subsection

# Polynuclear growth and the Toda lattice

Konstantin Matetski  Department of Mathematics
Michigan State University
619 Red Cedar Road
East Lansing, MI 48824
USA<EMAIL_ADDRESS>,  Jeremy Quastel  Department of Mathematics
University of Toronto
40 St. George Street
Toronto, Ontario
Canada M5S 2E4<EMAIL_ADDRESS>and  Daniel Remenik  Departamento de
Ingeniería Matemática and Centro de Modelamiento Matemático (IRL-CNRS 2807)
Universidad de Chile
Av. Beauchef 851, Torre Norte, Piso 5
Santiago
Chile<EMAIL_ADDRESS>
###### Abstract.

It is shown that the polynuclear growth model is a completely integrable
Markov process in the sense that its transition probabilities are given by
Fredholm determinants of kernels produced by a scattering transform based on
the invariant measures modulo the absolute height, continuous time simple
random walks. From the linear evolution of the kernels, it is shown that the
$n$-point distributions are determinants of $n\times n$ matrices evolving
according to the _two dimensional non-Abelian Toda lattice_.

###### Contents

  1. 1 Introduction
  2. 2 Main results
    1. 2.1 Model and setting
    2. 2.2 Toda
      1. 2.2.1 Non-Abelian 2D Toda equation
      2. 2.2.2 1D Toda lattice
      3. 2.2.3 Boundary conditions
      4. 2.2.4 Related work
    3. 2.3 Fredholm determinant formula
    4. 2.4 KPZ universality
      1. 2.4.1 Scalar KP equation
      2. 2.4.2 Matrix KP equation
    5. 2.5 Narrow wedge and flat examples, discrete Painlevé II
      1. 2.5.1 Narrow wedge initial data
      2. 2.5.2 Flat initial data
  3. 3 Derivation of the non-Abelian Toda equation
    1. 3.1 General kernel
    2. 3.2 PNG kernel
    3. 3.3 The forcing term
  4. 4 PNG as a Markov process
    1. 4.1 Semigroup
    2. 4.2 Generator
    3. 4.3 Invariant measures
    4. 4.4 Adjoint generator
    5. 4.5 Backward equation
    6. 4.6 Why hit kernels solve the Kolmogorov equation
  5. A Initial data for non-Abelian Toda
  6. B Trace class estimates
    1. B.1 Continuity
    2. B.2 The kernel
    3. B.3 Differentiability
    4. B.4 Proof of (4.65)

##  1\. Introduction

One-dimensional polynuclear growth (PNG) is a model for randomly growing
interfaces which has been extensively studied as a solvable model in the
Kardar-Parisi-Zhang universality class. The KPZ class is characterized by non-
standard fluctuations, which are universal for models in the class, but do
remember the initial data. Most previous work on PNG has considered the
_droplet_ or _narrow wedge_ initial condition. For this special initial
condition, the one-point distributions of the model can be identified with a
Poissonized version of the longest increasing subsequence of a random
permutation. Ulam’s problem of determining the law of large numbers can be
understood through the connection with the Hammersley process [Ham72, AD95,
Sep96, AD99], and a representation as a Toeplitz, then later a Fredholm,
determinant led to the identification of the fluctuations around this limit as
coinciding with those of the top eigenvalue of a matrix from the Gaussian
unitary ensemble [Ges90, Joh98, TW99, BDJ99, Joh00, Joh01]. Multipoint
distributions in this geometry can be computed based on a multilayer
extension, through which the Airy process was discovered [PS02, Joh03]. Flat
and stationary initial distributions were also considered in [BR00, SI04,
IS04, PS04, Fer04, FS11]. Recently [JR21, JR22] were able to produce certain
multitime formulas for some of these special initial conditions. One thing all
such results had in common, for PNG as well as other related models, was the
reliance on tricks particular to a few special initial conditions in order to
obtain Fredholm determinant formulas suitable for asymptotics.

