establishing an a priori
correspondence between the height function and the edge statistics of a free
fermion at positive temperature. Moreover, methods developed in this paper can
be applied not only to the study of the mSHE/KPZ equation, but also to a class
of discrete models in the KPZ class. These are the ASEP, the $q$-PushTASEP, or
the Log Gamma polymer and we will discuss this aspect more in section 1.4.
More remarkably our methods can be naturally adapted to the more challenging
task of treating the mSHE/KPZ equation in half space, as we discuss in the
next subsection. See also section 1.6 for more detailed account about novelty
of our work.

###  1.3. The KPZ equation in half space with narrow wedge initial conditions.

In this case the problem is phrased in terms of the mSHE as

(1.10)
$\begin{cases}\partial_{t}\mathscr{Z}=\frac{1}{2}\partial_{x}^{2}\mathscr{Z}+\mathscr{Z}\dot{\mathscr{W}},\qquad
x\in\mathbb{R}_{\geq 0},t\in\mathbb{R}_{+},\\\
\mathscr{Z}(x,0)=\delta_{0}(x),\qquad\left(\partial_{x}-\omega\right)\mathscr{Z}(x,t)\bigr{|}_{x=0}=0.\end{cases}$

The study of the well-posedness of (1.10) was done in [Wu20, Par19b] and in
particular a mild solution of (1.10) exists, is unique and it is almost surely
positive. This justifies the definition of
$\mathcal{H}^{\mathrm{hs}}=\log\mathscr{Z}^{\mathrm{hs}}$ as a solution of the
KPZ equation in half space with initial conditions consisting in a narrow
wedge peaked at the origin. The Robin boundary conditions of (1.10) become,
for the height function $\mathcal{H}^{\mathrm{hs}}$ the Neumann boundary
conditions

(1.11) $\partial_{x}\mathcal{H}^{\mathrm{hs}}(x,t)\big{|}_{x=0}=\omega.$

A physical realization of such boundary condition was proposed in [IT18]. As
in the full space case the solution $\mathscr{Z}^{\mathrm{hs}}$ of (1.10) can
also be interpreted as a point-to-point partition function of the continuum
random polymer propagating in the half line $[0,+\infty)$ with an interaction
produced by a wall at the origin. Again, this is evident from the Feynman-Kac
representation

(1.12) $\mathscr{Z}^{\mathrm{hs}}(x,t)\\\
=\mathbb{E}\left[:\exp:\left\\{\int_{0}^{t}\left(\dot{\mathscr{W}}(\mathscr{B}^{\mathrm{hs}}(s),s)-\omega\delta_{0}(\mathscr{B}^{\mathrm{hs}}(s))\right)\mathop{}\\!\mathrm{d}s\right\\}\bigg{|}\mathscr{B}^{\mathrm{hs}}(0)=0,\,\mathscr{B}^{\mathrm{hs}}(t)=x\right],$

where the expectation is taken over the reflected Brownian bridge
$\mathscr{B}^{\mathrm{hs}}$, i.e. a Brownian bridge conditioned to remain
nonnegative at all times. From (1.12) we observe that the interaction with the
wall, modulated by the parameter $\omega$, is attractive when $\omega<0$ and
repulsive for $\omega>0$.

####  1.3.1. Fredholm pfaffian solution.

The truly novel part of our results consists in the derivation of Fredholm
pfaffian formulas for solvable models related to the half space KPZ equation.
The typical approach to this problem has been that of leveraging integrable
strucures of the continuum directed polymer model, which will be discussed in
the next subsection, to obtain exact formulas for the partition funciton. This
route involves the use of complicated and mathematically non-rigorous Bethe
ansatz computations that only recently have led to manageable, although
conjectural, pfaffian formulas; see [GLD12, BBC16, KLD20, DNKLDT20].
Alternatively, in the more general case of the Log Gamma polymer model, exact
formulas for the distribution of the partition function have been found
through combinatorial techniques in [OSZ14] or in [BBC20] using eigenrelations
of Macdonald polynomials. These last results, although mathematically rigorous
have the disadvantage to be hard to manage and did not lead to a proof of
asymptotic results. The only mathematically rigorous derivation of a pfaffian
solution of (1.10), prior to this paper, was given in [BBCW18], where the
authors managed to study the case $\omega=-\frac{1}{2}$, through a matching
between the Laplace transform of the KPZ equation and a nonlocal average of a
pfaffian point process. In this paper we overcome these technical difficulties
by taking a different point of view and establishing a direct and a priori
corresondence between KPZ models in half space and explicit pfaffian point
processes. This allows us to study the full range $\omega\in\mathbb{R}$.

For the next theorem we recall the standard notation

(1.13) $J=\left(\begin{matrix}0&1\\\ -1&0\end{matrix}\right),$

and we use the convention that $J(x,y)=\mathbf{1}_{x=y}J$.

######  Theorem 1.3.

Consider $\mathscr{Z}^{\mathrm{hs}}(x,t)$ the solution of the half space
stochastic heat equation (1.10) with initial conditions
$\mathscr{Z}^{\mathrm{hs}}(x,0)=\delta_{x}$ and boundary parameter
$\omega>-1/2$. Then, we have

(1.14)
$\mathbb{E}\left[e^{-e^{-s+\log\mathscr{Z}^{\mathrm{hs}}(0,t)+t/24}}\right]=\mathrm{Pf}\left[J-\mathscr{L}_{\mathrm{mSHE}}\right]_{\mathbb{L}^{2}(s,+\infty)},$

where $\mathscr{L}_{\mathrm{mSHE}}$ is the $2\times 2$ matrix linear operator
with integral kernel

(1.15)
$\mathscr{L}_{\mathrm{mSHE}}(X,Y)=\left(\begin{matrix}\mathscr{K}^{\mathrm{hs}}(X,Y)&-\partial_{y}\mathscr{K}^{\mathrm{hs}}(X,Y)\\\
-\partial_{x}\mathscr{K}^{\mathrm{hs}}(X,Y)&\partial_{x}\partial_{y}\mathscr{K}^{\mathrm{hs}}(X,Y)\end{matrix}\right),$

with

(1.16)
$\begin{split}\mathscr{K}^{\mathrm{hs}}(X,Y)=\int_{\mathrm{i}\mathbb{R}+d}\frac{\mathop{}\\!\mathrm{d}Z}{2\pi\mathrm{i}}\int_{\mathrm{i}\mathbb{R}+d}\frac{\mathop{}\\!\mathrm{d}W}{2\pi\mathrm{i}}&e^{\frac{t}{2}\left(\frac{Z^{3}}{3}+\frac{W^{3}}{3}\right)-ZX-
WY}\frac{\Gamma(\frac{1}{2}+\omega-Z)}{\Gamma(\frac{1}{2}+\omega+Z)}\frac{\Gamma(\frac{1}{2}+\omega-W)}{\Gamma(\frac{1}{2}+\omega+W)}\\\
&\times\Gamma(2Z)\Gamma(2W)\frac{\sin[\pi(Z-W)]}{\sin[\pi(Z+W)]}\end{split}$

and we assume $0<d<\min(1/2,1/2+\omega)$.

A few remarks are in order.

######  Remark 1.4.

In section 7.3 we report equivalent expressions for the Fredholm pfaffian in
the right hand side of (1.14). These had been previously derived in [KLD20],
although through non rigorous Bethe ansatz computations.

######  Remark 1.5.

In section 5 we will present pfaffian formulas analogous to that of (1.14)
characterizing the distribution of the rightmost particle in a $q$-PushTASEP
with particle creation and the free energy of the Log Gamma polymer model in
half space. Unlike for the case of the mSHE, pfaffian formulas for these more
general models had not appeared before in literature, not even at a
conjectural level.

######  Remark 1.6.

A theorem by Parekh [Par19a] relates the solution of the half space mSHE
(1.10), which we denote by $\mathscr{Z}^{\mathrm{hs}}$ and the solution of the
half space mSHE with certain Brownian initial conditions and Dirichlet
boundary conditions

(1.17)
$\begin{cases}\partial_{t}\mathscr{Z}=\frac{1}{2}\partial_{x}^{2}\mathscr{Z}+\mathscr{Z}\dot{\mathscr{W}},\qquad
x\in\mathbb{R}_{\geq 0},t\in\mathbb{R}_{+},\\\
\mathscr{Z}(x,0)=e^{\mathscr{B}(x)-(\omega+\frac{1}{2})x},\qquad\mathscr{Z}(0,t)=0,\end{cases}$

where $\mathscr{B}(x)$ is a standard Brownian motion. Denoting by
$\bar{\mathscr{Z}}^{\mathrm{hs}}$ the solution of (1.17), then [Par19a,
Theorem 1.1] states the equivalence in distribution

(1.18)
$\mathscr{Z}^{\mathrm{hs}}(0,t)\stackrel{{\scriptstyle\mathcal{D}}}{{=}}\lim_{x\to
0}\frac{1}{x}\bar{\mathscr{Z}}^{\mathrm{hs}}(x,t).$

By this equality, theorem 1.3 also provides an explicit formula for the
probability distribution of $\bar{\mathscr{Z}}^{\mathrm{hs}}(x,t)$ around
$x=0$, although we are not going to state such results in this paper.

Although expression (1.14), as stated, only holds for $\omega>-1/2$, Fredholm
pfaffian expressions for the expectation
$\mathbb{E}\left[e^{-e^{-s+\log\mathscr{Z}^{\mathrm{hs}}(0,t)+t/24}}\right]$
are available for any $\omega\in\mathbb{R}$. They can be derived by analytic
continuation, in the parameter $\omega$, although such procedure, especially
for $\omega<-1/2$, turns out to be not completely trivial. Such results are
presented in theorem 7.3.

####  1.3.2. Limiting distribution and phase transition.

Characterizing the large scale behavior of the free energy of directed
polymers in presence of a wall has been a problem considered in the physics
literature since the work of Kardar [Kar85]. Varying the value of $\omega$,
the partition function of the polymer is expected to undergo a phase
transition, which needs to be characterized. This an example of the so called
“depinning transition”; see [PS02, Section 7.3-7.4]. Non-rigorous explicit
computations [KLD20, BBC20, DNKLDT20] have allowed to conjecture the large
time behavior for the free energy of the polymer model, which stated in terms
of the height function $\mathcal{H}^{\mathrm{hs}}$ reads

(1.19) $\mathcal{H}^{\mathrm{hs}}(0,t)\approx-
f_{\omega}t+\sigma_{\omega}t^{\zeta(\omega)}\chi_{\omega},$

where $f_{\omega},\sigma_{\omega}$ are explicit constants depending on
$\omega$ and

(1.20) $\zeta(\omega)=\begin{cases}\frac{1}{3}\qquad&\text{if
}\omega\geq-\frac{1}{2},\\\ \frac{1}{2}\qquad&\text{if
}\omega<-\frac{1}{2},\end{cases}\qquad\text{and}\qquad\chi_{\omega}=\begin{cases}\chi_{\mathrm{GSE}}\qquad&\text{if
}\omega>-\frac{1}{2},\\\ \chi_{\mathrm{GOE}}\qquad&\text{if
}\omega=-\frac{1}{2},\\\ \mathcal{N}(0,1)\qquad&\text{if
}\omega<-\frac{1}{2}.\end{cases}$

Above $\chi_{\mathrm{GSE}},\chi_{\mathrm{GOE}}$ follow respectively the GSE
and the GOE Tracy-Widom distributions [TW96], while $\mathcal{N}(0,1)$ is a
standard Gaussian random variable. As mentioned above, the only rigorous
previous study of this problem was carried out in [BBCW18] where the case
$\omega=-\frac{1}{2}$ was confirmed.

In the next theorem we establish the phase transition (1.19), (1.20). Scaling
$\omega$ properly around the critical value $-\frac{1}{2}$ as a function of
another parameter $\xi$ we find that $\log\mathscr{Z}^{\mathrm{hs}}(0,t)$
possesses fluctuations described by a crossover distribution
$\mathrm{F}_{\mathrm{cross}}(\,\cdot\,|\xi)$. This interpolates between the
GOE Tracy-Widom distribution, which is recovered setting $\xi=0$ and the GSE
Tracy-Widom distribution, which is obtained as the limit of
$F_{\mathrm{cross}}(\,\cdot\,|\xi)$ when $\xi\to+\infty$. The explicit
expression of $F_{\mathrm{cross}}(\,\cdot\,|\xi)$ is given in section 6.1,
where we also prove, in corollary 6.8, that it corresponds to a distribution
described first by Baik and Rains in [BR01c].

######  Theorem 1.7.

Let $\mathscr{Z}^{\mathrm{hs}}(x,t)$ be the solution of the stochastic heat
equation in half space (1.10). Then, the following limits hold:

  * •

if $\omega\geq-1/2$, rescaling $\omega=-1/2+\frac{\xi}{2^{-1/3}t^{1/3}}$ we
have

(1.21)
$\lim_{t\to+\infty}\mathbb{P}\left[\frac{\log\mathscr{Z}^{\mathrm{hs}}(0,t)+t/24}{2^{-1/3}t^{1/3}}\leq
r\right]=F_{\mathrm{cross}}(r;\xi),$

where $F_{\mathrm{cross}}$ is the Baik-Rains crossover distribution of
definition 6.3;

  * •

if $\omega<-1/2$, we have

(1.22)
$\lim_{t\to+\infty}\mathbb{P}\left[\frac{\log\mathscr{Z}^{\mathrm{hs}}(0,t)+f_{\omega}t}{\sigma_{\omega}t^{1/2}}\leq
r\right]=\int_{-\infty}^{r}\frac{e^{-u^{2}/2}}{\sqrt{2\pi}}\mathop{}\\!\mathrm{d}u,$

where $f_{\omega}=\frac{1}{6}+\frac{\omega}{2}+\frac{\omega^{2}}{2}$ and
$\sigma_{\omega}=\sqrt{-\omega-\frac{1}{2}}$.

###  1.4. Hierarchies of solvable models

The solvability of the mSHE comes from the fact that the $n$ point average

$\mathbb{E}[\mathscr{Z}(x_{1},t)\cdots\mathscr{Z}(x_{n},t)]$

solves the imaginary time delta Bose gas with $n$ particles. This idea, first
formulated in [Kar87], was used directly by [Dot10, CLDR10], to implement the
celebrated “replica method” and to derive an explicit formula for the moment
generating function of $\mathscr{Z}(x,t)$. Since moments of $\mathscr{Z}(x,t)$
are highly diverging (see [BCS14, Section A.4]), treating directly the mSHE
through replica method yields expressions that cannot be justified
mathematically and therefore other routes to produce explicit formulas are
needed. The standard procedure to circumvent this problem is that of
considering, in place of $\mathscr{Z}$ or $\mathcal{H}$, certain more regular
discretizations which preserve integrability [BCS14]. Such regularized models
usually possess a “temperature” parameter111Here the use of the word
“temperature” is motivated by the appearance of the Fermi-Dirac distribution.
See section 4.2 The same Fermi-Dirac distribution is present already in the
kernel (1.8). $q$ and a height function $h^{(q)}$, which under a proper high
temperature limit converges in distribution to the solution of the KPZ
equation $h^{(q)}\to\mathcal{H}$.

The most well studied of these reguralizations of the KPZ equation is arguably
the Asymmetric Simple Exclusion Process (ASEP), in which case the parameter
$q$ modulates the asymmetry in the jump rate of particles; see section 3.3 for
more details. In this case the height function $h^{(q)}$ is given by the
integrated current through a location and as $q\to 1$ convergence to the Cole-
Hopf solution of the KPZ equation was proven in [BG97]. Considering such weak
noise scaling was the approach adopted in [ACQ11, SS10], where authors were
able to leverage Fredholm determinant formulas found earlier for the ASEP by
Tracy and Widom [TW08b, TW09a, TW09b].

In the last decade, other discrete solvable models have been introduced in
literature. The ones possessing the richest structure admit a formulation in
terms of stochastic vertex models such as Corwin and Petrov’s Higher Spin
Vertex Model [CP16]. For all these models we can apply regularizations of the
replica method, which consist in a combination of stochastic duality [Sch97,
IS11, BCS14, CP16] and Bethe Ansatz [BCG16, BCPS15, BP18]. A more indirect
approach is that of using Macdonald processes [BC14], where KPZ solvable
models arise as certain marginals of a more general 2+1 growth process [BC14,
OP13, BP16, MP17, BP19, BMP21, MP22, BBW18, BBC20, BBCW18, CD21]. In this last
case the common approach is that of leveraging combinatorial properties of
Macdonald functions to derive informations about the model.

In this paper we will mainly consider a particular regularization of the KPZ
equation, that is the $q$-PushTASEP [BP16, MP17]. See Figure 2 below for an
illustration of the model in continuous time. In its most general formulation
this is, admittedly, a rather artificial model, whose definition is postponed
to section 3.1, which nevertheless turns out to be the right one to focus on
for three key reasons enumerated below.

_Multivariate model._ The $q$-PushTASEP depends on a number of parameters
which in the text we denote by $a_{1},a_{2},\dots,b_{1},b_{2},\dots$.
Considering proper scalings of $q,a_{i},b_{j}$, we can transform the
$q$-PushTASEP into several interesting other models, as schematically
described in fig. 1. For instance taking a certain $q\to 1$ limit one obtains
the Log Gamma polymer model introduced by Seppalainen [Sep12]. This model is
of interest on its own and in the last decade has received considerable
attention; see [Zyg18] and references therein.

_Half space variant._ The $q$-PushTASEP possesses a solvable variant that
regularizes the mSHE in half space, called _$q$ -PushTASEP with particle
creation_, introduced in [BBC20]. This is also a multivariate model depending
on parameters denoted by $\gamma,a_{1},a_{2},\dots$, where $\gamma$ governs
the creation of new particles and in the KPZ picture plays the role of
boundary parameter $\omega$. As we will show in the text we will be able to
solve, i.e. write down exact formulas, for this half space model, for general
choices of these parameters. This is remarkable since exact formulas for the
ASEP in half space, that also approximates the mSHE (1.10), are only known for
the particular choice $\omega=-\frac{1}{2}$ and were found in [BBCW18].

_Relation with $q$-Whittaker measures._ Both versions of the $q$-PushTASEP
hinted above have joint law described respectively by the $q$-Whittaker
measure [BC14] and by the half space $q$-Whittaker measure [BBC20], which are
probability measures over partitions $\mu$ of the form respectively

(1.23)
$\mathbb{P}(\mu)\propto\mathscr{P}_{\mu}(a;q)\mathscr{Q}_{\mu}(b;q),\qquad\mathbb{P}(\mu)\propto\mathbf{1}_{\mu^{\prime}\,\text{even}}\mathscr{Q}_{\mu}(\gamma,a;q).$

Here $\mathscr{P}_{\mu},\mathscr{Q}_{\mu}$ are the $q$-Whittaker polynomials;
see section 2 for precise definitions. This is important since the
$q$-Whittaker polynomials have a nice combinatorial structure. In particular
adopting a bijective approach, we were recently able to discover and prove
certain new summation identities [IMS21b] relating the $q$-Whittaker measures
with the periodic Schur measure [Bor07] and the Free boundary Schur measure
[BBNV18]; see (2.26), (2.27) in the text. These are other measures over
partitions $\lambda$ having respectively the form

(1.24)
$\mathbb{P}(\lambda)\propto\sum_{\rho}q^{|\rho|}s_{\lambda/\rho}(a)s_{\lambda/\rho}(b),\qquad\mathbb{P}(\lambda)\propto\mathbf{1}_{\lambda^{\prime}\,\text{even}}\sum_{\rho^{\prime}\,\text{even}}q^{|\rho|/2}s_{\lambda/\rho}(a),$

where $s_{\lambda/\rho}$ are the skew Schur polynomials, defined below in
section 2. They have notable determinantal and pfaffian structures which we
will recall in section 4.

Figure 1. Schematic representation of hierarchy of models related to the
$q$-PushTASEP.

###  1.5. Methods

The location of a tagged particle $\mathsf{x}_{N}(T)$ in the $q$-PushTASEP is
equivalent in distribution to the length of the first row of a random
partition $\mu$ sampled according to the $q$-Whittaker measure [MP17]

(1.25) $\mathsf{x}_{N}(T)\stackrel{{\scriptstyle\mathcal{D}}}{{=}}\mu_{1}+N.$

A summation identity discovered in [IMS21a, IMS21b] relates the first row in
the $q$-Whittaker measure with the first row $\lambda_{1}$ in the periodic
Schur measure, up to a shift

(1.26) $\mu_{1}+\chi\stackrel{{\scriptstyle\mathcal{D}}}{{=}}\lambda_{1}.$

Here $\chi$ is an independent random variable with explicit distribution and
this result is proven below in theorem 4.10. The combination of the previous
two relations immediately grants Fredholm determinant formulas describing the
distribution of the tagged particle $\mathsf{x}_{N}(T)$, borrowing known
results on the periodic Schur measure [Bor07, BB19].

Once formulas are obtained for the most general model, i.e. the $q$-PushTASEP,
we will consider various scaling limits. A change of specializations of
symmetric functions in the $q$-Whittaker measure will give us Fredholm
determinant formulas for the ASEP, thanks to a notable relation between the
Hall-Littlewood measure (seen as a particular case of the $q$-Whittaker
measure) and the ASEP (seen as a particular specialization of the Stochastic
Six Vertex Model) discovered in [BBW18]. A Fredholm determinant representation
for the probability distribution of the free energy of the log Gamma polymer
model will be derived through a $q\to 1$ limit of formulas for the
$q$-PushTASEP. In doing so a number of delicate estimates of decay for (ratio
of) $q$-Pochhammer symbols will be required and we will present them in
appendix B. In literature, bounds for $q$-Pochhammer symbols are usually
derived using certain complicated integral representations involving other
special functions; see, for instance, the recent work [CK21] and references
therein. In our case, all new results given in appendix B will have completely
elementary and self contained proofs, which is also a novel, although somewhat
technical, aspect of our work. Considering a scaling limit, called
_intermediate disorder regime_ [AKQ14], of formula for the Log Gamma polymer
model we will prove theorem 1.1.

As anticipated, we will also derive exact formulas for KPZ models in half
space. Conceptually, the approach we follow mirrors that used to derive
formulas in full space setting. Our starting point is the identity in
distribution [BBC20]

(1.27)
$\mathsf{x}^{\mathrm{hs}}_{N}(T)\stackrel{{\scriptstyle\mathcal{D}}}{{=}}\mu_{1}^{\mathrm{hs}}+N,$

between the $q$-PushTASEP with particle creation on the left hand side and the
half space $q$-Whittaker measure on the right hand side. We then relate,
through a summation identity discovered in [IMS21b], a particular case of the
half space $q$-Whittaker measure with the free boundary Schur measure
[BBNV18]. In formulas our relation reads

(1.28)
$\mu_{1}^{\mathrm{hs}}+\chi\stackrel{{\scriptstyle\mathcal{D}}}{{=}}\lambda_{1}^{\mathrm{hs}}.$

Combining (1.27) and (1.28) we are able to relate the rightmost particle
$\mathsf{x}^{\mathrm{hs}}_{T}(T)$ of the $q$-PushTASEP with particle creation
with an explicit pfaffian point process. Strictly speaking such correspondence
only holds when the parameter governing the creation of new particles, which
plays the role of boundary parameter in the KPZ picture, is set to $\gamma=0$.
To extend the correspondence to all non-negative values of $\gamma$ we will
use a special symmetry of the half space $q$-Whittaker measure found in
[BBC20]; see proposition 2.7 in the text. In this way we find a Fredholm
pfaffian representation for the probability distribution of
$\mathsf{x}_{T}^{\mathrm{hs}}(T)$ valid for a general choice of parameters. We
then move to consider scaling limits of our formula for the $q$-PushTASEP.
Taking the $q\to 1$ limit already considered in full space setting we derive a
Fredholm pfaffian representation for the probability distribution of the
point-to-point partition function of the Log Gamma polymer in half space. The
asymptotic analysis in this case builds up on the study made for the full
space model, although it ends up being slightly more involved; also in this
case technical estimates are needed and they are reported in appendix B.
Formulas we get for the Log Gamma polymer model are amenable to a subsequent
scaling limit, considered in [Wu20], under which the polymer partition
function reduces to the solution $\mathscr{Z}^{\mathrm{hs}}$ of the half space
mSHE (1.10) and in this way we prove theorem 1.3.

In section 6 and section 7 we will perform the large time-space asymptotic
limits of our Fredholm determinant and pfaffian formulas. In this way we
recover GUE limiting fluctuations results for the Log Gamma polymer model in
full space. The emphasis in the proof we provide should be on the simplicity
of the asymptotic analysis, given by the “free fermionic” nature of our
Fredholm operator. This should be compared with other existing techniques (see
e.g. [BCD21]), where a rigorous application of the saddle point method
requires involved manipulations of integral contours of the Fredholm kernel.

The asymptotic analysis of our Fredholm pfaffian formulas will be performed
employing a standard saddle point analysis. We will focus on three special
cases. In section 6.3 we will consider the asymptotic limit of Fredholm
pfaffian formulas for the half space $q$-Whittaker measure for $q=0$. Doing so
we will lay out the generic structure of the asymptotic analysis of our
pfaffian formulas and we will additionally prove a new representation of the
Baik-Rains crossover distribution originally defined in [BR01c]. Subsequently
we will compute the thermodynamic limit of the polymer partition function for
the half space Log Gamma polymer model, proving Baik-Rains phase transition.
Analogous computations will be reported in section 7.4, where we report the
proof of theorem 1.7.

###  1.6. Novelty of our approach.

Lastly, let us emphasize how our techniques differ from those available in the
pre-existing literature and what are the advantages of our approach. In simple
terms we have implemented a bijective approach to the study of the
$q$-Whittaker measure, and considering symmetries of our construction we have
established correspondences between solvable KPZ models in full and half space
with explicit determinantal and pfaffian point processes of free fermionic
origin. At the level of formulas, correspondences between KPZ models and free
fermions at positive temperature had been known for a long time, since the
works of [ACQ11, Dot10, CLDR10, SS10] in 2010, because the kernel (1.8) for
the solution of the KPZ equation (1.4) can be interpreted as the correlation
kernel for a positive temperature free fermion (see for instance [DLDMS15] for
an early application). The correspondence is formulated in a slightly
different manner by Borodin and Gorin [BG16], who understood the Fredholm
determinant solution of the KPZ equation as a _multiplicative nonlocal
statistics_ of the Airy point process. The latter holds more in general for
solvable regularizations of the KPZ equation, as found by Borodin in [Bor18]
and was extended in half space setting, for the particular case of ASEP, by
Barraquand, Borodin, Corwin, Wheeler [BBCW18] to study the KPZ equation on the
half line with critical boundary parameter $\omega=-\frac{1}{2}$.

The correspondence we establish and use in this paper are conceptually
different. The bijection connecting $q$-Whittaker measure and periodic Schur
measure, which we found in [IMS21b], translates to a connection between KPZ
models and free fermion at positive tempereture, and gives precise relations
between _local observables_ in both models. Once the connection is made, one
immediately concludes the existence of Fredholm determinant and pfaffian
formulas, without doing any further calculation.

Our new approach proposed in this paper to study KPZ models can be considered
as “positive temperature” version of the works of Johansson [Joh00] and of
Baik and Rains [BR01a, BR01b, BR01c] around 2000. Johansson applied the
Robinson-Schensted-Knuth (RSK) correspondence to map the problem of directed
polymer at zero temperature, or the totally asymmetric simple exclusion
process (TASEP), in full space to a free fermion at zero temperature. Baik and
Rains used the symmetries of RSK correspondence to study the above models in
half space. The ideas and methods in these works are quite powerful and have
been applied to many models in various settings but are not suited for
studying positive temperature models, for which different methodologies using
Bethe ansatz or Macdonald operators have been developed [BC14, BCS14]. In
[IMS21b] we succeeded in constructing a generalization of the RSK algorithm,
which allows us to study positive temperature KPZ models by connecting them to
free fermions at finite temperature, both in full and half space. Currently,
our methods only work for the study of “single point” functions, as in the
works of [Joh00, BR01a, BR01c, BR01b]. It would be of great interest to
understand if these can be extended to extract information for “multi point”
observables. For this task a more careful analysis of our bijective
construction seems to be required.

###  1.7. Outline

In section 2 we recall basic notions in combinatorics and symmetric functions.
There, we also define the $q$-Whittaker measure and the half space
$q$-Whittaker measure. In section 3 we introduce the solvable models in the
KPZ class we study in the rest of the paper, i.e. the $q$-PushTASEP, the ASEP
and the Log Gamma polymer. In section 4 we define the periodic Schur measure
and the free boundary Schur measure giving also Fredholm determinant and
pfaffian formulas for them. In section 4.5 we state the fundamental identities
relating $q$-Whittaker measures with periodic and free boundary Schur measure.
In section 5 we state Fredholm determinant and pfaffian formulas for all the
models introduced in section 3. Their asymptotic limit is computed in section
6 where we prove convergence to the GUE Tracy-Widom distribution and to the
Baik-Rains crossover. In section 7 we will state and prove formulas for the
solution of the KPZ/mSHE equation in full and half space. In section 7.4 we
will compute the large time fluctuations of the solution
$\mathscr{Z}^{\mathrm{hs}}(0,t)$ of the half space mSHE (1.10) at the origin
establishing Baik-Rains phase transition. In appendix A we recall basic
definitions and properties of Fredholm determinants and pfaffians. Finally, in
appendix B we will prove useful bounds needed to establish formulas for Log
Gamma polymer model in full and half space.

###  1.8. Acknowledgments

The work of TS has been supported by JSPS KAKENHI Grant Nos. JP16H06338,
JP18H03672, JP21H04432, No. JP22H01143. The work of TI has been supported by
JSPS KAKENHI Grant Nos. JP16K05192, JP19H01793, JP20K03626, and JP22H01143.
The work of MM has been supported by the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No. 101030938.

##  2\. $q$-Whittaker Measure and its half space variant

Here we define the $q$-Whittaker symmetric functions and recall some of their
properties and notable summation identities. In sections 2.4 and 2.5 we define
the $q$-Whittaker measures, which are important in the study of solvable
models in the KPZ class.

###  2.1. $q$-Whittaker and Schur polynomials

Symmetric functions we consider in this paper are labeled by partitions, i.e.
weakly decreasing lists of nonnegative integers
$\lambda=(\lambda_{1}\geq\lambda_{2}\geq\cdots\geq 0)$ such that
$\lambda_{n}=\lambda_{n+1}=\cdots=0$, for $n$ large enough. For a given
partition $\lambda$ we define its transpose
$\lambda^{\prime}=(\lambda_{1}^{\prime}\geq\lambda_{2}^{\prime}\geq\cdots)$
setting $\lambda_{i}^{\prime}=\\#\\{j:\lambda_{j}\geq i\\}$. Skew partitions,
denoted as $\lambda/\rho$, are also used and these are pairs $\lambda,\rho$,
satisfying $\lambda_{i}\geq\rho_{i}$ for all $i$. Two partitions $\lambda$ and
$\rho$ are said to interlace if
$\lambda_{1}\geq\rho_{1}\geq\lambda_{2}\geq\rho_{2}\geq\cdots$ and this
condition is denoted by the notation $\rho\prec\lambda$.

For the next definition we recall the notion of complete homogeneous symmetric
polynomials in $n$ variables $x_{1},\dots,x_{n}$,

(2.1) $h_{\ell}(x)=\sum_{1\leq i_{1}\leq\cdots\leq i_{\ell}\leq
n}x_{i_{1}}\cdots x_{i_{\ell}},$

where $\ell\in\mathbb{Z}_{\geq 0}$. By convention we set $h_{0}(x)=1$ and
$h_{\ell}(x)=0$ for $\ell<0$.

######  Definition 2.1 (Schur polynomials).

For any skew partition $\lambda/\rho$ and a set of $n$ variables
$x=(x_{1},\dots,x_{n})$, we define the _skew Schur polynomial_

(2.2)
$s_{\lambda/\rho}(x)=\det\left[h_{\lambda_{i}-\rho_{j}-i+j}(x)\right]_{i,j=1}^{N},$

where the number $N$ is large enough so that $\lambda_{N}=\rho_{N}=0$. Setting
$\rho=(0,0,\dots)$ we obtain the Schur polynomial $s_{\lambda}(x)$.

Next we present a notable $q$-deformation of the Schur polynomials. For a
fixed parameter $q\in(0,1)$, we recall the notion of $q$-Pochhammer symbols

(2.3) $(z;q)_{n}=\prod_{i=0}^{n-1}(1-zq^{i})\qquad\text{for
}n=0,1,\dots,\qquad(z;q)_{\infty}=\prod_{i=0}^{\infty}(1-zq^{i})$

and of $q$-binomials

(2.4) $\binom{n}{j}_{q}=\frac{(q;q)_{n}}{(q;q)_{j}(q;q)_{n-j}}.$

######  Definition 2.2 ($q$-Whittaker polynomials).

Given a skew partition $\mu/\varkappa$, the _skew $q$-Whittaker polynomial_ in
$n$ variables $\mathscr{P}_{\mu/\varkappa}(x_{1},\dots,x_{n};q)$ is defined by
the recursive relation

(2.5)
$\mathscr{P}_{\mu/\varkappa}(x_{1},\dots,x_{n};q)=\sum_{\eta}\mathscr{P}_{\eta/\varkappa}(x_{1},\dots,x_{n-1};q)\mathscr{P}_{\mu/\eta}(x_{n};q),$

where

(2.6) $\mathscr{P}_{\mu/\eta}(z;q)=\mathbf{1}_{\eta\prec\mu}\prod_{i\geq
1}z^{\mu_{i}-\eta_{i}}\binom{\mu_{i}-\mu_{i+1}}{\mu_{i}-\eta_{i}}_{q}.$

We also define the _dual $q$-Whittaker polynomials_

(2.7)
$\mathscr{Q}_{\mu}(x;q)=\mathdutchcal{b}_{\mu}(q)\mathscr{P}_{\mu}(x;q)\qquad\text{and}\qquad\mathscr{Q}_{\mu/\eta}(x;q)=\frac{\mathdutchcal{b}_{\mu}(q)}{\mathdutchcal{b}_{\eta}(q)}\mathscr{P}_{\mu/\eta}(x;q),$

where

(2.8) $\mathdutchcal{b}_{\mu}(q)=\prod_{i\geq
1}\frac{1}{(q;q)_{\mu_{i}-\mu_{i+1}}}.$

Setting $q=0$ both polynomials $\mathscr{P}_{\mu/\varkappa}$ and
$\mathscr{Q}_{\mu/\varkappa}$ become the skew Schur polynomials
$s_{\mu/\varkappa}$ and this gives a well known alternative definitions of
$s_{\mu/\varkappa}$.

The $q$-Whittaker polynomials can be seen as a special case of the more
general Macdonald polynomials [Mac95, Section VI]. These are special symmetric
polynomials depending on two parameters $q,t$ and denoted commonly as
$P_{\mu}(x;q,t)$. Setting $t=0$ we recover the $q$-Whittaker polynomial
$\mathscr{P}_{\mu}(x;q)=P_{\mu}(x;q,t=0)$. Another interesting particular case
of the Macdonald polynomials is given setting $q=0$ and keeping $t$ positive.
These are the Hall-Littlewood polynomials, which are also of interest in the
context of integrable probability and we will introduce them in the next
subsection, albeit through a different route.

###  2.2. Specializations of symmetric functions

The $q$-Whittaker polynomials enjoy the stability property

(2.9)
$\mathscr{P}_{\mu/\eta}(x_{1},\dots,x_{n-1},x_{n}=0;q)=\mathscr{P}_{\mu/\eta}(x_{1},\dots,x_{n-1};q)$

and this allows, through an inverse limit procedure, to define them in the
algebra of symmetric functions $\mathbf{\Lambda}_{\mathbb{C}}$. In [Mac95,
Section VI.4], this is explained in the more general case of Macdonald
functions. Implications of this observation include the extension of the
notion of “variables” of a symmetric function, which then become evaluation
homomorphisms of the algebra $\mathbf{\Lambda}_{\mathbb{C}}$.

######  Definition 2.3.

A _specialization_ $\varrho$ of the algebra of symmetric functions is an
algebra homomorphism $\varrho:\mathbf{\Lambda}_{\mathbb{C}}\mapsto\mathbb{C}$.
If $f\in\mathbf{\Lambda}_{\mathbb{C}}$ we will use the notation
$f(\varrho)=\varrho(f)$.

When dealing with explicit specializations it is convenient to define their
action on the algebraic basis of $\mathbf{\Lambda}_{\mathbb{C}}$ of power sum
symmetric functions $\\{p_{n}:n\in\mathbb{Z}_{\geq 0}\\}$, setting the values
$\varrho(p_{n})$. Rules for addition and scalar multiplication of
specializations are straightforward from the definition and we have

(2.10)
$(\varrho_{1}+\varrho_{2})(p_{n})=\varrho_{1}(p_{n})+\varrho_{2}(p_{n}),\qquad(c\varrho)(p_{n})=c^{n}\varrho(p_{n}),$

for all specializations $\varrho,\varrho_{1},\varrho_{2}$ and
$c\in\mathbb{C}$. Three types of specializations will be considered mainly in
this paper. These are:

  1. (1)

_alpha specializations_ of complex parameters $x=(x_{1},x_{2},\dots)$, where

(2.11) $\varrho:p_{n}\mapsto p_{n}(x)=x_{1}^{n}+x_{2}^{n}+\cdots,$

evaluates the power sum symmetric functions numerically at $x_{1},x_{2},\dots$
and we always assume that $\sum_{i\geq 1}|x_{i}|<\infty$;

  2. (2)

_$q$ -beta specializations_ of complex parameters $x=(x_{1},x_{2},\dots)$,
where one sets

(2.12) $\varrho(p_{n})=(-1)^{n-1}(1-q^{n})p_{n}(x),$

assuming again $\sum_{i\geq 1}|x_{i}|<\infty$. For brevity we will denote the
$q$-beta specialization of a symmetric function $f$ by $f(\widehat{x})$;

  3. (3)

_Plancherel specializations_ of parameter $\tau\in\mathbb{C}$ defined by
$\varrho(p_{n})=\delta_{n,1}\tau$.

Evaluating $q$-beta specializations of $q$-Whittaker functions yields another
remarkable family of symmetric polynomials (or functions)

(2.13)
$\mathscr{H}_{\mu^{\prime}/\eta^{\prime}}(x;q)=\mathscr{Q}_{\mu/\eta}(\widehat{x};q),$

called _Hall-Littlewood polynomials_ ; see [Mac95, Section VI eq. (5.1)]. In
literature the Hall-Littlewood polynomials usually depend on a parameter $t$,
which is $q$ in the notation (2.13). Because of such simple relation between
$\mathscr{P}$ and $\mathscr{H}$ functions, in this paper we will not emphasize
their difference.