In [MQR21, QR22] we introduced the _KPZ fixed point_ , a scaling invariant
Markov process expected to be the universal limit of 1:2:3 scaled height
functions, or their analogues, within the entire KPZ class. The transition
probabilities are given by Fredholm determinants of certain kernels based on a
Brownian scattering transform of the initial data. Furthermore, the $n$-point
distributions, starting from an essentially arbitrary function, come from
matrix Kadomtsev-Petviashvilli (KP) equations: In the one-point case, they
simply satisfy the KP-II equation; for higher $n$, they are given as traces of
matrix solutions of such equations.

A natural question is whether such a structure can be seen in one of the
discrete or semi-discrete models in the KPZ class. The question is
particularly relevant as it may shed light on the physical basis of these
equations. The KPZ fixed point formulas were derived through asymptotic
analysis of similar determinantal formulas for the totally asymmetric simple
exclusion process (TASEP). However, the TASEP transition probabilities appear
not to satisfy any standard completely integrable equation.

In this article we consider PNG as a Markov process, with general initial
data. Its transition probabilities, again from essentially arbitrary initial
data, are given by Fredholm determinants of kernels produced by a scattering
transform based on continuous time symmetric random walks—the invariant
measures of the process modulo the absolute height. For PNG, the $n$-point
distributions turn out to be associated to the most appealing discretization
of the matrix KP: In the one-point case, they are solutions of the _two-
dimensional Toda lattice_ ; for higher $n$, they are given as determinants of
$n\times n$ matrices solving the _non-Abelian Toda lattice_. All of these are
well-known [UT84] completely integrable extensions of the Toda lattice. In
this sense, PNG is shown to be a _completely integrable Markov process._

##  2\. Main results

###  2.1. Model and setting

The _polynuclear growth model_ (PNG) is a Markov process whose state space
$\operatorname{\mathscr{U\\!C}}$ consists of upper semi-continuous height
functions $h$ mapping $\mathbb{R}$ into the integers $\mathbb{Z}$ compactified
at $\\{-\infty\\}$. The topology on $\operatorname{\mathscr{U\\!C}}$ is that
of local Hausdorff convergence of hypographs, which is natural for models of
random growth (see Sec. 4.1 for a precise definition). The dynamics consists
of two parts; a continuous, deterministic dynamics
$h\mapsto\mathop{\mathop{\text{{sup}}}}\displaylimits_{|y-x|\leq t}h(y)$ and a
discontinuous, stochastic dynamics. In the stochastic part of the dynamics,
there is a Poisson point process in space-time with rate $2$ (this choice of
rate is arbitrary, but convenient in formulas). If $(t^{*},x^{*})$ is such a
point, then $h(t^{*},x)=h((t^{*})^{-},x)+\mathbf{1}(x=x^{*})$. In other words,
in each interval $[x,x+\mathrm{d}x]$, with probability $2\mathrm{d}x\hskip
0.50003pt\mathrm{d}t$, a _nucleation_ appears in $[t,t+\mathrm{d}t]$ and the
height function is increased by $1$ at $x$. Both dynamics are running
simultaneously111Note that the dynamics makes sense starting from _any_
function in $\operatorname{\mathscr{U\\!C}}$, as
$\mathop{\mathop{\text{{sup}}}}\displaylimits_{|y-x|\leq t}h(y)$ is finite for
any function in $\operatorname{\mathscr{U\\!C}}$, at any $x$, since the
supremum is achieved, and $h$ is not allowed to take the value $+\infty$ at
any point., so the continuous part of the dynamics implies that the
nucleations spread in both directions deterministically at speed $1$.

Almost equivalently, the model can be thought of as a collection of one-
dimensional ‘kinks’ (down steps) and ‘antikinks’ (up steps) moving on
$\mathbb{R}$. They move ballistically at speed one: kinks to the right,
antikinks to the left. They annihilate upon collision [Ben+81], and are also
produced, in pairs, a kink at $x^{+}$, an antikink at $x^{-}$, according to a
rate $2$ space-time Poisson process. The height $h(t,0)$ is just its value
$h(0,0)$ at time $0$, plus the number of kinks and antikinks which have
crossed the spatial point $0$ up to time $t$. One doesn’t have to quibble
about whether to count as crossings events like creations or annihilations
exactly at $0$, as these happen with probability $0$.