######  Remark 2.4 (Positive specializations).

For probabilistic applications we will be interested in $q$-Whittaker-
_positive specializations_ $\varrho$, which are defined by the property that
$\varrho(\mathscr{P}_{\mu})\geq 0$ for all partitions $\mu$. These were
characterized by Matveev in [Mat19] for the more general case of Macdonald
functions. Adapting [Mat19, Theorem 1.4] to our case, we see that a
specialization $\varrho$ is $q$-Whittaker-positive if and only if it can be
expressed as

(2.14)
$\varrho=\varrho_{x}+\varrho_{\widehat{y}}+\varrho^{\mathrm{pl}}_{\tau}$

where $\varrho_{x},\varrho_{\widehat{y}},\varrho^{\mathrm{pl}}_{\tau}$ are
respectively an alpha, $q$-beta and Plancherel specializations of parameters
$x=(x_{1},x_{2},\dots)$, $y=(y_{1},y_{2},\dots)$ and $\tau$ such that
$x_{i},y_{i},\tau\geq 0$ for all $i$ and

(2.15) $\sum_{i=1}^{\infty}(x_{i}+y_{i})<\infty.$

Analogously _Schur-positive_ specializations $\varrho$ are defined imposing
$\varrho(s_{\mu})\geq 0$ for all $\mu$. In this case the Edrei-Thoma’s theorem
[ASW52, Edr53, Whi52], says that a specialization is Schur positive if and
only if it is of the form (2.14), setting $q=0$ in $\varrho_{\widehat{y}}$.

###  2.3. Summation identities

The $q$-Whittaker functions possess a number of interesting properties and
satisfy notable summation identities. Summing over products of two
$q$-Whittaker functions one obtains the _Cauchy identity_ that reads

(2.16)
$\sum_{\mu}\mathscr{P}_{\mu/\nu}(a;q)\mathscr{Q}_{\mu/\varkappa}(b;q)=\Pi(a,b)\sum_{\eta}\mathscr{P}_{\varkappa/\eta}(a;q)\mathscr{Q}_{\nu/\eta}(b;q),$

where

(2.17) $\Pi(a,b)=\prod_{k\geq 0}H(q^{k}a,b)\qquad\text{and}\qquad
H(a,b)=\exp\left\\{\sum_{n\geq 1}\frac{p_{n}(a)p_{n}(b)}{n}\right\\}.$

Setting $\nu=\varkappa=(0,0,\dots)$, identity (2.16) simplifies as

(2.18) $\sum_{\mu}\mathscr{P}_{\mu}(a;q)\mathscr{Q}_{\mu}(b;q)=\Pi(a;b).$

Here $a$ and $b$ are generic specializations for which the function $H(a,b)$
is numerically convergent. In the simplest case, when $a$ and $b$ are alpha
specializations of finitely many parameters $(a_{1},\dots,a_{n})$,
$(b_{1},\dots,b_{t})$, such that $|a_{i}b_{j}|<1$ for all $i,j$, we have

(2.19)
$\Pi(a,b)=\prod_{i=1}^{n}\prod_{j=1}^{t}\frac{1}{(a_{i}b_{j};q)_{\infty}}.$

Summations over single $q$-Whittaker functions yield the _Littlewood identity_

(2.20)
$\sum_{\mu}\mathdutchcal{b}^{\mathrm{el}}_{\mu}(q)\mathscr{P}_{\mu}(a;q)=\widetilde{\Pi}(a),$

where $\mathdutchcal{b}^{\mathrm{el}}_{\mu}$ is defined as

(2.21)
$\mathdutchcal{b}_{\mu}^{\mathrm{el}}(q)=\prod_{i=2,4,6,\dots}\frac{\mathbf{1}_{\mu_{i-1}=\mu_{i}}}{(q;q)_{\mu_{i}-\mu_{i+1}}},$

and

(2.22) $\widetilde{\Pi}(a)=\prod_{k\geq
0}\widetilde{H}(q^{k}a),\qquad\qquad\widetilde{H}(a)=\exp\left\\{\sum_{n\geq
1}\frac{p_{2n-1}(a)}{2n-1}+\frac{p_{n}(a)^{2}}{2n}\right\\}.$

In (2.21) the superscript el means “even length” as
$\mathdutchcal{b}^{\mathrm{el}}_{\mu}$ vanishes unless $\mu^{\prime}$ is even,
i.e. $\mu^{\prime}_{i}$ is even for all $i$. When $a$ is an alpha
specialization of finitely many parameters $(a_{1},\dots,a_{n})$ we have

(2.23) $\widetilde{\Pi}(a)=\prod_{1\leq i<j\leq
n}\frac{1}{(a_{i}a_{j};q)_{\infty}},$

as it can be verified by evaluating the summations in the expression of
$\widetilde{H}(a)$.

Interestingly, using homogeneity of Schur functions and their skew Cauchy
identities it is possible to reproduce $\Pi(a,b),\widetilde{\Pi}(a)$ appearing
on the right hand side of (2.18), (2.20). We have [SS90, Corollary 6.7]

(2.24)
$\sum_{\lambda,\rho}q^{|\rho|}s_{\lambda/\rho}(a)s_{\lambda/\rho}(b)=\frac{1}{(q;q)_{\infty}}\Pi(a;b)$

and [SS90, Corollary 6.10]

(2.25) $\sum_{\begin{subarray}{c}\lambda,\rho:\\\
\lambda^{\prime},\rho^{\prime}\text{
even}\end{subarray}}q^{|\rho|/2}s_{\lambda/\rho}(a)=\frac{1}{(q;q)_{\infty}}\widetilde{\Pi}(a).$

Indeed, similarities between identities (2.24), (2.25) and (2.18), (2.20) are
not merely a coincidence and their mutual relations were understood in
[IMS21a, IMS21b]. We recall these results in the following theorems.

######  Theorem 2.5.

Fix $|q|<1$ and specializations of symmetric functions $a$ and $b$. Then, for
all $k=0,1,2,\dots$, we have

(2.26)
$\sum_{\ell=0}^{k}\frac{q^{\ell}}{(q;q)_{\ell}}\sum_{\mu:\mu_{1}=k-\ell}\mathscr{P}_{\mu}(a;q)\mathscr{Q}_{\mu}(b;q)=\sum_{\lambda,\rho:\lambda_{1}=k}q^{|\rho|}s_{\lambda/\rho}(a)s_{\lambda/\rho}(b).$

Notice that equality (2.26) represents a mutual refinement of (2.18) and
(2.24). A similar mutual refinement between (2.20) and (2.25) is given below.

######  Theorem 2.6.

Fix $|q|<1$ and a specialization of symmetric functions $a$. Then, for all
$k=0,1,2,\dots$, we have

(2.27)
$\sum_{\ell=0}^{k}\frac{q^{\ell}}{(q;q)_{\ell}}\sum_{\mu:\mu_{1}=k-\ell}\mathdutchcal{b}_{\mu}^{\mathrm{el}}(q)\mathscr{P}_{\mu}(a;q)=\sum_{\begin{subarray}{c}\lambda,\rho:\lambda^{\prime},\rho^{\prime}\,\mathrm{even}\\\
\lambda_{1}=k\end{subarray}}q^{|\rho|/2}s_{\lambda/\rho}(a).$

In [IMS21b], theorems 2.5 and 2.6 were proven bijectively only in the case
where $a$ and $b$ are alpha specializations. Nevertheless, as pointed out in
[IMS21b, Remark 10.8 and Remark 10.14] the same identities hold for general
specializations following simple stability considerations such as (2.9).

###  2.4. $q$-Whittaker measure

The _$q$ -Whittaker measure_ [BC14] is the probability measure on the set of
partitions given by

(2.28)
$\mathbb{W}^{(q)}_{a;b}(\mu)=\frac{1}{\Pi(a;b)}\mathdutchcal{b}_{\mu}(q)\mathscr{P}_{\mu}(a;q)\mathscr{P}_{\mu}(b;q),$

where $a,b$ are ($q$-Whittaker) positive specializations such that the
function $H(a,b)$ is numerically convergent. In case $a$ and $b$ are $q$-beta
specializations $\widehat{a},\widehat{b}$ of parameters $(a_{1},\dots,a_{n})$,
$(b_{1},\dots,b_{t})$, the $q$-Whittaker polynomials
$\mathscr{P}_{\mu}(\widehat{a};q),\mathscr{P}_{\mu}(\widehat{b};q)$ turn into
the _dual_ Hall-Littlewood polynomials labeled by the transposed partitions
$\mu^{\prime}$ and the measure $\mathbb{W}^{(q)}_{\widehat{a};\widehat{b}}$
turns into the Hall-Littlewood measure [Mac95, Chapter VI eq. (5.1)]. This is
another important measure in integrable probability and it is related to KPZ
models as we will discuss below; for more on the subject see [CSV11, BBW18,
BM18, BBCW18, BP19]. In this paper we will not introduce a different notation
for the Hall-Littlewood measure and we will consider it as a particular
specialization of the $q$-Whittaker measure.

###  2.5. Half space $q$-Whittaker measure

Introducing the function

(2.29)
$\mathscr{E}_{\mu}(\gamma;q)=\sum_{\eta}\mathdutchcal{b}_{\eta}^{\mathrm{el}}(q)\mathscr{Q}_{\mu/\eta}(\gamma;q),$

we define, as in [BBC20], the _half space $q$-Whittaker measure_

(2.30)
$\mathbb{HW}^{(q)}_{a;\gamma}(\mu)=\frac{1}{\widetilde{\Pi}(a)\Pi(a;\gamma)}\mathscr{E}_{\mu}(\gamma,q)\mathscr{P}_{\mu}(a,q).$

Here $a$ and $\gamma$ are $q$-Whittaker positive specializations such that
functions $\widetilde{H}(a)$ and $H(a,\gamma)$ are numerically convergent.
Combining Cauchy identities (2.16) and (2.20) one can verify that the
normalization constant in the right hand side of (2.30) is correct.

We will mostly be interested in the case when $\gamma$ is a single variable
alpha specialization. In such situation, the half space $q$-Whittaker measure
possesses a symmetry that allows to distribute the parameter $\gamma$ of the
function $\mathscr{E}$ on the specialization of the $\mathscr{P}$ function.

######  Proposition 2.7 ([BBC20] Proposition 2.6).

Let $\bm{a}$ be a $q$-Whittaker positive specialization and $\gamma$ be a
single variable alpha specialization. Consider a random partition $\mu$
distributed according to $\mathbb{HW}^{(q)}_{\bm{a};\gamma}$ and another
random partition $\lambda$ distributed according to
$\mathbb{HW}^{(q)}_{\bm{a}^{\prime};0}$, where $\bm{a}^{\prime}=(a,\gamma)$.
Then $\\{\mu_{1},\mu_{3},\dots\\}$ and $\\{\lambda_{1},\lambda_{3},\dots\\}$
have the same distribution.

We will be interested in the marginal $\mu_{1}$ of the half space
$q$-Whittaker measure as it relates to solvable KPZ models; see section 3. For
this purpose the symmetry reported in proposition 2.7 is very convenient as it
reduces the study of the measure with specializations $\bm{a}$ and $\gamma$ to
the simpler case when $\gamma=0$.

##  3\. Models in the KPZ class and $q$-Whittaker measures

In this section, we explain the relation between KPZ models and $q$-Whittaker
measure, both in the whole space case and in the half space.

###  3.1. $q$-PushTASEP

The _geometric $q$-PushTASEP_ is a stochastic interacting particle system
introduced in [MP17]. In order to describe the random jumps of particles, we
recall the _$q$ -deformed beta binomial distribution_

(3.1)
$\bm{\varphi}_{q,\xi,\eta}(s|m)=\xi^{s}\frac{(\eta/\xi;q)_{s}(\xi;q)_{m-s}}{(\eta;q)_{m}}\frac{(q;q)_{m}}{(q;q)_{s}(q;q)_{m-s}},$

which is a probability distribution on $s\in\\{0,\dots,m\\}$, for any
$m\in\mathbb{Z}_{\geq 0}$, whenever parameters $q,\xi,\eta$ make the above
expression nonnegative. Two such choices are given by
$q,\xi\in[0,1),\eta\in[0,\xi]$ and $q>1,\xi\in q^{\mathbb{Z}_{\leq
0}},\eta=0$. Setting $\eta=0,m=\infty$ and taking $q,\xi\in[0,1)$, we obtain
the _$q$ -Geometric_ distribution of parameter $\xi$,

(3.2)
$\bm{\varphi}_{q,\xi,0}(s|\infty)=\xi^{s}\frac{(\xi;q)_{\infty}}{(q;q)_{s}},\qquad
s\in\mathbb{Z}_{\geq 0},$

which we denote by $q$-Geo$(\xi)$.

In the $q$-PushTASEP, a finite number $N$ of particles occupy, following the
exclusion rule, sites of the lattice $\mathbb{Z}$ and their position is
recorded in an array
$\mathsf{X}(T)=(\mathsf{x}_{1}(T)<\cdots<\mathsf{x}_{N}(T))$, where
$T\in\mathbb{Z}_{\geq 0}$ denotes the time. During the time step $T\to T+1$,
the array $\mathsf{X}(T)$ is updated sequentially from left to right and

(3.3) $\mathsf{x}_{k}(T+1)=\mathsf{x}_{k}(T)+V_{k,T}+W_{k,T},\qquad\text{for
}k=1,\dots,N,$

where $V_{k,T}\sim q$-$\mathrm{Geo}(a_{k}b_{T+1})$ and

(3.4)
$W_{k,T}\sim\bm{\varphi}_{q^{-1},q^{\mathrm{gap}_{k}(T)},0}(\bullet\,|\,\mathsf{x}_{k-1}(T+1)-\mathsf{x}_{k-1}(T)).$

In the previous expression we assume
$\mathrm{gap}_{j}(T)=\mathsf{x}_{j}(T)-\mathsf{x}_{j-1}(T)-1$ and we set, by
convention $\mathsf{x}_{0}(T)=-\infty$. In (3.3) we see that the random
movement of a particle splits into two independent contributions: $W_{k,T}$
consists in a random pushing that the particle $\mathsf{x}_{k-1}$ impresses on
$\mathsf{x}_{k}$, while $V_{k,T}$ is an independent jump of $\mathsf{x}_{k}$.

Markov rules (3.3), although explicit are far from intuitive or natural. An
interesting simplification can be obtained considering a _continuous time_
limit, setting $b_{i}=\varepsilon$, $T=\tau/\varepsilon$ and $\varepsilon\to
0$. This procedure yields the continuous time $q$-PushTASEP first defined in
[BP16]; for a visualization see fig. 2.

The $q$-PushTASEP is deeply connected to marginals of the $q$-Whittaker
measure, as recalled in the next proposition.

######  Proposition 3.1 ([MP17], Section 6.3).

Consider $\bm{a},\bm{b}$ alpha specializations of parameters respectively
$(a_{1},\dots,a_{N})\in(0,1)^{N}$ and $(b_{1},\dots,b_{T})\in(0,1)^{T}$. Let
$\mu\sim\mathbb{W}_{\bm{a};\bm{b}}^{(q)}$ and let $\mathsf{X}(T)$ be a
$q$-PushTASEP under initial conditions $\mathsf{x}_{k}(0)=k$ for
$k=1,\dots,N$. Then, we have the following equality in distribution

(3.5) $\mathsf{x}_{N}(T)\stackrel{{\scriptstyle\mathcal{D}}}{{=}}\mu_{1}+N.$

Figure 2. Visualization of the dynamics in the continuous time $q$-PushTASEP.
Each particle $\mathsf{x}_{k}$ possesses an exponential clock with parameter
$a_{k}$ and once the clock rings at time $\tau$ it performs a jump to its
right $\mathsf{x}_{k}(\tau_{+})=\mathsf{x}_{k}(\tau)+1$. The movement of the
$k$-th particle triggers an instantaneous pushing mechanism and
$\mathsf{x}_{j}(\tau_{+})=\mathsf{x}_{j}(\tau)+1$ with probability
$q^{\mathrm{gap}_{j}(\tau)}\mathbf{1}_{\mathsf{x}_{j-1}(\tau_{+})=\mathsf{x}_{j-1}(\tau)+1}$,
for all $j>k$.

###  3.2. $q$-PushTASEP with particle creation

The $q$-PushTASEP admits a solvable variant where the stochastic dynamics do
not preserve the number of particles, which are added to the system at each
time step. This model was introduced in [BBC20] and we will refer to it as
_$q$ -PushTASEP with particle creation_. At any time $T\in\mathbb{Z}_{\geq
0}$, the system consists of exactly $T+1$ particles located in the lattice
$\mathbb{Z}_{\geq 0}$ and whose positions are

(3.6)
$0=\mathsf{x}_{0}^{\mathrm{hs}}(T)<\mathsf{x}_{1}^{\mathrm{hs}}(T)<\cdots<\mathsf{x}_{T}^{\mathrm{hs}}(T)<\infty,$

recorded in the array $\mathsf{X}^{\mathrm{hs}}(T)$. During the time step from
$T$ to $T+1$, the configuration gets updated sequentially from the particle
with label 1 to the right following the $q$-PushTASEP rules described by
(3.3), where we assume $V_{k,T}\sim q\text{-}\mathrm{Geo}(a_{k}a_{T+1})$. Once
the position of the rightmost particle $\mathsf{x}_{T}^{\mathrm{hs}}(T+1)$ has
been determined a new particle is added to its right and
$\mathsf{x}_{T+1}^{\mathrm{hs}}(T+1)=\mathsf{x}_{T}^{\mathrm{hs}}(T+1)+1+\widetilde{V}_{T+1}$,
where $\widetilde{V}_{T+1}\sim q$-$\mathrm{Geo}(\gamma a_{T+1})$ is an
independent increment. Notice that the $0$-th particle never moves. Here we
have assumed that $a_{i},\gamma$ are positive numbers such that
$a_{i}a_{j},\gamma a_{i}<1$ for all $1\leq i<j$. By construction, we will only
consider _empty initial conditions_ , that is
$\mathsf{X}^{\mathrm{hs}}(0)=\\{\mathsf{x}_{0}(0)=0\\}$.

The salient feature of this model is the fact that during the evolution, new
particles are added at the right of the system with spacing modulated by the
parameter $\gamma$. Such property, makes the $q$-pushTASEP with particle
creation a regularization of the KPZ equation in half space, although to see
this several non trivial scalings are necessary. As for the “full space”
version of the $q$-PushTASEP, this is, admittedly not necessarily the most
intuitive model to consider. More canonical models of growth with boundary
effects have been considered in the past. These include the Asymmetric Simple
Exclusion Process [BBCW18, TW13a, TW13b] or other discrete polymer models
[OSZ14], but unfortunately their solvability, for generic values of boundary
strength, does not lead directly to manageable or rigorous exact formulas.

The next proposition provides the relation between the $q$-PushTASEP with
particle creation and the half space $q$-Whittaker measure.