Figure. In the PNG dynamics, flat islands with integer heights expand
deterministically in both directions with speed $1$, merging when they collide
(red), and with new islands (green) appearing above existing ones at the times
of a space-time Poisson process with rate $2$.

If one thinks of the $\frac{1}{2}(t+x)$ direction as time and the
$\frac{1}{2}(t-x)$ direction as space, or vice-versa, the dynamics becomes
that of the Hammersley process [AD95]. Note however, that this transformation
does not take our initial data into the standard initial data for the
Hammersley process.

We will denote the PNG height function at time $t$ by $h(t,x)$, or sometimes
$h(t,x;h_{0})$ if we want to indicate the initial data. The _narrow wedge at
$y\in\mathbb{R}$_ is defined by222 To see that starting from
$\mathfrak{d}_{0}$ the one-point marginals are equivalent to a Poissonized
version of the longest increasing subsequence problem, consider the problem of
finding Lipschitz-$1$ paths going from $(0,0)$ to $(t,0)$ which pick up a
maximal number of space-time Poisson points along the way. All these paths lie
inside the rhombus $R$ with vertices $(0,0)$, $(t/2,t/2)$, $(t/2,-t/2)$ and
$(t,t)$, and the maximal number of points which they can pick up is exactly
$h(t,0)$. Now rotate the picture by $-45^{\circ}$, let $N$ denote the number
of Poisson points inside the square corresponding to $R$ (so that $N$ is a
Poisson$[t]$ random variable), and order these $N$ points according to their
$x$ coordinate. The $y$ coordinates of these points define a random
permutation $\sigma$ of $\\{1,\dotsc,N\\}$, which is clearly chosen uniformly
from $S_{N}$, and $h(t,0)$ is nothing but the length of the longest increasing
subsequence in $\sigma$.

$\mathfrak{d}_{y}(y)=0\qquad\text{and}\qquad\mathfrak{d}_{y}(x)=-\infty\text{{
for $x\neq y$}}.$ (2.1)

If we start PNG with a narrow wedge at $y$, $\mathfrak{d}_{y}$, we could call
the resulting solution at time $t$, as a function of $x$,
$\mathscr{A}_{t}(x,y)$. The PNG dynamics (under the natural coupling where one
starts with two or more different initial conditions and runs them using the
same Poisson noise) preserve the max operation,

$h(t,x;\mathop{\mathop{\text{{max}}}}\displaylimits\\{h_{1},h_{2}\\})=\mathop{\mathop{\text{{max}}}}\displaylimits\\{h(t,x;h_{1}),h(t,x;h_{2})\\},$
(2.2)

and hence one has the variational formula

$h(t,x;h_{0})=\mathop{\mathop{\text{{sup}}}}\displaylimits_{y}\\{\mathscr{A}_{t}(x,y)+h_{0}(y)\\}.$
(2.3)

We record a few other key properties. Typically for KPZ class models, PNG is
statistically invariant under spatial shifts

$T_{-y}h(t,x;T_{y}h_{0})\stackrel{{\scriptstyle\text{{dist}}}}{{=}}h(t,x;h_{0}),\qquad
T_{y}f(x)=f(x-y),$ (2.4)

and under reflections

$Rh(t,x;Rh_{0})\stackrel{{\scriptstyle\text{{dist}}}}{{=}}h(t,x;h_{0}),\qquad
Rf(x)=f(-x).$ (2.5)

A more special property, presumably connected to solvability, is _skew time
reversal invariance_ ,

$\mathbb{P}(h(t,x;g)\leq-f)=\mathbb{P}(h(t,x;f)\leq-g).$ (2.6)

Unusually among continuous time random growth models, PNG has a _finite
propagation speed_ : Given everything up to time $t$, for $s>0$, $h(t+s,x)$
only depends on $h(t,y)$, $|y-x|\leq s$, and on the points in the background
Poisson process inside the space-time region
$\\{(u,y)\in\mathbb{R}^{2}\\!:t\leq u\leq t+s,\,|y-x|\leq t+s-u$}.