######  Proposition 3.2 ([BBC20] Proposition 4.24).

Consider parameters $a_{1},\dots,a_{N},\gamma\in(0,1)$ and let $\bm{a}$ be the
alpha specialization with parameters $(a_{1},\dots,a_{N})$. Let
$\mu\sim\mathbb{HW}^{(q)}_{\bm{a};\gamma}$ and let
$\mathsf{X}^{\mathrm{hs}}(T)$ be the $q$-PushTASEP with particle creation and
empty initial conditions with parameters $\bm{a},\gamma$. Then, we have the
following equality in distribution

(3.7) $\mathsf{x}^{\mathrm{hs}}_{N}(N)=\mu_{1}+N.$

###  3.3. ASEP

The asymmetric simple exclusion process is a canonical model of transport of
particles in one dimension. It was introduced in [Spi70] and its stochastic
integrable stucture has been investigated intensively in the last two decades
[Sch97, TW08a, TW08b, TW09a, IS11, BCS14, BO17]. On the lattice $\mathbb{Z}$
at time $\tau\geq 0$ we consider a set of particles, with locations222Notice
that in this case the particle ordering is _opposite_ to the one adopted for
the $q$-PushTASEP in sections 3.1 and 3.2.
$\mathsf{x}_{1}(\tau)>\mathsf{x}_{2}(\tau)>\cdots$. Each particle possesses
two exponential clocks of rate respectively $q\in[0,1)$ and $1$. When the
first clock rings the particle attempts a jump to its left neighboring site
and when the second one rings it attempts a jump to its right. Jumps are
successful provided the target location is empty; see fig. 3. Particle
configurations we consider always possess a rightmost particle, whose location
is $\mathsf{x}_{1}$. For this we can define the integrated current through a
location $x$ as

(3.8) $\mathsf{J}(x,\tau)=\sum_{k\geq
1}\mathbf{1}_{\mathsf{x}_{k}(\tau)>x}-\sum_{k\geq
1}\mathbf{1}_{\mathsf{x}_{k}(0)>x}.$

The ASEP is deeply related to the Hall-Littlewood measure. This was
essentially discovered in [BBW18], where authors established a more general
correspondence between a marginal of the Hall-Littlewood measure and the
stochastic six vertex model. The convergence of the stochastic six vertex
model to the ASEP was proven in [Agg16].

######  Proposition 3.3.

Consider the $q$-Whittaker measure
$\mathbb{W}_{\widehat{a};\widehat{b}}^{(q)}$ where $\widehat{a},\widehat{b}$
are $q$-beta specializations of parameters $(a_{1},\dots,a_{N})$ and
$(b_{1},\dots,b_{T})$. Let $\mu\sim\mathbb{W}_{\widehat{a};\widehat{b}}^{(q)}$
and consider the scaling

(3.9) $T=\frac{\tau}{\varepsilon},\qquad N=\frac{\tau}{\varepsilon}+x,\qquad
a_{i}=b_{j}=1-\frac{(1-q)\varepsilon}{2},\qquad\mu_{1}=N-\mathsf{J}^{(\varepsilon)}(x,\tau).$

Then, as $\varepsilon\to 0$, the random variable
$\mathsf{J}^{(\varepsilon)}(x,\tau)$ converges in law to the integrated
current $\mathsf{J}(x,\tau)$ of the ASEP with initial conditions
$\mathsf{x}_{k}(0)=-k$ for $k=1,2\dots$.

Figure 3. Visualization of the dynamics in the ASEP.

As hinted earlier, the ASEP can be seen as a particular case of a more general
stochastic solvable model, i.e. the stochastic six vertex model [GS92, BCG16].
In fact, combinatorial techniques allow to transform the stochastic six vertex
model into many more solvable systems, including the $q$-PushTASEP; see [BP18,
CP16, BMP21]. Since the stochastic six vertex model is related to the Hall-
Littlewood measure [BBW18], correspondences as those of proposition 3.1 or
proposition 3.3, leading to results we will elaborate in the following
sections, could be given for a full hierarchy of solvable models. To keep our
argument concise we will not report such results here.

###  3.4. Log gamma polymer

The _Log Gamma polymer_ model is a discrete solvable random polymer model
introduced by Seppaläinen in [Sep12]. We recall that an _inverse gamma random
variable of parameter $\alpha$_, which we will denote as
$\mathrm{Gamma}^{-1}(\alpha)$, has probability density function

(3.10)
$p_{\mathrm{Gamma}^{-1}(\alpha)}(x)=\frac{x^{-\alpha-1}e^{-1/x}}{\Gamma(\alpha)}.$

On each site $(i,j)$ of the planar lattice $\Lambda=\mathbb{Z}_{\geq
0}\times\mathbb{Z}_{\geq 0}$ we place a random variable
$w_{i,j}\sim\mathrm{Gamma}^{-1}(A_{i}+B_{j})$, where $A_{i},B_{j}$ are
positive real numbers. For a fixed point $(T,N)$, we consider the set
$\Pi_{(1,1)\to(T,N)}$ of up-right paths
$\varpi=(\varpi_{1},\dots,\varpi_{N+T-1})$ on $\Lambda$ with endpoints
$\varpi_{1}=(1,1)$ and $\varpi_{N+T-1}=(T,N)$. Here by up-right path we mean
that the increments $\varpi_{i+1}-\varpi_{i}$ are either $(1,0)$ or $(0,1)$
for any $i$; see fig. 4, left panel. To any path $\varpi$ we associate the
random weight

(3.11) $w(\varpi)=\prod_{\ell=1}^{N+T-1}w_{\varpi_{\ell}}$

and we define the partition function

(3.12) $Z(N,T)=\sum_{\varpi\in\Pi_{(1,1)\to(T,N)}}w(\varpi).$

######  Remark 3.4.

Let us explain the origin of the name “Log Gamma” polymer. Canonically the
partition function of a directed polymer with fixed endpoints at $(1,1)$ and
$(T,N)$, has the form

(3.13) $Z(N,T)=\sum_{\varpi\in\Pi_{(1,1)\to(T,N)}}e^{\beta E(\varpi)},$

where $E$ is the energy of the polymer $\varpi$ and $\beta$ is the inverse
temperature of the system. In case the energy is taken as a sum of independent
terms $E(\varpi)=\sum_{\ell}E_{\varpi_{\ell}}$ and
$E_{(i,j)}\sim-\log\mathrm{Gamma}(A_{i}+B_{j})$ and the parameter $\beta=1$ we
recover the partition function (3.12).

(a)  (b)

Figure 4. Visualizations of the Log Gamma polymer model in full space (a) and
in half space (b).

The Log Gamma polymer model arises as a scaling limit of the $q$-PushTASEP, as
it was proven by Matveev and Petrov in [MP17, Theorem 8.7]. A particular case
of such result is stated in the next proposition.

######  Proposition 3.5.

Let $\mathsf{X}(T)$ be a $q$-PushTASEP with initial conditions
$\mathsf{x}_{k}(0)=k$ for $k=1,\dots,N$. Consider the scaling
$q=e^{-\varepsilon}$, $a_{i}=e^{-\varepsilon A_{i}}$, $b_{j}=e^{-\varepsilon
B_{j}}$ and

(3.14)
$\mathsf{x}_{N}(T)=(N+T-1)\varepsilon^{-1}\log\varepsilon^{-1}+\varepsilon^{-1}\log
Z^{(\varepsilon)}(T,N).$

Then, as $\varepsilon\to 0$, the random variable $Z^{(\varepsilon)}(T,N)$
converges in distribution to $Z(T,N)$.

###  3.5. Half space Log Gamma Polymer

Here we consider a variant of the Log Gamma polymer model in the restricted
lattice $\Lambda^{\mathrm{hs}}=\\{(i,j)\in\mathbb{Z}_{\geq
1}\times\mathbb{Z}_{\geq 1}:i\geq j\\}$. In this case to each site $(i,j)$ of
$\Lambda^{\mathrm{hs}}$ we associate a random variable

(3.15)
$w_{i,j}\sim\begin{cases}\mathrm{Gamma}^{-1}(A_{i}+\varUpsilon)\qquad&\text{if
}i=j,\\\ \mathrm{Gamma}^{-1}(A_{i}+A_{j})\qquad&\text{if }i>j,\end{cases}$

where $\varUpsilon>-\min\\{A_{i}:i=1,2,\dots\\}$ is the _boundary strength_.
For any $N\leq T$, define, as in the previous subsection, the set
$\Pi^{\mathrm{hs}}_{(1,1)\to(T,N)}$ of up-right paths
$\varpi=(\varpi_{1},\dots,\varpi_{T+N-1})$ on $\Lambda^{\mathrm{hs}}$ with
endpoints $\varpi_{1}=(1,1)$ and $\varpi_{T+N-1}=(T,N)$. The weight of a path
$\varpi$ is defined as in (3.11) and the partition function of the half space
Log Gamma polymer is the random function

(3.16)
$Z^{\mathrm{hs}}(N,T)=\sum_{\varpi\in\Pi^{\mathrm{hs}}_{(1,1)\to(T,N)}}w(\varpi).$

As in the whole space case, the random function $Z^{\mathrm{hs}}$ can be
recovered as a scaling limit of the $q$-PushTASEP with particle creations, as
recalled next.

######  Proposition 3.6.

Let $\mathsf{X}^{\mathrm{hs}}(T)$ be a $q$-PushTASEP with particle creation
and empty initial conditions. Consider the scaling $q=e^{-\varepsilon}$,
$a_{i}=e^{-\varepsilon A_{i}},\gamma=e^{-\varepsilon\varUpsilon}$ and

(3.17)
$\mathsf{x}^{\mathrm{hs}}_{N}(T)=(N+T-1)\varepsilon^{-1}\log\varepsilon^{-1}+\varepsilon^{-1}\log
Z^{\mathrm{hs}}(T,N;\varepsilon),$

with $T\geq N$. Then, as $\varepsilon\to 0$, the random variable
$Z^{\mathrm{hs}}(T,N;\varepsilon)$ converges in distribution to
$Z^{\mathrm{hs}}(T,N)$.

###### Proof.

This is proven in [BBC20], combining Proposition 6.27 and Proposition 4.24 of
the same paper. ∎

##  4\. Periodic and free boundary Schur measures

In this section we define two important objects: the periodic and the free
boundary Schur measures. These are discrete models of free fermions at
positive temperature and as such their law is described in terms of
determinantal and pfaffian point processes. The fundamental identities (2.26)
and (2.27) relate them precisely to the KPZ models discussed in the previous
section.

###  4.1. Periodic Schur measure

The _periodic Schur measure_ was first introduced by Borodin in [Bor07] as a
generalization of Okounkov’s Schur measure [Oko01]. It is a measure on the set
of partitions given by

(4.1)
$\mathbb{PS}_{a,b}^{(q)}(\lambda)=\frac{1}{(q;q)_{\infty}^{-1}\Pi(a;b)}\sum_{\rho}q^{|\rho|}s_{\lambda/\rho}(a)s_{\lambda/\rho}(b),$

where $s_{\lambda/\rho}(x)$ is the skew Schur function (2.2). Here $a,b$ are
Schur-positive specializations and the normalization constant is a consequence
of the Cauchy identity (2.24). While the Schur measure is a model of free
fermions at _zero temperature_ , the periodic Schur measure, after a simple
trasformation describes the grand canonical ensemble of free fermions at
_positive temperature_ , modulated by the parameter $q$; see [BB19]. As found
in [Bor07] a certain independent shift of coordinates of $\lambda$ transforms
the periodic Schur measure into a determinantal point process. We say that an
integer valued random variable $S$ has _theta_ distribution of parameters
$q\in(0,1),\zeta>0$, in short $S\sim\mathrm{Theta}(\zeta;q)$, if

(4.2)
$\mathbb{P}(S=k)=\frac{1}{\vartheta_{3}(\zeta;q)}q^{k^{2}/2}\zeta^{k},\qquad
k\in\mathbb{Z},$

where

(4.3) $\vartheta_{3}(\zeta;q)=(q,-\sqrt{q}\zeta,-\sqrt{q}/\zeta;q)_{\infty}$

is the Jacobi theta function. The factorized form of the normalizaztion
constant is the celebrated Jacobi triple product identity [Ism05, Theorem
12.3.2]. We can now define the _shift-mixed periodic Schur measure_ as the
point process

(4.4)
$\mathfrak{S}(\lambda,S)=\left(\lambda_{i}-i+\frac{1}{2}+S\right)_{i=1,2,\dots}\subset\mathbb{Z}^{\prime},$

where we used the notation $\mathbb{Z}^{\prime}=\mathbb{Z}+\frac{1}{2}$, and
$\lambda\sim\mathbb{PS}_{a,b}^{(q)}$, $S\sim\mathrm{Theta}(\zeta;q)$ are
independent.

######  Theorem 4.1 ([Bor07]).

The shift-mixed periodic Schur measure is a determinantal point process with
correlation kernel

(4.5)
$K(x,y)=\frac{1}{(2\pi\mathrm{i})^{2}}\oint_{|z|=r}\frac{\mathop{}\\!\mathrm{d}z}{z^{x+1}}\oint_{|w|=r^{\prime}}\frac{\mathop{}\\!\mathrm{d}w}{w^{-y+1}}\frac{F(z)}{F(w)}\kappa(z,w),$

where $x,y\in\mathbb{Z}^{\prime}$ and

(4.6) $F(z)=\prod_{\ell\geq 0}\frac{H(q^{\ell}a;z)}{H(q^{\ell}b;z^{-1})},$
(4.7)
$\kappa(z,w)=\sqrt{\frac{w}{z}}\frac{(q;q)_{\infty}^{2}}{(w/z,qz/w;q)_{\infty}}\frac{\vartheta_{3}(\zeta
z/w;q)}{\vartheta_{3}(\zeta;q)}.$

The function $H$ was defined in (2.17) and the integration contour’s radii
satisfy $1<r/r^{\prime}<q^{-1}$ and $R^{-1}\leq r,r^{\prime}\leq R$, where $R$
is the minimum of convergence radii of $H(a;z)$ and $H(b;z)$.

We will be interested in the rightmost coordinate of the shift-mixed periodic
Schur measure $\lambda_{1}+S$. Using basic notions of determinantal point
processes its probability distribution can be written in terms of the Fredholm
determinant of the kernel $K$ of (4.5). We have, for any $s\in\mathbb{R}$,

(4.8) $\mathbb{P}[\lambda_{1}+S\leq
s]=\det(1-K)_{\ell^{2}(\mathbb{Z}_{>s}^{\prime})},$

where $\mathbb{Z}_{>s}^{\prime}$ denotes the set of half integers larger than
to $s$.

###  4.2. Alternative representation of the Fredholm determinant via
Ramanujan’s bilateral sum.

To state the main result of the subsection we recall the identity

(4.9)
$\sum_{m\in\mathbb{Z}^{\prime}}\frac{z^{m}}{1+\zeta^{-1}q^{-m}}=z^{-1/2}\frac{(q;q)_{\infty}^{2}}{(1/z,qz;q)_{\infty}}\frac{\vartheta_{3}(\zeta
z;q)}{\vartheta_{3}(\zeta;q)},$

which holds for $1<|z|<q^{-1}$ and it is a particular case of Ramanujan’s
${}_{1}\psi_{1}$ sum [Ism05, Theorem 12.3.1].

######  Proposition 4.2.

[IMS21a, Proposition 2.3] Let $\lambda$ be distributed according to the
periodic Schur measure (4.1) and let $S\sim\mathrm{Theta}(\zeta;q)$. Then, for
any $s\in\mathbb{Z}$, we have

(4.10) $\mathbb{P}\left[\lambda_{1}+S\leq
s\right]=\det\left[1-f_{\zeta,s}\cdot
M\right]_{\ell^{2}(\mathbb{Z}^{\prime})},$

where $M$ is given by

(4.11)
$M(m,n)=\oint_{|z|=r}\frac{\mathop{}\\!\mathrm{d}z}{2\pi\mathrm{i}}\oint_{|w|=r^{\prime}}\frac{\mathop{}\\!\mathrm{d}w}{2\pi\mathrm{i}}\frac{F(z)}{F(w)}\frac{z^{n-1/2}}{w^{m+1/2}}\frac{1}{z-w},$

and $f_{\zeta,s}$ is the Fermi-Dirac distribution333Setting
$q=e^{-\varepsilon/k_{B}T}$ and $\zeta=e^{\mu/k_{B}T}$, where $T$ is the
temperature, $\varepsilon$ is the quantized energy, $k_{B}$ is Boltzmann
constant and $\mu$ is the chemical potential, then $f_{\zeta,s}(m)$ assumes
the canonical form of the Fermi-Dirac distribution.

(4.12) $f_{\zeta,s}(m)=\frac{1}{1+\zeta^{-1}q^{-s-m}}.$

###### Proof.

Using (4.9), the kernel $K$ can be rewritten as

(4.13)
$K(x,y)=\sum_{m\in\mathbb{Z}^{\prime}}\frac{1}{1+\zeta^{-1}q^{-m}}\oint_{|z|=r}\frac{\mathop{}\\!\mathrm{d}z}{2\pi\mathrm{i}}\oint_{|w|=r^{\prime}}\frac{\mathop{}\\!\mathrm{d}w}{2\pi\mathrm{i}}\frac{w^{y-1-m}}{z^{x+1-m}}\frac{F(z)}{F(w)}.$

Defining

(4.14)
$A(x,m)=\oint_{|z|=r}\frac{\mathop{}\\!\mathrm{d}z}{2\pi\mathrm{i}}\frac{F(z)}{z^{x+1-m}}\qquad\text{and}\qquad
B(n,y)=\oint_{|w|=r^{\prime}}\frac{\mathop{}\\!\mathrm{d}w}{2\pi\mathrm{i}}\frac{w^{y-1-m}}{F(w)},$

we observe that

(4.15)
$\begin{split}K(x+s,y+s)&=\sum_{m\in\mathbb{Z}^{\prime}}\frac{1}{1+\zeta^{-1}q^{-m}}A(x+s,m)B(m,y+s)\\\
&=\sum_{m\in\mathbb{Z}^{\prime}}\frac{1}{1+\zeta^{-1}q^{-m}}A(x,m-s)B(m-s,y)\\\
&=\sum_{m^{\prime}\in\mathbb{Z}^{\prime}}f_{\zeta,s}(m^{\prime})A(x,m^{\prime})B(m^{\prime},y).\end{split}$

In this way, from the notable symmetry of the Fredholm determinant
$\det[1-AfB]=\det[1-fBA]$, we find that

(4.16)
$\det[1-K]_{\ell^{2}(\mathbb{Z}^{\prime}_{>s})}=\det\left[1-f_{\zeta,s}\cdot
M\right]_{\ell^{2}(\mathbb{Z}^{\prime})},$

where we noticed that

(4.17) $M(m,n)=\sum_{x=0}^{\infty}B(m,x)A(x,n).$

Relation (4.8) now concludes the proof. ∎

The determinantal expression stated in proposition 4.2 relates the probability
distribution of the rightmost particle in the shift mixed periodic Schur
measure to an average of a multiplicative observable of a determinantal point
process with correlation kernel $M$.

######  Corollary 4.3.

Under the hypothesis of proposition 4.2, we have

(4.18) $\mathbb{P}\left[\lambda_{1}+S\leq
s\right]=\mathbb{E}_{M}\left[\prod_{i=1}^{\infty}\frac{1}{1+\zeta
q^{s+\mathfrak{a}_{i}}}\right],$

where $\\{\mathfrak{a}_{i}:i\in\mathbb{Z}_{\geq 1}\\}$ is a determinantal
point process with correlation kernel $M$.