Modulo a non-trivial upward height shift, the invariant measures for the
process consist of the following one-parameter family of two-sided continuous
time random walks. Let $h^{+}(x)$, $h^{-}(x)$ be independent Poisson process
on $\mathbb{R}$ with rates $\rho$ and $\rho^{-1}$. Then
$h(x)=h^{+}(x)-h^{-}(x)$ is invariant modulo height shifts, in the sense that,
starting PNG from such an initial state, with $h(0)=0$, at time $t>0$,
$h(t,x)-h(t,0)$ will have the same distribution as the process in $x$. This is
proved in Sec. 4.3. A concrete choice is the symmetric invariant state with
$\rho=1$, which we denote by $\mathsf{N}(x)$. In the hitting probabilities
below, the determinant turns out to be unaffected by the choice of $\rho$.

The transition probabilities of a Markovian growth model such as PNG are
completely determined by the finite dimensional distributions

$F(t,x_{1},\ldots,x_{n},r_{1},\ldots,r_{n})=\mathbb{P}_{h_{0}}(h(t,x_{i})\leq
r_{i},i=1,\ldots,n),$ (2.7)

for an initial condition $h_{0}\in\operatorname{\mathscr{U\\!C}}$, $t>0$,
$x_{1}<\cdots<x_{n}$, and $r_{1},\ldots,r_{n}\in\mathbb{Z}$.

The PNG dynamics is surprisingly similar to _ballistic aggregation_ (sometimes
called _ballistic deposition_) where the height functions live over
$\mathbb{Z}$ instead of $\mathbb{R}$ and $h(t,x)$ jumps to
$\mathop{\mathop{\text{{max}}}}\displaylimits\\{h(t,x-1),h(t,x)+1,h(t,x+1)\\}$
at rate one. One of the challenges in the field of random growth is that we
know almost nothing about many natural and basic models such as ballistic
aggregation (in this case, only that it grows at some rate $Ct$ [Sep00] and
plenty of computer simulations). In contrast, for PNG we are now going to
explain how the entire family (2.7) comes from known integrable non-linear
wave equations.

###  2.2. Toda

We need to introduce some notation. For $a\leq b$ consider the _hit operator_
, with kernel

$P^{\operatorname{\text{{hit}}}(h_{0})}_{a,b}(u,v)=\mathbb{P}_{\mathsf{N}(a)=u}(\tau\in[a,b],\,\mathsf{N}(b)=v),\qquad\tau=\mathop{\mathop{\text{{inf}}}}\displaylimits\\{x\geq
a:\mathsf{N}(x)\in\operatorname{\text{{hypo}}}(h_{0})\\},$ (2.8)

where
$\operatorname{\text{{hypo}}}(h_{0})=\\{(x,g)\in\mathbb{R}\times\mathbb{Z}\\!:g\leq
h_{0}(x)\\}$ is the _hypograph_ of $h_{0}$, closed since
$h_{0}\in\operatorname{\mathscr{U\\!C}}$, and $\mathsf{N}$ is the continuous
time random walk invariant measure introduced in the paragraph before (2.7).
If $a>b$ we set $P^{\operatorname{\text{{hit}}}(h_{0})}_{a,b}\equiv 0$. We
need the symmetric first and second order discrete difference operators

$\displaystyle\nabla f(u)$
$\displaystyle=\tfrac{1}{2}(f(u+1)-f(u-1)),\qquad\Delta
f(u)=f(u+1)+f(u-1)-2f(u).$ (2.9)

The latter is the infinitesimal generator of the random walk $\mathsf{N}$,
that is, for $x\geq 0$ and $u,v\in\mathbb{Z}$ we have

$\mathbb{P}_{\mathsf{N}(0)=u}(\mathsf{N}(x)=v)=e^{x\Delta}(u,v).$ (2.10)

The following should be thought of as a _scattering transform_ of $h_{0}$ by
random walks,

$\mathscr{T}^{h_{0}}_{a,b}=e^{a\Delta}P^{\operatorname{\text{{hit}}}(h_{0})}_{a,b}e^{-b\Delta}.$
(2.11)