###### Proof.

Notable properties of determinantal point process imply that, for any function
$g$ decaying “sufficiently well”, we have

(4.19)
$\mathbb{E}_{M}\left[\prod_{i=1}^{\infty}(1+g(\mathfrak{a}_{i}))\right]=\det[1+g\cdot
M]_{\ell^{2}(\mathbb{Z}^{\prime})},$

so that setting $g=-f_{\zeta,s}$ and using (4.10) we prove relation (4.18). ∎

######  Remark 4.4.

Comparing kernels $M$ with $K$, evaluated at $q=0$, we see that $M$ is the
correlation kernel of “holes” in the Schur measure $\mathbb{PS}^{(q=0)}_{a,b}$
(see [BO17]), with specializations $a,b$ consisting of geometric progressions
of single parameter alpha specializations of ratio $q$, i.e.
$a=(a_{1},qa_{1},q^{2}a_{1},\dots,a_{2},qa_{2},q^{2}a_{2},\dots)$,
$b=(b_{1},qb_{1},q^{2}b_{1},\dots,b_{2},qb_{2},q^{2}b_{2},\dots)$.

###  4.3. Free boundary Schur measure

The _free boundary Schur measure_ was introduced by Betea, Bouttier, Nejjar
and Vuletic in [BBNV18]. Rather than the general process considered in the
original paper [BBNV18], we will only focus on a particular case that turns
out to be very important in connection to KPZ models in half space. We define
the probability measure on the set of partitions with _even column_ length

(4.20)
$\mathbb{FBS}_{a}^{(q)}(\lambda)=\frac{\mathbf{1}_{\lambda^{\prime}\text{ is
even}}}{(q;q)_{\infty}^{-1}\widetilde{\Pi}(a)}\sum_{\rho:\rho^{\prime}\text{
is even}}q^{|\rho|/2}s_{\lambda/\rho}(a).$

Here $a$ is a Schur positive specialization and the normalization term
$\widetilde{\Pi}(a)$ was introduced in (2.22). As in the case of the periodic
Schur measure, the free boundary Schur measure (4.20) itself does not possess
a nice determinantal nor pfaffian structure, but its shift-mixed version is
shown to be a pfaffian point process. Let
$S\sim\mathrm{Theta}(\zeta^{2};q^{2})$ be independent of $\lambda$. Then, the
_shift-mixed free boundary Schur measure_ is the point process

(4.21)
$\mathfrak{S}(\lambda,2S)=\left(\lambda_{i}-i+\frac{1}{2}+2S\right)_{i=1,2,\dots}\subset\mathbb{Z}^{\prime}.$

In the next theorem we give a characterization of $\mathfrak{S}(\lambda,2S)$.
For this we need to define the function $\Delta^{\prime}$ as

(4.22) $\Delta^{\prime}(x,y)=\mathbf{1}_{y=x+1}-\mathbf{1}_{y=x-1}$

and the difference operator $\nabla_{x}$ as

(4.23) $\nabla_{x}f(x)=\frac{1}{2}[f(x+1)-f(x-1)].$

######  Theorem 4.5 ([BBNV18, BBNV19]).

The shift-mixed free boundary Schur measure is a pfaffian point process with
$2\times 2$ matrix correlation kernel

(4.24) $K^{\mathrm{hs}}(x,y)=\left(\begin{matrix}k(x,y)&-2\nabla_{y}k(x,y)\\\
-2\nabla_{x}k(y,x)&4\nabla_{x}\nabla_{y}k(x,y)+\Delta^{\prime}(x,y)\end{matrix}\right),$

where

(4.25)
$k(x,y)=\frac{1}{(2\pi\mathrm{i})^{2}}\oint_{|z|=r}\frac{\mathop{}\\!\mathrm{d}z}{z^{x+3/2}}\oint_{|w|=r}\frac{\mathop{}\\!\mathrm{d}w}{w^{y+5/2}}F(z)F(w)\kappa^{\mathrm{hs}}(z,w),$
(4.26) $F(z)=\prod_{\ell\geq 0}\frac{H(q^{\ell}a;z)}{H(q^{\ell}a;z^{-1})}$

and

(4.27)
$\kappa^{\mathrm{hs}}(z,w)=\frac{(q,q,w/z,qz/w;q)_{\infty}}{(1/z^{2},1/w^{2},1/zw,qwz;q)_{\infty}}\frac{\vartheta_{3}(\zeta^{2}z^{2}w^{2};q^{2})}{\vartheta_{3}(\zeta^{2};q^{2})}.$

The function $H(a,z)$ was defined in (2.17), while the integration contours
are positively oriented and their radius satisfies $1<r<\min(q^{-1/2},R)$,
where $R$ is the convergence radius of $H(a;z)$.

###### Proof.

Result of theorem 4.5, although implicitly derived in [BBNV18], is more
explicitly stated in [BBNV19] and the expression of the $2\times 2$ matrix
correlation kernel $K^{\mathrm{hs}}$ is a reformulation of a particular case
of equations (4.2)-(4.4) of [BBNV19], where in their notation we set
$u=\sqrt{q},v=1,a_{1}=a_{2}=0$. Let us show how it is derived. From [BBNV19]
eq. (4.2)-(4.4) one sees that the shift mixed free boundary Schur measure is a
pfaffian point process with $2\times 2$ matrix correlation kernel

(4.28)
$\widetilde{K}(x,y)=\left(\begin{matrix}\widetilde{K}_{1,1}(x,y)&\widetilde{K}_{1,2}(x,y)\\\
-\widetilde{K}_{1,2}(y,x)&\widetilde{K}_{2,2}(x,y)+\Delta^{\prime}(x,y)\end{matrix}\right),$

defined as

(4.29) $\displaystyle\widetilde{K}_{1,1}(x,y)$
$\displaystyle=\oint_{|z|=r}\frac{\mathop{}\\!\mathrm{d}z}{2\pi\mathrm{i}}\oint_{|w|=r^{\prime}}\frac{\mathop{}\\!\mathrm{d}w}{2\pi\mathrm{i}}\frac{F(z)F(w)}{z^{x+3/2}w^{y+5/2}}\kappa^{\mathrm{hs}}(z,w),$
(4.30) $\displaystyle\widetilde{K}_{1,2}(x,y)$
$\displaystyle=\oint_{|z|=r}\frac{\mathop{}\\!\mathrm{d}z}{2\pi\mathrm{i}}\oint_{|w|=1/r^{\prime}}\frac{\mathop{}\\!\mathrm{d}w}{2\pi\mathrm{i}}\frac{F(z)F(1/w)}{z^{x+3/2}w^{-y+1/2}}\kappa^{\mathrm{hs}}(z,1/w)(1-w^{2}),$
(4.31) $\displaystyle\widetilde{K}_{2,2}(x,y)$
$\displaystyle=\oint_{|z|=1/r}\frac{\mathop{}\\!\mathrm{d}z}{2\pi\mathrm{i}}\oint_{|w|=1/r^{\prime}}\frac{\mathop{}\\!\mathrm{d}w}{2\pi\mathrm{i}}\frac{F(1/z)F(1/w)}{{z^{-x+3/2}w^{-y+1/2}}}\kappa^{\mathrm{hs}}(1/z,1/w)(1-z^{2})(1-w^{2}),$

with $1<r,r^{\prime}<\min(q^{-1},R)$. Notice that in [BBNV19] the element
$\widetilde{K}_{2,2}$ has integration contours $|z|=r$ and $|w|=r^{\prime}$
and the function $\Delta^{\prime}$ is absent. To reach our expression we shift
both contours and doing so the only pole one encounters is at $z=1/w$, whose
contribution is

(4.32)
$\begin{split}\oint_{|w|=1/r^{\prime}}\frac{\mathop{}\\!\mathrm{d}w}{2\pi\mathrm{i}}w^{y-x}\left(\frac{1}{w^{2}}-1\right)&=\mathbf{1}_{y=x+1}-\mathbf{1}_{y=x-1}=\Delta^{\prime}(x,y).\end{split}$

With a change of variable $w\to 1/w$ in (4.30) and $z\to 1/z,w\to 1/w$ in
(4.31) we can now make integration contours to be the same for each component
$\widetilde{K}_{i,j}$. Finally we notice that, after such change of variables,
the $\widetilde{K}_{i,j}$’s differ from each other by the sole presence of
factors $(w-1/w)$ and $(z-1/z)$ in the integrand, so that each term can be
easily matched with discrete derivatives $\nabla_{y},\nabla_{x}$ of the
function $k(x,y)=\widetilde{K}_{1,1}(x,y)$ as in (4.24), using the simple
relation

(4.33)
$\nabla_{x}\,\frac{1}{z^{x}}=-\frac{1}{2}\left(z-\frac{1}{z}\right)\frac{1}{z^{x}}.$

This concludes the proof. ∎

######  Remark 4.6.

Setting $q=0$, the free boundary Schur measure becomes the _Pfaffian Schur
measure_ defined in [BR05] (see also an earlier work [SI04]) and studied in
[Gho19, BBCS18, BBNV18]. Below in section 6.3 we will denote this measure by
$\mathbb{FBS}^{(q=0)}$ and we will present an asymptotic analysis for the
first row $\lambda_{1}$ of a random partition $\lambda$.

As in case of the periodic Schur measure, we will be interested in the
rightmost coordinate of the shift-mixed free boundary Schur measure. Its
probability distribution function is explicitly written as the Fredholm
pfaffian

(4.34) $\mathbb{P}[\lambda_{1}+2S\leq
s]=\mathrm{Pf}(J-K^{\mathrm{hs}})_{\ell^{2}(\mathbb{Z}_{>s}^{\prime})}.$

Recall that $J$ is the matrix (1.13). In the following proposition we provide
an alternative expression for the probability law of $\lambda_{1}$, which is
slightly more convenient to consider degenerations and asymptotic limits.

######  Proposition 4.7.

Let $\lambda$ be a random partition distributed according to the free boundary
Schur measure and $S\sim\mathrm{Theta}(\zeta^{2};q^{2})$. Then the probablity
distribution of $\lambda_{1}+2S$ admits the Fredholm pfaffian representation

(4.35) $\mathbb{P}[\lambda_{1}+2S\leq
s]=\mathrm{Pf}(J-L)_{\ell^{2}(\mathbb{Z}^{\prime}_{>s})},$

where $L$ is the $2\times 2$ matrix kernel

(4.36) $L(x,y)=\left(\begin{matrix}k(x,y)&-\nabla_{y}k(x,y)\\\
-\nabla_{x}k(x,y)&\nabla_{x}\nabla_{y}k(x,y)\end{matrix}\right),$

and the function $k$ is given by (4.25), while $\nabla_{x},\nabla_{y}$ were
defined in (4.23).

###### Proof.

This is a slight generalization of [BBCS18, Proposition 5.2], where partial
derivatives $\partial_{x},\partial_{y}$ are replaced by discrete difference
operators $\nabla_{x},\nabla_{y}$. We find that the proof reported in [BBCS18]
adapts well to our situation and so here we are only going to sketch the
argument. Notation we introduce below will be close to that of [BBCS18].
Consider the expansion of Fredholm pfaffians of kernels $L$ and
$K^{\mathrm{hs}}$ as

(4.37) $\mathrm{Pf}[J-L]_{\ell^{2}(\mathbb{Z}^{\prime}_{>s})}=\sum_{\ell\geq
0}\frac{(-1)^{\ell}}{\ell!}a_{\ell},\qquad\mathrm{Pf}\left[J-K^{\mathrm{hs}}\right]_{\ell^{2}(\mathbb{Z}^{\prime}_{>s})}=\sum_{\ell\geq
0}\frac{(-1)^{\ell}}{\ell!}b_{\ell},$

where

(4.38)
$a_{\ell}=\sum_{x_{1},\dots,x_{\ell}>s}\mathrm{Pf}[L(x_{i},x_{j})]_{i,j=1}^{\ell},\qquad
b_{\ell}=\sum_{x_{1},\dots,x_{\ell}>s}\mathrm{Pf}\left[K^{\mathrm{hs}}(x_{i},x_{j})\right]_{i,j=1}^{\ell}.$

The $2\times 2$ matrix kernel $K^{\mathrm{hs}}$ can be written as a sum

(4.39) $K^{\mathrm{hs}}(x,y)=A(x,y)+\left(\begin{matrix}0&0\\\
0&\Delta^{\prime}(x,y)\end{matrix}\right),$

with $A_{1,1}(x,y)=k(x,y)$, $A_{1,2}(x,y)=-2\nabla_{y}k(x,y)$,
$A_{1,2}(x,y)=-2\nabla_{x}k(x,y)$, $A_{2,2}(x,y)=4\nabla_{x}\nabla_{y}k(x,y)$
. Using a notable minor expansion for pfaffians of sums of $n\times n$
matrices $N,M$ (see [Ste90, Lemma 4.2])

(4.40) $\mathrm{Pf}[M+N]=\sum_{\begin{subarray}{c}I\subset\\{1,\dots n\\}\\\
\\#I\text{
even}\end{subarray}}(-1)^{|I|-\\#I/2}\mathrm{Pf}[M_{I}]\mathrm{Pf}[N_{I^{c}}]$

where $\\#I$ is the cardinality of $I$, while $|I|$ is the sum of its
elements, we can express $b_{\ell}$ as

(4.41)
$\frac{(-1)^{\ell}}{\ell!}b_{\ell}=\sum_{m=\lceil\ell/2\rceil}^{\ell}b_{\ell}^{(m)}.$

In the above expression we have condensated in the term $b_{\ell}^{(m)}$ all
summations involving exactly a product of degree $\ell-m$ of $\Delta^{\prime}$
functions originating from the evaluation of minors of the matrix
$\left(\begin{smallmatrix}0&0\\\
0&\Delta^{\prime}(x_{i},x_{j})\end{smallmatrix}\right)_{i,j=1}^{m}$. A
computation shows that terms $b_{\ell}^{(m)}$ have the explicit expression

(4.42)
$\begin{split}b_{\ell+i}^{(\ell)}&=\frac{(-1)^{\ell+i}}{(\ell+i)!}\sum_{x_{1},\dots,x_{\ell+i}>s}\left(\sum_{1\leq
j_{1}<\cdots<j_{2i}\leq\ell+i}\mathrm{Pf}[\Delta^{\prime}(x_{j_{r}},x_{j_{s}})]_{r,s=1}^{i}(-1)^{i}\mathrm{Pf}\left[A_{\ell+i}^{\widehat{2j_{1}},\dots,\widehat{2j_{2i}}}\right]\right)\\\
&=\frac{(-1)^{\ell}}{(\ell+i)!}\binom{\ell+i}{2i}\sum_{x_{1},\dots,x_{\ell+i}>s}\mathrm{Pf}[\Delta^{\prime}(x_{r},x_{s})]_{r,s=\ell-i+1}^{\ell+i}\mathrm{Pf}\left[A_{\ell+i}^{\widehat{2(\ell-i+1)},\dots,\widehat{2(\ell+i)}}\right],\end{split}$

where we have denoted by $A_{n}^{\widehat{2j_{1}},\dots,\widehat{2j_{r}}}$ is
the $2(n-r)\times 2(n-r)$ matrix resulting from removing rows and columns
$2j_{1},\dots,2j_{r}$ from the matrix
$(A(x_{\alpha},x_{\beta}))_{\alpha,\beta=1}^{n}$. Notice that the second
equality in (4.42) follows from the fact that, through a change of
$x$-variables all terms in the sum inside the parentheses become equal, since
the $x_{i}$ are all summed over the same domain. We now evaluate the sum in
the right hand side of (4.42), expanding one of the Pfaffians as in (A.7). We
find

(4.43)
$\begin{split}&\sum_{x_{1},\dots,x_{\ell+i}>s}\mathrm{Pf}[\Delta^{\prime}(x_{r},x_{s})]_{r,s=\ell-i+1}^{\ell+i}\mathrm{Pf}\left[A_{\ell+i}^{\widehat{2(\ell-i+1)},\dots,\widehat{2(\ell+i)}}\right]\\\
&=\sum_{\begin{subarray}{c}\alpha=\\{(r_{1},s_{1}),\dots,(r_{i},s_{i})\\}\\\
r_{j},s_{j}\in\\{\ell-i+1,\dots,\ell+i\\}\\\
r_{1}<\cdots<r_{i},\,\,r_{j}<s_{j}\end{subarray}}\epsilon(\alpha)\sum_{x_{1},\dots,x_{\ell+i}>s}\prod_{j=1}^{i}\Delta^{\prime}(x_{r_{j}},x_{s_{j}})\mathrm{Pf}\left[A_{\ell+i}^{\widehat{2(\ell-i+1)},\dots,\widehat{2(\ell+i)}}\right]\\\
&=\frac{(2i)!}{2^{i}i!}\sum_{x_{1},\dots,x_{\ell+i}>s}\prod_{j=1}^{i}\Delta^{\prime}(x_{\ell-i+2j-1},x_{\ell-i+2j})\mathrm{Pf}\left[A_{\ell+i}^{\widehat{2(\ell-i+1)},\dots,\widehat{2(\ell+i)}}\right]\end{split}$

where in the third line we recognized that, changing $x$-variables and
absorbing the sign $\varepsilon(\alpha)$ into the pfaffian all terms of the
summation over permutations $\alpha$ in the second line are equal. We can now
evaluate the action of operators $\Delta^{\prime}$ on the pfaffian and, after
taking summations over variables $x_{\ell+1},\dots x_{\ell+i}$, we find

(4.44)
$\begin{split}&2^{i}\,\nabla_{x_{\ell}}\cdots\nabla_{x_{\ell+i}}\mathrm{Pf}\left[A_{\ell+i}^{\widehat{2(\ell-i+1)},\dots,\widehat{2(\ell+i)}}\right]\bigg{|}_{x_{\ell}=\cdots=x_{\ell+i}}\\\
&=2^{i}\,\mathrm{Pf}\left[\nabla_{x_{\ell}}\cdots\nabla_{x_{\ell+i}}A_{\ell+i}^{\widehat{2(\ell-i+1)},\dots,\widehat{2(\ell+i)}}\bigg{|}_{x_{\ell}=\cdots=x_{\ell+i}}\right]\\\
&=(-1)^{i}2^{\ell}\mathrm{Pf}[L(x_{i},x_{j})]_{i,j=1}^{\ell}.\end{split}$

Substituting the last relation and (4.43) into (4.42) we obtain

(4.45)
$\begin{split}b_{\ell+i}^{(\ell)}=\frac{(-1)^{\ell+i}}{(\ell+i)!}\binom{\ell+i}{2i}\frac{(2i)!}{2^{i}i!}2^{\ell}a_{\ell}=\frac{(-1)^{\ell}a_{\ell}}{\ell!}\binom{\ell}{i}2^{k-i}(-1)^{i},\end{split}$

so that

(4.46)
$\sum_{i=0}^{\ell}b_{\ell+i}^{(\ell)}=\frac{(-1)^{\ell}}{\ell!}a_{\ell}\sum_{i=0}^{\ell}\binom{\ell}{i}2^{k-i}(-1)^{i}=\frac{(-1)^{\ell}}{\ell!}a_{\ell}.$

We can now turn the Fredholm pfaffian expansion of $J-K^{\mathrm{hs}}$ into
that of $J-L$ as

(4.47) $\begin{split}\mathrm{Pf}[J-K^{\mathrm{hs}}]&=\sum_{\ell\geq
0}\frac{(-1)^{\ell}}{\ell!}b_{\ell}=\sum_{m\geq
0}\sum_{i=0}^{m}b_{m+i}^{(m)}=\sum_{\ell\geq
0}\frac{(-1)^{\ell}}{\ell!}a_{\ell}=\mathrm{Pf}[J-L],\end{split}$

where in the second equality we used (4.41) and operated an exchange of
summation indices and in the third equality we renamed $m$ back to $\ell$.
This manipulations are justified by absolute convergence of all terms
involved, which is consequence of exponential decay of coefficients of matrix
kernels $L(x,y)$ and $K^{\mathrm{hs}}(x,y)$ as $x,y\to\infty$. This completes
the proof. ∎

###  4.4. Alternatve representation of Fredholm pfaffian via a
$q$-trigonometric identity by Gosper

Following Gosper [Gos01], we define the $q$-analog of the sine and cosine
functions as

(4.48) $\sin_{q}(\pi
z)=q^{\left(z-\frac{1}{2}\right)^{2}}\frac{(q^{2z},q^{2-2z};q^{2})_{\infty}}{(q,q;q^{2})_{\infty}}\qquad\text{and}\qquad\cos_{q}(z)=\sin_{q}(\frac{\pi}{2}-z).$

In the same paper, Gosper conjectured, via numerical experiments a number of
properties of his $q$-trigonometric functions. Among these we have the
following $q$-analog of the addition formula for angles, proven more recently
in [EB18],

(4.49)
$\sin_{q}(x+y)=\frac{\sin_{q}(x-y)}{\sin_{q^{2}}(x-y)}\left[\sin_{q^{2}}(x)\cos_{q^{2}}(y)+\sin_{q^{2}}(y)\cos_{q^{2}}(x)\right].$

Using this remarkable relation and Ramanujan’s bilateral sum (4.9) we obtain
an alternative representation of the Fredholm pfaffian in the right hand side
of (4.35).