Out of it, and for a fixed choice of $t\geq 0$ and $x_{1}<\dotsm<x_{n}$, we
build an _extended kernel_ acting on
$\ell^{2}(\\{x_{1},\dotsc,x_{n}\\}\times\mathbb{Z})$:

$\displaystyle K^{\text{{ext}}}(x_{i},\cdot;x_{j},\cdot)$
$\displaystyle=-\mathbf{1}_{x_{i}<x_{j}}e^{(x_{j}-x_{i})\Delta}+e^{-2t\nabla-
x_{i}\Delta}\mathscr{T}^{h_{0}}_{x_{i}-t,x_{j}+t}e^{2t\nabla+x_{j}\Delta}.$
(2.12)

Now given $r=(r_{1},\dotsc,r_{n})\in\mathbb{Z}^{n}$, we use the extended
kernel $K^{\text{{ext}}}$ to build an $n\times n$ matrix kernel
$K=K(t;x_{1},\dotsc,x_{n};r_{1},\dotsc,r_{n};h_{0})$ acting on
$\oplus_{n}\ell^{2}(\mathbb{Z}_{>0})\coloneqq\ell^{2}(\mathbb{Z}_{>0})\oplus\dotsm\oplus\ell^{2}(\mathbb{Z}_{>0})$,
the $n$-fold direct sum of $\ell^{2}(\mathbb{Z}_{>0})$:

$(K_{r})_{ij}(u,v)=K^{\text{{ext}}}(x_{i},u+r_{i};x_{j},v+r_{j}).$ (2.13)

We stress that, although we only include the vector $r$ in the notation,
$K_{r}$ depends on $t$, $x_{1},\dotsc,x_{n}$ and $h_{0}$ as well. Let

$r_{0}(t,x)=\mathop{\mathop{\text{{sup}}}}\displaylimits_{|y-x|\leq
t}h_{0}(y),$ (2.14)

which is simply the level reached at $x$ by the deterministic part of the
dynamics, starting with $h_{0}$, so

$F(t,x_{1},\dotsc,x_{n},r_{1},\dotsc,r_{n})=0\quad\text{{if
$r_{i}<r_{0}(t,x_{i})~{}~{}$ for some $i\in\\{1,\dotsc,n\\}$}}.$

We will show that if $r_{i}\geq r_{0}(t,x_{i})$ for each $i$, then $I-K_{r}$
is invertible; we will write

$R_{r}=(I-K_{r})^{-1}$ (2.15)

for this inverse. Introduce also the variables

$\zeta=\tfrac{1}{2}(t-\overline{x})\quad\text{and}\quad\eta=\tfrac{1}{2}(t+\overline{x}),\qquad\overline{x}=x_{1}+\cdots+x_{n},$
(2.16)

write

$r\pm
1=(r_{1},\dotsc,r_{n})\pm(1,\dotsc,1)\qquad\text{and}\qquad\partial_{\overline{x}}=\partial_{x_{1}}+\dotsm+\partial_{x_{n}},$

and consider the following element of $GL(n)$, depending on $\zeta,\eta$, and
$r$:

$\displaystyle Q_{r}=R_{r}(1,1).$ (2.17)

####  2.2.1. Non-Abelian 2D Toda equation

We can now state the main result.

######  Theorem 2.1.

Let $h_{0}\in\operatorname{\mathscr{U\\!C}}$ be deterministic. For
$r=(r_{1},\dotsc,r_{n})$ such that $r_{i}>r_{0}(t,x_{i})$ for each $i$, the
matrix function $Q_{r}$ satisfies the non-Abelian 2D Toda equation

$\partial_{\zeta}(\partial_{\eta}Q_{r}Q_{r}^{-1})+Q_{r}Q_{r-1}^{-1}-Q_{r+1}Q_{r}^{-1}=0,$
(2.18)

while

$\frac{F(t,x_{1},\ldots,x_{n},r_{1}+1,\ldots,r_{n}+1)}{F(t,x_{1},\ldots,x_{n},r_{1},\ldots,r_{n})}=\mathop{\mathop{\text{{det}}}}\nolimits
Q_{r}.$ (2.19)

The non-Abelian 2D Toda equation (2.18) arises from the special dependence of
the kernel $K_{r}$ on $r$, $\eta$ and $\zeta$ (or, more precisely, the
$r_{i}$’s, $t$ and the $x_{i}$’s). The theorem is proved in Sec. 3. It was
inspired by [BP87], where a special case is considered in an abstract setting
(see Rem. 3.2).