######  Proposition 4.8.

Let $\lambda$ be distributed according to the free boundary Schur measure
(4.20) and $S\sim\mathrm{Theta}(\zeta^{2};q^{2})$. Then we have

(4.50) $\mathbb{P}\left[\lambda_{1}+2S\leq
s\right]=\mathrm{Pf}\left[J-f_{\zeta^{2},s}\cdot
M^{\mathrm{hs}}\right]_{\ell^{2}(\mathbb{Z}^{\prime})},$

where $M^{\mathrm{hs}}$ is the $2\times 2$ matrix kernel

(4.51)
$M^{\mathrm{hs}}(m,n)=\left(\begin{matrix}M^{\mathrm{hs}}_{1,1}(m,n)&M^{\mathrm{hs}}_{1,2}(m,n)\\\
-M^{\mathrm{hs}}_{1,2}(n,m)&M^{\mathrm{hs}}_{2,2}(m,n)\end{matrix}\right),$

with coefficients

(4.52)
$\begin{split}M^{\mathrm{hs}}_{1,1}(m,n)=\oint_{|z|=r}\frac{\mathop{}\\!\mathrm{d}z}{2\pi\mathrm{i}}\oint_{|z|=r}\frac{\mathop{}\\!\mathrm{d}w}{2\pi\mathrm{i}}&\frac{F(z)}{z^{m+5/2}}\frac{F(w)}{w^{n+5/2}}\frac{z-w}{1-zw}\frac{1+zw}{2}\\\
&\times\frac{z(1/z^{2},q^{2}z^{2};q^{2})_{\infty}}{(1/z^{2};q)_{\infty}}\frac{w(1/w^{2},q^{2}w^{2};q^{2})_{\infty}}{(1/w^{2};q)_{\infty}},\end{split}$
(4.53)
$\begin{split}M^{\mathrm{hs}}_{1,2}(m,n)=\oint_{|z|=r}\frac{\mathop{}\\!\mathrm{d}z}{2\pi\mathrm{i}}\oint_{|z|=r}\frac{\mathop{}\\!\mathrm{d}w}{2\pi\mathrm{i}}&\frac{F(z)}{z^{m+5/2}}\frac{F(w)}{w^{n+5/2}}\frac{z-w}{1-zw}\frac{1+zw}{2}\\\
&\times\frac{z(1/z^{2},q^{2}z^{2};q^{2})_{\infty}}{(1/z^{2};q)_{\infty}}\frac{(qw^{2},q/w^{2};q^{2})_{\infty}}{(q;q^{2})_{\infty}^{2}(1/w^{2};q)_{\infty}},\end{split}$
(4.54)
$\begin{split}M^{\mathrm{hs}}_{2,2}(m,n)=\oint_{|z|=r}\frac{\mathop{}\\!\mathrm{d}z}{2\pi\mathrm{i}}\oint_{|z|=r}\frac{\mathop{}\\!\mathrm{d}w}{2\pi\mathrm{i}}&\frac{F(z)}{z^{m+5/2}}\frac{F(w)}{w^{n+5/2}}\frac{z-w}{1-zw}\frac{1+zw}{2}\\\
&\times\frac{(qz^{2},q/z^{2};q^{2})_{\infty}}{(q;q^{2})_{\infty}^{2}(1/z^{2};q)_{\infty}}\frac{(qw^{2},q/w^{2};q^{2})_{\infty}}{(q;q^{2})_{\infty}^{2}(1/w^{2};q)_{\infty}}\end{split}$

and $f_{\zeta^{2},s}$ is given in (4.12). The notation adopted above is the
same as in theorem 4.5.

###### Proof.

The addition formula (4.49) implies, setting $z=q^{-2x},w=q^{2y}$ and
rescaling $q\to q^{1/2}$, the identity

(4.55)
$(w/z,qz/w;q)_{\infty}=\frac{1}{z}\frac{(1/zw,qzw;q)_{\infty}}{(1/z^{2}w^{2},q^{2}z^{2}w^{2};q^{2})_{\infty}}\left[g_{1}(z)g_{2}(w)-g_{1}(w)g_{2}(z)\right],$

where we defined

(4.56) $g_{1}(z)=z(1/z^{2},q^{2}z^{2};q^{2})_{\infty},\qquad
g_{2}(w)=\frac{(qw^{2},q/w^{2};q^{2})_{\infty}}{(q;q^{2})^{2}_{\infty}}.$

Using (4.55) and (4.9) we can rewrite the function $\kappa^{\mathrm{hs}}$ of
(4.27) as

(4.57)
$\kappa^{\mathrm{hs}}(z,w)=w\frac{g_{1}(z)g_{2}(w)-g_{1}(w)g_{2}(z)}{(1/z^{2},1/w^{2};q)_{\infty}}\sum_{m\in\mathbb{Z}^{\prime}}\frac{(zw)^{2m}}{1+\zeta^{-2}q^{-2m}}.$

Defining

(4.58)
$a(x)=\oint_{|z|=r}\frac{\mathop{}\\!\mathrm{d}z}{2\pi\mathrm{i}}\frac{F(z)}{z^{x+3/2}}\frac{g_{1}(z)}{(1/z^{2};q)_{\infty}},\qquad
b(y)=\oint_{|w|=r}\frac{\mathop{}\\!\mathrm{d}w}{2\pi\mathrm{i}}\frac{F(w)}{w^{y+3/2}}\frac{g_{2}(w)}{(1/w^{2};q)_{\infty}},$

we observe that

(4.59)
$k(x,y)=\sum_{m\in\mathbb{Z}^{\prime}}\frac{1}{1+\zeta^{-2}q^{-2m}}\left[a(x-2m)b(y-2m)-a(y-2m)b(x-2m)\right],$

where $k(x,y)$ was given in (4.25). Now, defining

(4.60) $A(x,m)=\left(\begin{matrix}-\nabla_{x}a(x+2m)&\nabla_{x}b(x+2m)\\\
-a(x+2m)&b(x+2m)\end{matrix}\right),\qquad
B(m,y)=\left(\begin{matrix}b(y+2m)&-\nabla_{y}b(y+2m)\\\
a(y+2m)&-\nabla_{y}a(y+2m)\end{matrix}\right),$

we can express the matrix kernel $L$ as

(4.61)
$\begin{split}JL(x+s,y+s)&=\sum_{m\in\mathbb{Z}^{\prime}}\frac{1}{1+\zeta^{-2}q^{-2m}}A(x+s,m)B(m,y+s)\\\
&=\sum_{m\in\mathbb{Z}^{\prime}}\frac{1}{1+\zeta^{-2}q^{-2m}}A(x,m-\frac{s}{2})B(m-\frac{s}{2},y)\\\
&=\sum_{m^{\prime}\in\mathbb{Z}^{\prime}}f_{\zeta^{2},s}(m^{\prime})A(x,\frac{m^{\prime}}{2})B(\frac{m^{\prime}}{2},y).\end{split}$

Using the notable relation $\mathrm{Pf}[J-J^{T}AfB]=\mathrm{Pf}[J-fBAJ^{T}]$,
we find that

(4.62)
$\mathrm{Pf}[J-L]_{\ell^{2}(\mathbb{Z}_{>s}^{\prime})}=\mathrm{Pf}\left[J-f_{\zeta^{2},s}\cdot
M^{\mathrm{hs}}\right]_{\ell^{2}(\mathbb{Z}^{\prime})},$

since

(4.63)
$M^{\mathrm{hs}}(m,n)=\sum_{x=1}^{\infty}B(\frac{m}{2},x)A(x,\frac{n}{2})J^{T}.$

This completes the proof. ∎

The identity (4.50) implies that the probability distribution of the right
edge particle of the shift mixed free boundary Schur measure is equivalent to
the expectation of the multiplicative Fermi-Dirac factor over a pfaffian point
process with correlation kernel $M^{\mathrm{hs}}$.

######  Corollary 4.9.

Under the hypothesis of proposition 4.8, we have

(4.64) $\mathbb{P}\left[\lambda_{1}+2S\leq
s\right]=\mathbb{E}_{M^{\mathrm{hs}}}\left[\prod_{i=1}^{\infty}\frac{1}{1+\zeta^{2}q^{s+\mathfrak{a}_{i}}}\right],$

where $\\{\mathfrak{a}_{i}:i\in\mathbb{Z}_{\geq 1}\\}$ is a pfaffian point
process with correlation kernel $M^{\mathrm{hs}}$.

###### Proof.

Notable properties of pfaffian point process imply that, for any function $g$
with “nice” decay properties, we have

(4.65)
$\mathbb{E}_{M^{\mathrm{hs}}}\left[\prod_{i=1}^{\infty}(1+g(\mathfrak{a}_{i}))\right]=\mathrm{Pf}\left[J+g\cdot
M^{\mathrm{hs}}\right]_{\ell^{2}(\mathbb{Z}^{\prime})},$

so that setting $g=-f_{\zeta^{2},s}$ and using (4.50) we prove relation
(4.64). ∎

We think of (4.64) as a pfaffian variant of (4.18). In remark 4.4 we pointed
out the relation between the correlation kernel $M$ and the Schur measure and
one could naively expect a similar relation between the correlation kernel
$M^{\mathrm{hs}}$ and the Pfaffian Schur measure. However, from a simple
inspection of formulas given in proposition 4.8, this does not appear to be
true. It is in fact an interesting problem to give a physical interpretation
of the pfaffian point process associated to $M^{\mathrm{hs}}$.

###  4.5. Equivalence with marginals of $q$-Whittaker measures

Here we state the fundamental connections between models of determinantal and
pfaffian measures presented above in this section and discrete KPZ solvable
models discussed in section 3.

######  Theorem 4.10 ([IMS21a]).

Let $\mu\sim\mathbb{W}^{(q)}_{a;b}$ and $\chi\sim q\text{-}\mathrm{Geo}(q)$ be
indipendent random variables. Consider also
$\lambda\sim\mathbb{PS}^{(q)}_{a,b}$. Then the following equivalence in
distribution holds

(4.66) $\mu_{1}+\chi\stackrel{{\scriptstyle\mathcal{D}}}{{=}}\lambda_{1}.$

Theorem 4.10 is an immediate consequence of the summation identity (2.26). In
[IMS21a] we proved the same theorem, along with identity (2.26), by matching
Fredholm determinant expressions for shift-mixed periodic Schur measure with
those of $q$-Whittaker measure found in [IS19].

The following theorem is new and it is a half space extension of theorem 4.10.

######  Theorem 4.11.

Let $\mu\sim\mathbb{HW}^{(q)}_{a;0}$ and $\chi\sim q\text{-}\mathrm{Geo}(q)$
be indipendent random variables. Consider also
$\lambda\sim\mathbb{FBS}^{(q)}_{a}$. Then the following equivalence in
distribution holds

(4.67) $\mu_{1}+\chi\stackrel{{\scriptstyle\mathcal{D}}}{{=}}\lambda_{1}.$

###### Proof.

This is straightforward consequence of the refined Littlewood identity (2.27).
∎

Unlike for theorem 4.10, a proof of equality (4.67) based on integrable
probabilistic techniques is at the moment not available. This is because
rigorous methods using Bethe ansatz have proven to be difficult to implement
for the study of half space $q$-Whittaker measure and through such techniques
it remains a challenge to obtain explicit and manageable pfaffian formulas for
interesting physical observables; see [BBC20].

##  5\. Fredholm determinant and pfaffian formulas for KPZ models

Leveraging equalities in distribution established in theorem 4.10 and theorem
4.11 along with the simple structure of the periodic and free boundary Schur
measures we are able to produce determinantal and pfaffian formulas for
observables of KPZ solvable models. Determinantal formulas reported below,
although new, add to a number of analogous results discovered in the last 15
years [TW08a, TW09a, BC14, BCS14, BCR13, MP17] and in many cases appear to be
simpler. Pfaffian formulas, on the other hand, are completely original and
have no analog in the existing mathematical literature.

###  5.1. Fredholm determinant formulas for $q$-PushTASEP and ASEP in full
space

Let us start presenting a representation for the probability distribution of a
tagged particle in $q$-PushTASEP in terms of a Fredholm determinant of a free-
fermionic type kernel.

######  Corollary 5.1.

Let $\mathsf{X}(T)$ be the $q$-PushTASEP with initial conditions
$\mathsf{x}_{k}(0)=k$, for $k=1,\dots,N$ and $\chi\sim q$-$\mathrm{Geo}(q)$,
$S\sim\mathrm{Theta}(\zeta,q)$ independent random variables. Then we have

(5.1) $\mathbb{P}\left(\mathsf{x}_{N}(T)+\chi+S\leq
N+s\right)=\det\left(1-K_{\mathrm{push}}\right)_{\ell^{2}(\mathbb{Z}_{>s}^{\prime})},$

where the kernel $K_{\mathrm{push}}$ is obtained from (4.5) setting

(5.2)
$F(z)=\frac{\prod_{i=1}^{T}(b_{i}/z;q)_{\infty}}{\prod_{i=1}^{N}(a_{i}z;q)_{\infty}}.$

###### Proof.

This is obtained combining theorem 4.10, proposition 3.1 and theorem 4.1. ∎

######  Remark 5.2.

The left hand side of (5.1) admits an alternative expression as $q$-Laplace
transform

(5.3) $\mathbb{E}\left(\frac{1}{(-\zeta
q^{\frac{1}{2}+s+N-\mathsf{x}_{N}(T)};q)_{\infty}}\right)$

of the probability density function of $\mathsf{x}_{N}(T)$. This is a
consequence of the identity

(5.4) $\mathbb{P}(\chi+S\leq n)=\frac{1}{(-\zeta
q^{\frac{1}{2}+n};q)_{\infty}},$

where $\chi$ and $S$ are as in corollary 5.1. For a direct proof of (5.4) see
[IMS21a, Lemma 2.4].

The first instance of a determinantal formula for the $q$-PushTASEP was given
in [BCFV15, Theorem 3.3] and it was based on clever manipulations of integral
formulas for $q$-moments; see [BC14]. The kernel presented in [BCFV15], as
others obtained starting from $q$-moments formulas, does not appear to be
related with determinantal point processes or free fermions. As such,
asymptotic analysis is rather involved, although doable, as one has to deal
with a complicated pole structure of the kernel. It would be very useful to
obtain our expression (5.1) by a proper summation of $q$-moments.

Next we present a Fredholm determinant formula of free fermionic type for the
ASEP.

######  Corollary 5.3.

Let $\mathsf{J}(x,\tau)$ be the integrated current of the ASEP with initial
conditions $\mathsf{x}_{k}(0)=-k$, for $k=1,2,\dots$ and let $\chi\sim
q$-$\mathrm{Geo}(q)$ and $S\sim\mathrm{Theta}(\zeta,q)$ be independent random
variables. Then, we have

(5.5) $\mathbb{P}\left(-\mathsf{J}(x,\tau)+\chi+S\leq
s\right)=\det(1-K_{\mathrm{ASEP}})_{\ell^{2}(\mathbb{Z}^{\prime}_{>s})},$

where the kernel $K_{\mathrm{ASEP}}$ is obtained from (4.5) setting

(5.6) $F(z)=e^{-(1-q)\frac{z\tau}{1+z}}(1+1/z)^{x}.$

###### Proof.

Consider the $q$-Whittaker measure $\mathbb{W}^{(q)}_{a^{\prime},b^{\prime}}$,
where $a^{\prime},b^{\prime}$ are $q$-beta specializations with parameters
$(a_{1},\dots,a_{N})$, $(b_{1},\dots,b_{T})$. Then, combining theorem 4.10 and
theorem 4.1 we have

(5.7) $\mathbb{P}(\mu_{1}+\chi+S\leq
r)=\det(1-\tilde{K})_{\ell^{2}(\mathbb{Z}^{\prime}_{>r})},$

where $\tilde{K}$ is obtained from (4.5) setting

(5.8) $F(z)=\frac{\prod_{i=1}^{N}(1+za_{i})}{\prod_{i=1}^{T}(1+b_{i}/z)}.$

This is simply a result of the definition of $q$-beta specialization (2.12).
Considering the scaling (3.9), under which $\mu_{1}-N$ approaches in law the
current of the ASEP with step initial data, the left hand side of (5.7)
becomes

(5.9) $\mathbb{P}(-\mathsf{J}^{(\varepsilon)}(x,\tau)+\chi+S\leq r-N),$

while the function $F(z)$ becomes

(5.10)
$F(z)=z^{N}(1+1/z)^{x}e^{\tau\frac{(1-q)}{2}\left(1-\frac{2z}{1+z}\right)+O(\varepsilon)}.$

From the explicit expression (4.5) we see that rescaling $\tilde{K}(x+N,y+N)$
and $r=s+N$ one can prove the convergence

(5.11)
$\det(1-\tilde{K})_{\ell^{2}(\mathbb{Z}^{\prime}_{>r})}\xrightarrow[\varepsilon\to
0]{}\det(1-K_{\mathrm{ASEP}})_{\ell^{2}(\mathbb{Z}^{\prime}_{>s})},$

which combined with proposition 3.3, proves (5.5). ∎

Using corollary 5.3, along with corollary 4.3 one can recover results found by
Borodin and Olshanski in [BO17, Theorem 10.2] relating the integrated current
of the ASEP with the discrete Laguerre ensemble. Similarly to the observation
made at the end of section 3.3, one could use the correspondence between the
$q$-Whittaker measure and the periodic Schur measure to deduce determinantal
formula for the full hierarchy of solvable models descending from the
stochastic six vertex model. Nevertheless we will not take this route in this
paper having decided to treat only the most representative cases.

###  5.2. Fredholm determinant representation for Log Gamma polymer in full
space

We report a Fredholm determinant formula for the Laplace transform of the
probability density function of the Log Gamma polymer partition function
$Z(N,N)$. In literature a resembling expression was derived originally in
[BCR13], where the integral kernel possessed a single contour integral
representation; see also [BCD21]. Compared to such knows formulas, the one we
present below is noticeably simpler and the kernel possesses a double contour
integral representation. This feature is a result of the direct relation
between Log Gamma polymers, seen as a scaling limit of $q$-PushTASEP as in
proposition 3.5, with free fermions at positive temperature.

######  Theorem 5.4.

Let $Z(N,N)$ be the partition function of the Log Gamma polymer model on the
quadrant. Then we have

(5.12) $\mathbb{E}\left[e^{-e^{-\varsigma+\log
Z(N,N)}}\right]=\det(1-\mathbf{K})_{\mathbb{L}^{2}(\varsigma,+\infty)},$

where the kernel is given by

(5.13)
$\mathbf{K}(X,Y)=\int_{\mathrm{i}\mathbb{R}-d}\frac{\mathop{}\\!\mathrm{d}Z}{2\pi\mathrm{i}}\int_{\mathrm{i}\mathbb{R}+d^{\prime}}\frac{\mathop{}\\!\mathrm{d}W}{2\pi\mathrm{i}}\frac{\pi}{\sin(\pi(W-Z))}\frac{g_{A,B}(Z)}{g_{A,B}(W)}e^{ZX-
WY},$

where $d,d^{\prime}>0$, such that $\frac{1}{2N}<d^{\prime}+d<1$ and

(5.14) $g_{A,B}(Z)=\prod_{i=1}^{N}\frac{\Gamma(A_{i}+Z)}{\Gamma(B_{i}-Z)}.$

The proof of theorem 5.4 is based on a scaling limit of the Fredholm
determinant formula (5.1). We will present it below in the subsection.

######  Remark 5.5.

A Fredholm determinant representation analogous to (5.12) should be also
available for the partition function $Z(T,N)$ for generic $T,N$. In this case
the kernel $\mathbf{K}$ gets modified by changing the factor $g_{A,B}(Z)$ into
$\prod_{i=1}^{N}\Gamma(A_{i}+Z)\prod_{i=1}^{T}\Gamma(B_{i}-Z)^{-1}$. This
claim should follow from an adaptation of arguments presented in the proof of
theorem 5.4 and technical results developed in appendix B. We will not discuss
any further such generalizations in this paper.

######  Remark 5.6.

The Laplace transform in the left hand side of (5.12) admits an alternative
expression using the identity

(5.15)
$\mathbb{E}\left[e^{-e^{-\varsigma+U}}\right]=\mathbb{P}\left[U+\mathcal{G}\leq\varsigma\right],$

which holds for any random variable $U$ and $\mathcal{G}$ is an independent
standard Gumbel random variable. This can be seen as a continuous variant of
the identity shown in remark 5.2.

######  Lemma 5.7.

The Fredholm determinant of the kernel $\mathbf{K}$ is absolutely convergent.

###### Proof.

Using the parameterization of integration variables $Z=-d+\mathrm{i}u$,
$W=d+\mathrm{i}v$, with $d\in(\frac{1}{4},\frac{1}{2})$, we easily find the
estimate

(5.16) $\begin{split}\left|\mathbf{K}(X,Y)\right|\leq
e^{-d(X+Y)}\int_{\mathbb{R}}\frac{\mathop{}\\!\mathrm{d}u}{2\pi}\int_{\mathbb{R}}\frac{\mathop{}\\!\mathrm{d}v}{2\pi}&\left|\frac{\pi}{\sin\left[\pi(2d+\mathrm{i}(v-u))\right]}\right|\\\
&\times\prod_{i=1}^{N}\left|\frac{\Gamma(A_{i}-d+\mathrm{i}u)}{\Gamma(B_{i}+d-\mathrm{i}u)}\frac{\Gamma(B_{i}-d-\mathrm{i}v)}{\Gamma(A_{i}+d+\mathrm{i}v)}\right|.\end{split}$

The integral in the previous expression is bounded by a constant. To show this
we use the estimates

(5.17) $\left|\frac{1}{\sin[\pi(\alpha+\mathrm{i}u)]}\right|\leq
C_{\alpha}e^{-\pi|u|},$

and

(5.18) $\left|\frac{\Gamma(a+\mathrm{i}u)}{\Gamma(b-\mathrm{i}u)}\right|\leq
C_{a,b}\left(1+|u|\right)^{a-b}.$

In the first inequelity $C_{\alpha}$ is a constant depending on
$\alpha\notin\mathbb{Z}$, while the second one holds for
$a\notin\mathbb{Z}_{\leq 0}$ and is a consequence of the following adaptation
of the Stirling’s approximation

(5.19)
$|\Gamma(a+\mathrm{i}u)|=\sqrt{2\pi}\,|u|^{a-\frac{1}{2}}e^{-a-\frac{|u|\pi}{2}}\left(1+O(|u|^{-1})\right).$

Then, the integral in the right hand side of (5.16) is bounded by

(5.20)
$C\int_{\mathbb{R}}\mathop{}\\!\mathrm{d}u\int_{\mathbb{R}}\mathop{}\\!\mathrm{d}ve^{-\pi|u-v|}\left(\frac{1+|u|}{1+|v|}\right)^{\sum_{i=1}^{N}A_{i}-B_{i}}(1+|u|)^{-2Nd}(1+|v|)^{-2Nd},$

which is again bounded by a constant since $4Nd>1$, as it can be clearly seen
after a change of variables
$u\mapsto\frac{\varphi+\theta}{2},v\mapsto\frac{\varphi-\theta}{2}$. We have
then shown that

(5.21) $\left|\mathbf{K}(X,Y)\right|\leq Ce^{-d(X+Y)},$

where $C$ is a constant independent of $X,Y$. We can now estimate the series
expansion of the Fredholm determinant of $\mathbf{K}$ as

(5.22) $\begin{split}&\sum_{\ell\geq
0}\frac{1}{\ell!}\int_{[\varsigma,+\infty)^{\ell}}\mathop{}\\!\mathrm{d}X_{1}\cdots\mathop{}\\!\mathrm{d}X_{\ell}\left|\det[\mathbf{K}(X_{i},X_{j})]_{i,j=1}^{\ell}\right|\\\
&\leq\sum_{\ell\geq
0}\frac{1}{\ell!}\int_{[\varsigma,+\infty)^{\ell}}\mathop{}\\!\mathrm{d}X_{1}\cdots\mathop{}\\!\mathrm{d}X_{\ell}\,\ell^{\ell/2}C^{\ell}e^{-d(X_{1}+\cdots+X_{\ell})}\\\
&=\sum_{\ell\geq 0}\frac{1}{\ell!}\ell^{\ell/2}\left(\frac{Ce^{-\varsigma
d}}{d}\right)^{\ell}<\infty,\end{split}$

where in the first inequality we used the Hadamard’s bound (A.8). This
completes the proof. ∎

We now present the proof of theorem 5.4. For this we introduce the _$q$ -Gamma
function_ of a complex variable $X\in\mathbb{C}\setminus\mathbb{Z}_{\leq 0}$

(5.23) $\Gamma_{q}(X)=(1-q)^{1-X}\frac{(q;q)_{\infty}}{(q^{X};q)_{\infty}}.$

Arguments used to prove our formula (5.12) use a number of technical bounds on
ratio of $q$-Pochhammer symbols and $q$-Gamma functions, which for the sake of
readability are reported in appendix B.

###### Proof of theorem 5.4.

Our starting point is the Fredholm determinant representation for the
probability distribution of a tagged particle $\mathsf{x}_{N}(T)$ in the
$q$-PushTASEP (5.1). Adopting the scaling

(5.24) $q=e^{-\varepsilon},\qquad a_{i}=e^{-\varepsilon A_{i}},\qquad
b_{i}=e^{-\varepsilon B_{i}},\qquad\zeta=(1-q)^{2N}e^{-\varsigma},$

we find that, assuming $\chi\sim q$-$\mathrm{Geo}(q)$ and
$S\sim\mathrm{Theta}(\zeta;q)$ and using proposition 3.5, lemmas B.1 and B.2,

(5.25) $\mathsf{x}_{N}(N)+\chi+S=\varepsilon^{-1}\log
Z(N,N)+\varepsilon^{-1}\mathcal{G}-\varepsilon^{-1}\varsigma+\varepsilon^{-1/2}\mathcal{N}(0,1)+O(1),$

where $\mathcal{G}\sim\mathrm{Gumbel}$ and $\mathcal{N}(0,1)$ is a standard
normal random variable. In this way, the left hand side of (5.1) becomes

(5.26) $\begin{split}\lim_{\varepsilon\to
0}\mathbb{P}(\mathsf{x}_{N}(N)+\chi+S<N+s)&=\mathbb{P}(\log
Z(N,N)+\mathcal{G}<\varsigma)\\\ &=\mathbb{E}\left(e^{-e^{-\varsigma+\log
Z(N,N)}}\right),\end{split}$

where the second equality was discussed in remark 5.6 and comes from a direct
evaluation of the probability in the middle term using the definition of the
Gumbel distribution.

We now move our attention to the right hand side of (5.1). Under the scaling
(5.24) the integral kernel $K_{\mathrm{push}}$, using the notion of $q$-Gamma
function (5.23), can be written as

(5.27)
$\begin{split}K_{\mathrm{push}}(\varepsilon^{-1}X,\varepsilon^{-1}Y)=\frac{\varepsilon}{(2\pi\mathrm{i})^{2}}&\int_{\begin{subarray}{c}Z\in\mathcal{D}(-d,\varepsilon)\\\
W\in\mathcal{D}(d,\varepsilon)\end{subarray}}\frac{\mathop{}\\!\mathrm{d}Z\mathop{}\\!\mathrm{d}W}{q^{(Z-W)/2}}e^{XZ-
YW}\\\
&\times\prod_{i=1}^{N}\frac{\Gamma_{q}(A_{i}+Z)\Gamma_{q}(B_{i}-W)}{\Gamma_{q}(B_{i}-Z)\Gamma_{q}(A_{i}+W)}\Gamma_{q}(Z-W)\Gamma_{q}(1+W-Z)\\\
&\times(1-q)^{2N(Z-W)}\frac{\vartheta_{3}(q^{Z-W}e^{\varsigma}(1-q)^{2N};q)}{\vartheta_{3}(e^{\varsigma}(1-q)^{2N};q)},\end{split}$

where, compared to (4.5), we operated a change of variables $z=q^{Z},w=q^{W}$,
taking as radii of the integration contours $r=q^{d},r^{\prime}=q^{-d}$. For
the integration contours we have used the notation

(5.28)
$\mathcal{D}(d,\varepsilon)=d+\mathrm{i}[-\frac{\pi}{\varepsilon},\frac{\pi}{\varepsilon}].$

By virtue of lemmas B.3 and B.4, the integrand in (5.27) converges pointwise
to the integrand of (5.13). To prove the pointwise convergence
$\varepsilon^{-1}K_{\mathrm{push}}(\varepsilon^{-1}X,\varepsilon^{-1}Y)\xrightarrow[]{\varepsilon\to
0}\mathbf{K}(X+\varsigma,Y+\varsigma)$ we notice that the integrand in (5.27)
is bounded in absolute value by

(5.29)
$C_{1}e^{-d(X+Y)}\prod_{i=1}^{N}(1+|u|)^{-(B_{i}-A_{i})-2d}(1+|v|)^{-(A_{i}-B_{i})-2d}e^{-C_{2}|u-v|},$

where $u=\Im\\{Z\\},v=\Im\\{W\\}$ and $C_{1},C_{2}$ are constants independent
of $q,Z,W,X,Y$; this can be shown combining lemmas B.3, B.6 and B.7. The
function in (5.29) is summable, as can be proven following the same ideas used
in lemma 5.7. Again bounding the integrand of (5.27) with (5.29) we obtain the
bound

(5.30)
$\left|\varepsilon^{-1}K_{\mathrm{push}}(\varepsilon^{-1}X,\varepsilon^{-1}Y)\right|\leq
Ce^{-d(X+Y)},$

which hold for a constant $C$ independent of $\varepsilon,X,Y$. This implies
that

(5.31) $\lim_{\varepsilon\to
0}\det(1-K_{\mathrm{push}})_{\ell^{2}(\mathbb{Z}_{>s}^{\prime})}=\det(1-\mathbf{K})_{\mathbb{L}^{2}(\varsigma,+\infty)},$

by dominated convergence, concluding the proof. ∎

###  5.3. Fredholm pfaffian representation for $q$-PushTASEP in half space

Here we present a Fredholm pfaffian formula for the distribution of the
rightmost particle in the $q$-PushTASEP with particle creation. With the
exception of [BBCW18], where a Pfaffian representation was found for the
current distribution of the ASEP with an open boundary kept at critical
density, corollary 5.8 below represents the first result of this kind for
discrete solvable models and it holds for any admissible choice of parameters
$\gamma,a_{1},a_{2},\dots$.

######  Corollary 5.8.

Let $\mathsf{X}^{\mathrm{hs}}(T)$ be the geometric $q$-PushTASEP with particle
creation with parameters $\gamma,a_{1},a_{2},\dots\in(0,1)$ and empty initial
conditions. Consider also $\chi\sim q$-$\mathrm{Geo}(q)$ and
$S\sim\mathrm{Theta}(\zeta^{2},q^{2})$ to be independent random variables.
Then we have

(5.32) $\mathbb{P}\left(\mathsf{x}^{\mathrm{hs}}_{N}(N)-N+\chi+2S\leq
s\right)=\mathrm{Pf}\left(J-L_{\mathrm{push}}\right)_{\ell^{2}(\mathbb{Z}^{\prime}_{>s})},$

where $L_{\mathrm{push}}$ is the $2\times 2$ matrix kernel obtained from
(4.36), setting

(5.33) $F(z)=\frac{(\gamma/z;q)_{\infty}}{(\gamma
z;q)_{\infty}}\prod_{i=1}^{N}\frac{(a_{i}/z;q)_{\infty}}{(a_{i}z;q)_{\infty}}.$

###### Proof.

As in section 3.2, we call $\gamma,a_{1},a_{2},\dots$ the parameters
describing the jump process in the $q$-PushTASEP with particle creation.
Defining $a^{\prime}=(a_{1},\dots,a_{N},\gamma)$ and taking
$\tilde{\mu}\sim\mathbb{HW}^{(q)}_{a^{\prime};0}$ we have the equality in
distribution

(5.34) $\mathsf{x}^{\mathrm{hs}}_{N}(N)=\tilde{\mu}_{1}+N,$

which follows combining proposition 3.2 with the symmetry of the half space
$q$-Whittaker measure of proposition 2.7. We can now use theorem 4.11 to
express the law of $\tilde{\mu}_{1}+\chi+2S$ in terms of the rightmost
particle in the Free boundary Schur measure whose Fredholm pfaffian
representation is reported in proposition 4.7, concluding the proof. ∎

######  Remark 5.9.

Notice that, unlike in the case of $q$-PushTASEP in full space, the left hand
side of (5.32) is not the $q$-Laplace transform of the probability
distribution of $\mathsf{x}^{\mathrm{hs}}_{N}(N)$.

The $q$-PushTASEP with particle creation was introduced and studied in
[BBC20]. In the same paper authors, using Macdonald difference operators,
recovered integral representations for joint $q$-moments of the model and a
representation of the $q$-Laplace transform of the distribution of a tagged
particle as a series of nested integrals; see [BBC20, Corollary 4.7]. Due to
the presence of certain cross terms it was not possible for the authors to
rearrange their expression as the expansion of a Fredholm pfaffian or
determinant. Our result (5.32) indicates that an observable different than the
$q$-Laplace transform of $\mathsf{x}^{\mathrm{hs}}_{N}(N)$ has the potential
to be studied through $q$-moment computations.

###  5.4. Fredholm pfaffian representation for Log Gamma polymer in half space

Here we consider the point-to-point partition function $Z^{\mathrm{hs}}(N,N)$
in the Log Gamma polymer in half space and we present Fredholm pfaffian
representations for the Laplace transform of its probability distribution.
First, in theorem 5.10, we will consider the case when the boundary parameter
$\varUpsilon$ is positive. To extend such solution to negative values of
$\varUpsilon$ requires an analytic continuation, which will lead to the result
of theorem 5.12, presented in the next subsection. Formulas reported below are
ameanable of asymptotic analysis, which will be performed in section 6.

######  Theorem 5.10.

Let $Z^{\mathrm{hs}}(N,N)$ be the partition function of the Log Gamma polymer
model in half space with parameters $A_{1},\dots,A_{N}>0$ and boundary
strength $\varUpsilon>0$. Take a number $d$ such that

(5.35)
$0<d<\min\left\\{\frac{1}{2},\varUpsilon,A_{1},\dots,A_{N}\right\\}\qquad\text{and}\qquad
d>\frac{1}{2N}.$

Then, we have

(5.36) $\mathbb{E}\left[e^{-e^{-\varsigma+\log
Z^{\mathrm{hs}}(N,N)}}\right]=\mathrm{Pf}(J-\mathbf{L})_{\mathbb{L}^{2}(\varsigma,+\infty)},$

where $\mathbf{L}$ is the $2\times 2$ matrix kernel

(5.37)
$\mathbf{L}(X,Y)=\left(\begin{matrix}\mathbf{k}(X,Y)&-\partial_{Y}\mathbf{k}(X,Y)\\\
-\partial_{X}\mathbf{k}(X,Y)&\partial_{X}\partial_{Y}\mathbf{k}(X,Y)\end{matrix}\right)$

with

(5.38)
$\mathbf{k}(X,Y)=\int_{\mathrm{i}\mathbb{R}-d}\frac{\mathop{}\\!\mathrm{d}Z}{2\pi\mathrm{i}}\int_{\mathrm{i}\mathbb{R}-d}\frac{\mathop{}\\!\mathrm{d}W}{2\pi\mathrm{i}}e^{XZ+YW}g_{\varUpsilon;A}(Z)g_{\varUpsilon;A}(W)\mathsf{k}^{\mathrm{hs}}(Z,W),$
(5.39)
$g_{\varUpsilon;A}(Z)=\frac{\Gamma(\varUpsilon+Z)}{\Gamma(\varUpsilon-Z)}\prod_{i=1}^{N}\frac{\Gamma(A_{i}+Z)}{\Gamma(A_{i}-Z)}$

and

(5.40)
$\mathsf{k}^{\mathrm{hs}}(Z,W)=\Gamma(-2Z)\Gamma(-2W)\frac{\sin\left[\pi(Z-W)\right]}{\sin\left[\pi(Z+W)\right]}.$

In literature explicit formulas for the expectation
$\mathbb{E}\left[e^{-e^{-\varsigma}Z^{\mathrm{hs}}(N,N)}\right]$ were derived
in [OSZ14, Corollary 5.6] and [BBC20, Corollary 6.40], although in none of
these cases they were expressed as Fredholm pfaffians. It would be interesting
to understand if it is possible to prove our formula (5.58) starting from
these other explicit results. Moreover, recently in [BR22] authors were able
to prove, using Bethe Ansatz, pfaffian formulas for a polymer model in half
space with certain Beta weights. It would be interesting to understand if our
techniques can be extended to prove identities for the Beta polymer model in
half space.

The proof of theorem 5.10 will be presented at the end of this subseciton. It
is based on a scaling limit of the Fredholm pfaffian formula of corollary 5.8
to establish which technical results reported in appendix B will be used. In
the next proposition, we confirm that the right hand side of (5.36) defines a
numerically convergent series and hence the relation is well posed.

######  Proposition 5.11.

The Fredholm pfaffian of the $2\times 2$ matrix kernel $\mathbf{L}$ is
absolutely convergent.