The non-Abelian 2D Toda equation forms a well-known integrable system [Mik81].
Lax and Zakharov-Shabat (zero-curvature) forms of the equation can be found in
Sec. 3.1 of [UT84] (see also [Tak18]). They arise from compatibility of the
equations for vectors $\Phi_{r}$ [NW97]

$\partial_{\eta}\Phi_{r}=V_{r}\Phi_{r}+\lambda_{r}\Phi_{r-1},\qquad\partial_{t}\Phi_{r}=U_{r}\Phi_{r+1}$
(2.20)

with

$U_{r}=Q_{r}Q_{r-1}^{-1},\qquad V_{r}=-\partial_{\eta}Q_{r}Q_{r}^{-1}.$ (2.21)

The non-Abelian 2D Toda equation is sometimes written in terms of $U_{r}$ and
$V_{r}$ as

$\partial_{\eta}U_{r}+V_{r}U_{r}-U_{r}V_{r-1}=0,\qquad\partial_{\zeta}V_{r}+U_{r+1}-U_{r}=0$
(2.22)

(the first equation is an elementary consequence of the definition (2.21) of
$U_{r}$ and $V_{r}$, while the second one is equivalent to (2.18)).

When $n=1$, there is no need to take the determinant in (2.19) and the
previous theorem reduces to the following more accessible statement:

######  Corollary 2.2.

Let $h_{0}\in\operatorname{\mathscr{U\\!C}}$ be deterministic. Then

$F_{r}(t,x)=\mathbb{P}_{h_{0}}(h(t,x)\leq r)$ (2.23)

satisfies the scalar 2D Toda equation

$\tfrac{1}{4}(\partial_{t}^{2}-\partial_{x}^{2})\mathop{\mathop{\text{{log}}}}\nolimits
F_{r}=\frac{F_{r+1}F_{r-1}}{F_{r}^{2}}-1$ (2.24)

for $t>0$ and $r>r_{0}(t,x)$ given by (2.14).

To see how the corollary follows from Thm. 2.1, note that from (2.19) and
(2.21) we have $U_{r}=\frac{F_{r+1}F_{r-1}}{F_{r}^{2}}$. On the other hand,
since in this case $U_{r}$ and $V_{r}$ are scalars, the first equation in
(2.22) reads $\partial_{\eta}\mathop{\mathop{\text{{log}}}}\nolimits
U_{r}+V_{r}-V_{r-1}=0$. Taking the $\zeta$ derivative of this identity and
using the second equation in (2.22) gives
$\partial^{2}_{\zeta\eta}\mathop{\mathop{\text{{log}}}}\nolimits
U_{r}=U_{r+1}-2U_{r}+U_{r-1}$. But $\mathop{\mathop{\text{{log}}}}\nolimits
U_{r}=\mathop{\mathop{\text{{log}}}}\nolimits
F_{r+1}-2\mathop{\mathop{\text{{log}}}}\nolimits
F_{r}+\mathop{\mathop{\text{{log}}}}\nolimits F_{r-1}$ and
$\partial^{2}_{\zeta\eta}=\tfrac{1}{4}(\partial_{t}^{2}-\partial_{x}^{2})$, so
this is just a second difference version of (2.24).