###### Proof.

We make use of the asymptotic expansion (5.19) to estimate the integrand in
(5.38). Following ideas already elaborated in the proof of lemma 5.7 we obtain
the bound

(5.41) $\begin{split}\left|\mathbf{k}(X,Y)\right|&\leq
Ce^{-d(X+Y)}\int_{\mathbb{R}}\mathop{}\\!\mathrm{d}u\int_{\mathbb{R}}\mathop{}\\!\mathrm{d}v[(1+|u|)(1+|v|)]^{-2Nd-1/2}e^{-\pi|u+v|}\\\
&\leq\frac{C}{2Nd}e^{-d(X+Y)},\end{split}$

where $C$ is a constant independent of $X,Y$. Conditions (5.35) on parameter
$d$ imply that the double integral in the right hand side of the inequality
above is convergent and bounded by a constant independent of $X,Y$. Through
similar argument we easily get

(5.42)
$|\mathbf{k}(X,Y)|,|\partial_{X}\mathbf{k}(X,Y)|,|\partial_{Y}\mathbf{k}(X,Y)|,|\partial_{X}\partial_{Y}\mathbf{k}(X,Y)|\leq
De^{-d(X+Y)},$

where $D$ is another constant. We can now estimate each term of the expansion
of the Fredholm pfaffian of $\mathbf{L}$ and, using the Hadamard’s bound
(A.8), we have

(5.43) $\begin{split}&\sum_{\ell\geq
0}\frac{1}{\ell!}\int_{[\varsigma,+\infty)^{\ell}}\mathop{}\\!\mathrm{d}X_{1}\cdots\mathop{}\\!\mathrm{d}X_{\ell}\left|\mathrm{Pf}[\mathbf{L}(X_{i},X_{j})]_{i,j=1}^{\ell}\right|\\\
&\leq\sum_{\ell\geq
0}\frac{1}{\ell!}\int_{[\varsigma,+\infty)^{\ell}}\mathop{}\\!\mathrm{d}X_{1}\cdots\mathop{}\\!\mathrm{d}X_{\ell}(2\ell)^{\ell/2}D^{\ell}e^{-d(X_{1}+\cdots+X_{\ell})}\\\
&=\sum_{\ell\geq
0}\frac{1}{\ell!}\ell^{\ell/2}\left(\frac{\sqrt{2}De^{-\varsigma
d}}{d}\right)^{\ell}<\infty,\end{split}$

which proves the result. ∎

###### Proof of theorem 5.10.

We proceed as in theorem 5.4. Our starting point in this case is the Fredholm
pfaffian representation (5.32) of the probability distribution for the
particle $\mathsf{x}^{\mathrm{hs}}_{N}(N)$ of a $q$-PushTASEP with particle
creation. Using the convergence result of proposition 3.6, along with lemmas
B.1 and B.2, imposing the scaling,

(5.44) $q=e^{-\varepsilon},\qquad a_{i}=e^{-\varepsilon
A_{i}},\qquad\gamma=e^{-\varepsilon\varUpsilon},\qquad\zeta=(1-q^{2})^{N}e^{-\varsigma/2},$

we have

(5.45) $\mathsf{x}_{N}^{\mathrm{hs}}(N)-N+\chi+2S=\varepsilon^{-1}\left(\log
Z^{\mathrm{hs}}(N,N)+\mathcal{G}-\varsigma+\sqrt{2\varepsilon}\mathcal{N}(0,1)\right)+O(1).$

This shows that

(5.46) $\begin{split}\lim_{\varepsilon\to
0}\mathbb{P}(\mathsf{x}_{N}^{\mathrm{hs}}(N)-N+\chi+2S<s)&=\mathbb{P}(\log
Z^{\mathrm{hs}}(N,N)+\mathcal{G}<\varsigma)\\\
&=\mathbb{E}\left(e^{-e^{-\varsigma+\log
Z^{\mathrm{hs}}(N,N)}}\right),\end{split}$

and it takes care of the left hand side of (5.32). Let us now take the same
limit on the right hand side. We can rewrite the $2\times 2$ matrix kernel
prescribed by corollary 5.8 as

(5.47)
$L_{\mathrm{push}}(\varepsilon^{-1}X,\varepsilon^{-1}Y)=\left(\begin{matrix}\varepsilon&-\nabla_{y}\\\
-\nabla_{x}&\varepsilon^{-1}\nabla_{x}\nabla_{y}\end{matrix}\right)k(\varepsilon^{-1}X,\varepsilon^{-1}Y),$

with

(5.48)
$k(\varepsilon^{-1}X,\varepsilon^{-1}Y)=\frac{1}{(2\pi\mathrm{i})^{2}}\frac{\varepsilon^{2}}{(1-q)^{2}}\int_{\begin{subarray}{c}Z\in\mathcal{D}(-d,\varepsilon)\\\
W\in\mathcal{D}(-d,\varepsilon)\end{subarray}}\frac{\mathop{}\\!\mathrm{d}Z\mathop{}\\!\mathrm{d}W}{q^{(Z+3W)/2}}e^{XZ+YW}f(Z)f(W)\tilde{\kappa}^{\mathrm{hs}}(Z,W).$

Here, with respect to (4.25) we operated the change of variables
$z=q^{Z},w=q^{W}$ and we made use of auxiliary functions

$\displaystyle\tilde{\kappa}^{\mathrm{hs}}(Z,W)=\frac{\Gamma_{q}(-2Z)\Gamma_{q}(-2W)\Gamma_{q}(1+Z+W)\Gamma_{q}(-Z-W)}{\Gamma_{q}(W-Z)\Gamma_{q}(1+Z+W)}E_{q}(\varsigma,Z+W),$
$\displaystyle
f(Z)=\frac{\Gamma_{q}(\varUpsilon+Z)}{\Gamma_{q}(\varUpsilon-Z)}\prod_{i=1}^{N}\frac{\Gamma_{q}(A_{i}+Z)}{\Gamma_{q}(A_{i}-Z)},$
$\displaystyle
E_{q}(\varsigma,U)=(1-q)^{2NU}\frac{\vartheta_{3}((1-q)^{2N}e^{-\varsigma}q^{2U};q^{2})}{\vartheta_{3}((1-q)^{2N}e^{-\varsigma};q^{2})}.$

For the integration contours we used again the notation (5.28). Notice that
the discrete derivatives $\nabla_{x}$ and $\nabla_{y}$ modify the function $k$
by inserting respectively factors $\frac{1}{2}(q^{-Z}-q^{Z})\approx\varepsilon
Z$ and $\frac{1}{2}(q^{-W}-q^{W})\approx\varepsilon W$ in the integrand. By
means of estimates reported in lemmas B.3 and B.7 and proposition B.8 we can
bound the absolute value of the integrand in the expression of $k$ with
summable functions proving that

(5.49) $\lim_{\varepsilon\to
0}k(\varepsilon^{-1}X,\varepsilon^{-1}Y)=\mathbf{k}(X+\varsigma,Y+\varsigma),$

by the bounded convergence theorem and moreover that

(5.50) $|k(\varepsilon^{-1}X,\varepsilon^{-1}Y)|\leq Ce^{-d(X+Y)},$

for some constant $C$ independent of $\varepsilon,X,Y$. Analogous arguments
are applicable to other elements of the matrix kernel $L_{\mathrm{push}}$ and
we find

$\lim_{\varepsilon\to
0}\left(\varepsilon^{-1}\nabla_{y}k(\varepsilon^{-1}X,\varepsilon^{-1}Y),\varepsilon^{-1}\nabla_{x}k(\varepsilon^{-1}X,\varepsilon^{-1}Y),\varepsilon^{-2}\nabla_{x}\nabla_{y}k(\varepsilon^{-1}X,\varepsilon^{-1}Y)\right),\\\
=\left(\partial_{Y}\mathbf{k}(X+\varsigma,Y+\varsigma),\partial_{X}\mathbf{k}(X+\varsigma,Y+\varsigma),\partial_{X}\partial_{Y}\mathbf{k}(X+\varsigma,Y+\varsigma)\right)$

and moreover

$|\varepsilon^{-1}\nabla_{y}k(\varepsilon^{-1}X,\varepsilon^{-1}Y)|,|\varepsilon^{-1}\nabla_{x}k(\varepsilon^{-1}X,\varepsilon^{-1}Y)|,|\varepsilon^{-2}\nabla_{x}\nabla_{y}k(\varepsilon^{-1}X,\varepsilon^{-1}Y)|\leq
Ce^{-d(X+Y)},$

where the constant $C$ can be chosen to be the same for all terms. These
results imply, by bounded convergence, that

(5.51) $\lim_{\varepsilon\downarrow
0}\mathrm{Pf}(J-L_{\mathrm{push}})_{\ell^{2}(\mathbb{Z}^{\prime}_{>s})}=\mathrm{Pf}(J-\mathbf{L})_{\mathbb{L}^{2}(\varsigma,+\infty)},$

proving the desired identity (5.36). ∎

###  5.5. Analytic extension of Fredholm pfaffian

We now want to characterize the distribution of the half space polymer
partition function $Z^{\mathrm{hs}}(N,N)$ for choices of boundary parameter
$\varUpsilon\leq 0$. This will require an analytic continuation of the
relation (5.36). From the explicit representation of the function
$\mathbf{k}$, we see that moving the parameter $\varUpsilon$ from the positive
to the negative real semi-axis several singularities for the $Z$ and $W$
variables cross the integration contours corresponding to values
$-\varUpsilon,-\varUpsilon-1,\dots,-\varUpsilon-m+1$, if we assume that
$\varUpsilon$ ends up in the interval $(-m,-m+1]$. Defining functions

(5.52)
$\mathbf{A}_{j}(X)=\frac{(-1)^{j}}{j!}e^{-(j+\varUpsilon)X}\frac{\Gamma(2\varUpsilon+2j)}{\Gamma(2\varUpsilon+j)}\prod_{i=1}^{N}\frac{\Gamma(A_{i}-\varUpsilon-j)}{\Gamma(A_{i}+\varUpsilon+j)},$
(5.53)
$\mathbf{B}(Y)=\int_{\mathrm{i}\mathbb{R}-d}\frac{\mathop{}\\!\mathrm{d}W}{2\pi\mathrm{i}}e^{YW}\Gamma(-2W)\frac{\Gamma(1-\varUpsilon+W)}{\Gamma(1-\varUpsilon-W)}\prod_{i=1}^{N}\frac{\Gamma(A_{i}+W)}{\Gamma(A_{i}-W)},$

we evaluate the $Z$-residue at $-\varUpsilon-j$ of the integrand of
$\mathbf{k}$ and from (5.38), we find it to be

(5.54)
$\mathrm{Res}_{Z=-\varUpsilon-j}\left\\{\parbox{62.59596pt}{\centering$Z$-integrand
in
\eqref{eq:k_hs_log_gamma}\@add@centering}\right\\}=\mathbf{A}_{j}(X)\mathbf{B}(Y).$

Analogously the $W$-residue at $-\varUpsilon-j$ of the integrand of
$\mathbf{k}$ is

(5.55)
$\mathrm{Res}_{W=-\varUpsilon-j}\left\\{\parbox{62.59596pt}{\centering$W$-integrand
in
\eqref{eq:k_hs_log_gamma}\@add@centering}\right\\}=-\mathbf{A}_{j}(Y)\mathbf{B}(X).$

These computations allow us to identify the analytic continuation of the
$2\times 2$ matrix kernel $\mathbf{L}$, by adding residues
$\mathbf{A}_{j}(X)\mathbf{B}(Y)-\mathbf{A}_{j}(Y)\mathbf{B}(X)$ to the
function $\mathbf{k}$ in each component of (5.37). However, when $\varUpsilon$
is negative and $j+\varUpsilon<0$ we see that functions $\mathbf{A}_{j}(X)$’s
are exponentially diverging as $X$ grows. This means that the analytic
continuation of $\mathbf{L}$ naively proposed above produces an operator which
is not well defined on
$\mathbb{L}^{2}(\varsigma,\infty)\times\mathbb{L}^{2}(\varsigma,\infty)$ and
therefore we cannot immediately make sense of its Fredholm pfaffian. In the
next theorem we resolve this issue and present the mathematically correct
extension of identity (5.36). To state our result we introduce the scalar
product $\langle(\cdot,\cdot)|(\cdot,\cdot)\rangle$ on the Hilbert space
$\mathbb{L}^{2}(\varsigma,\infty)\times\mathbb{L}^{2}(\varsigma,\infty)$ as

(5.56)
$\langle(f_{1},f_{2})|(g_{1},g_{2})\rangle=\int_{\varsigma}^{+\infty}\left[f_{1}(x)g_{1}(x)+f_{2}(x)g_{2}(x)\right]\mathop{}\\!\mathrm{d}x.$

######  Theorem 5.12.

Let $Z^{\mathrm{hs}}$ be the partition function of the Log Gamma polymer model
in half space with parameters $A_{1},\dots,A_{N}>0$ and boundary strength
$\varUpsilon$ such that $-\min\\{A_{1},\dots,A_{N}\\}<\varUpsilon\leq 0$. Let
$m\in\mathbb{Z}_{\geq 0}$ and $\varepsilon\in\\{0,1\\}$ be such that
$\varUpsilon\in(-m+\frac{1-\varepsilon}{2},-m+\frac{2-\varepsilon}{2}]$ and
take a number $d$ satisfying

(5.57)
$0<d<\min\left\\{\frac{1}{2},\varUpsilon+m,A_{1},\dots,A_{N}\right\\}\qquad\text{and}\qquad
d>\frac{1}{2N}.$

Then, there exists $N^{*}$ such that, for all $N\geq N^{*}$ we have

(5.58) $\begin{split}\mathbb{E}\left[e^{-e^{-\varsigma+\log
Z^{\mathrm{hs}}(N,N)}}\right]=&\mathrm{Pf}\left[J-\widehat{\mathbf{L}}\right]_{\mathbb{L}^{2}(\varsigma,+\infty)}\\\
&\qquad\times\bigg{(}1-\langle(\mathbf{B},-\mathbf{B}^{\prime})|\left(1-J^{T}\widehat{\mathbf{L}}\right)^{-1}|(\mathbf{A}^{\prime},\mathbf{A})\rangle\bigg{)},\end{split}$

where

(5.59)
$\widehat{\mathbf{L}}(X,Y)=\mathbf{L}(X,Y)-\left(\begin{matrix}1&-\partial_{Y}\\\
-\partial_{X}&\partial_{X}\partial_{Y}\end{matrix}\right)\sum_{j=m}^{2m+\varepsilon-2}\left[\mathbf{A}_{j}(X)\mathbf{B}(Y)-\mathbf{A}_{j}(Y)\mathbf{B}(X)\right],$
(5.60) $\mathbf{A}(X)=\sum_{j=0}^{2m+\varepsilon-2}\mathbf{A}_{j}(X).$

Notice that when $m=0$ we have $\widehat{\mathbf{L}}=\mathbf{L}$, since the
summations in (5.59) is empty.

######  Lemma 5.13.

Consider parameters $A_{1},\dots,A_{N}>0$ and
$\varUpsilon>-\min\\{A_{1},\dots,A_{N}\\}$. Then there exists $N^{*}$,
depending on $\varUpsilon$ such that for each $N\geq N^{*}$ the operator
$\mathbf{L}$ has norm $\|\mathbf{L}\|<1$.

###### Proof.

This is a straightforward consequence of bound (5.41) and of analogous
estimates for partial derivatives of $\mathbf{k}$, which one can easily
produce also when the parameter $\varUpsilon$ becomes negative. ∎

###### Proof of theorem 5.12.

As hinted above before the statement of the theorem, identity (5.58) will
follow from (5.36) after an analytic continuation in the parameter
$\varUpsilon$ to the region $\varUpsilon\leq 0$. We first observe that the
Laplace transform
$\mathbb{E}\left(e^{-e^{-\varsigma}Z^{\mathrm{hs}}(N,N)}\right)$ is clearly an
analytic function of the variable $\varUpsilon$ in the region
$\varUpsilon>-\min\\{A_{1},\dots,A_{N}\\}$, since the $\varUpsilon$ dependence
is given by boundary random variables (3.15). To show that the same is true
for the Fredholm pfaffian in the right hand side of (5.36), we show that the
right hand side of (5.58) is an analytic function of the variable
$\varUpsilon$ that is well defined for $\varUpsilon>-m$ and that it equals
$\mathrm{Pf}[J-\mathbf{L}]$, of (5.36) when $\Re\\{\varUpsilon\\}\in(0,1)$.

Let us start by assuming that parameters $A_{1},\dots,A_{N}>2m$. This
hypothesis can be removed at any time through analytic continuation and it
only serves to ensure that through the procedure we describe below we do not
have to worry about possible singularities coming from factors
$\Gamma(A_{i}+Z),\Gamma(A_{i}+W)$ in the function $\mathbf{k}$ in (5.38). As
discussed above, the problem with the argument presented prior to the
statement of the theorem is that perturbations of the kernel $\mathbf{L}$
given by combinations of functions $\mathbf{A}_{j},\mathbf{B}$ and their
derivatives define an unbounded operator when $\varUpsilon<0$. Let us then
assume that $\varUpsilon>0$ and define the kernel $\mathbf{R}(X,Y)$

(5.61)
$\mathbf{R}(X,Y)=\left(\begin{matrix}\mathbf{A}(X)\mathbf{B}(Y)-\mathbf{A}(Y)\mathbf{B}(X)&-\mathbf{A}(X)\mathbf{B}^{\prime}(Y)+\mathbf{A}^{\prime}(Y)\mathbf{B}(X)\\\
-\mathbf{A}^{\prime}(X)\mathbf{B}(Y)+\mathbf{A}(Y)\mathbf{B}^{\prime}(X)&\mathbf{A}^{\prime}(X)\mathbf{B}^{\prime}(Y)-\mathbf{A}^{\prime}(Y)\mathbf{B}^{\prime}(X)\end{matrix}\right).$

By simple estimates

(5.62) $|\mathbf{A}(X)|,|\mathbf{A}^{\prime}(X)|\leq De^{-\varUpsilon
X},\qquad|\mathbf{B}(Y)|,|\mathbf{B}^{\prime}(Y)|\leq De^{-dY},$

holding for a constant $D>0$, which can be made arbitrarily small choosing $N$
large, we deduce that, in case $\varUpsilon>0$, $\mathbf{R}(X,Y)$ defines a
rank 2 operator
$\mathbf{R}:\mathbb{L}^{2}(-\varsigma,+\infty)\times\mathbb{L}^{2}(-\varsigma,+\infty)\to\mathbb{L}^{2}(-\varsigma,+\infty)\times\mathbb{L}^{2}(-\varsigma,+\infty)$.
Applying proposition A.2 we produce the identity

(5.63)
$\begin{split}\mathrm{Pf}(J-\mathbf{L})&=\mathrm{Pf}(J-(\mathbf{L}-\mathbf{R})-\mathbf{R})\\\
&=\mathrm{Pf}(J-(\mathbf{L}-\mathbf{R}))\bigg{(}1-\langle(\mathbf{B},-\mathbf{B}^{\prime})|(1-J^{T}(\mathbf{L}-\mathbf{R}))^{-1}|(\mathbf{A}^{\prime},\mathbf{A})\rangle\bigg{)}.\end{split}$

In the right hand side the inverse operator is defined by the Neumann series
$(1-J^{T}(\mathbf{L}-\mathbf{R}))^{-1}=\sum_{\ell\geq
0}\left(J^{T}(\mathbf{L}-\mathbf{R})\right)^{\ell}$ and such series is
absolutely convergent for $N$ large enough as a result of lemma 5.13 and
(5.62). We now show that the right hand side of (5.63) remains well defined
and analytic even when we drag the parameter $\varUpsilon$ in the region