####  2.2.2. 1D Toda lattice

In the case of flat initial data, $h_{0}\equiv 0$, $F_{r}$ is independent of
$x$, the $x$ derivative drops out of (2.24), and taking
$g_{r}=\mathop{\mathop{\text{{log}}}}\nolimits
F_{r}-\mathop{\mathop{\text{{log}}}}\nolimits F_{r-1}$ and $t\longmapsto 2t$
we obtain the classic Toda lattice,

$\ddot{g}_{r}=e^{g_{r+1}-g_{r}}-e^{g_{r}-g_{r-1}}.$ (2.25)

####  2.2.3. Boundary conditions

One needs to supplement (2.24) with boundary conditions at $t=0$ and
$r=r_{0}(t,x)$. At $t=0$ we clearly have from the definition (2.23) of
$F_{r}(t,x)$ that

$F_{r}(0,x)=\mathbf{1}_{r\geq h_{0}(x)}.$ (2.26)

Since we are dealing with a wave equation, we require an extra piece of data
at $t=0$, which in simple cases is given by

$\partial_{t}F_{r}(0,x)=-\textstyle\sum_{y}\\!\left((h_{0}(y)-h_{0}(y^{-}))\mathbf{1}_{h_{0}(y^{-})\leq
r<h_{0}(y)}+(h_{0}(y)-h_{0}(y^{+}))\mathbf{1}_{h_{0}(y^{+})\leq
r<h_{0}(y)}\right)\delta_{y}(x).$ (2.27)

This comes from the deterministic part of the dynamics: The random part does
not contribute to the derivative because a jump at $x$ coming from a
nucleation in a time interval of length $\varepsilon$ has probability of order
$\varepsilon^{2}$. From the dynamics of the process, $h(t,x)$ can never be
smaller than $r_{0}(t,x)$. Hence $F_{r}(t,x)=0$ for $r<r_{0}(t,x)$. This
leaves open the fate of $F_{r_{0}(t,x)}(t,x)$. We clearly have
$\frac{F_{r_{0}+1}F_{r_{0}-1}}{F_{r_{0}}^{2}}=0$, since $F_{r_{0}-1}=0$, and
$F_{r_{0}}>0$, since if there are no nucleations in the backward light cone of
$(t,x)$, then $h(t,x)=r_{0}(t,x)$, and this has positive probability (note
however, that other scenarios can also lead to $h(t,x)=r_{0}(t,x)$ so
$F_{r_{0}}$ is not trivial to compute.) This would appear to give
$\frac{1}{4}(\partial_{t}^{2}-\partial_{x}^{2})\mathop{\mathop{\text{{log}}}}\nolimits
F_{r}(t,x)\mid_{r=r_{0}(t,x)}=-1$, but it is not quite true because
$F_{r_{0}}$ has jumps when $r_{0}$ has jumps. These can be worked out in the
following way. First of all, $r_{0}(t,x)$ is computed in an elementary way
from the initial data $h_{0}(x)$. It has discontinuities of the first kind
(jumps) on (annihilating) lines of slope either $1$ or $-1$ emerging from the
initial jumps of $h_{0}$ and at each discontinuity point $(t,x)$ we have
$r_{0}(t^{-},x)<r_{0}(t^{+},x)$. It is not hard to see that
$F_{r_{0}(t,x)}(t,x)$ must jump from $F_{r_{0}(t^{-},x)}(t^{-},x)$ to
$F_{r_{0}(t^{+},x)}(t^{-},x)$ as we cross the discontinuity line. In the
interior of the regions bounded by the discontinuity lines, we do have
$\frac{1}{4}(\partial_{t}^{2}-\partial_{x}^{2})\mathop{\mathop{\text{{log}}}}\nolimits
F_{r}(t,x)\mid_{r=r_{0}(t,x)}=-1$. Because the jump is actually computed using
the value of $F$ at $t^{-}$, $F_{r_{0}(t,x)}(t,x)$ can now be computed
everywhere. Again one requires an extra boundary condition which is provided
by $\mathop{\mathop{\text{{lim}}}}\displaylimits_{r\to\infty}F_{r}(t,x)=1$. We
believe that under these conditions the 2D Toda equation is well posed, but
this is very far from anything available in the literature, and we leave it
for future work. In the special narrow wedge and flat cases, these boundary
conditions are very easy to state:

_Narrow wedge_. $F_{r}(0,x)=\mathbf{1}_{x\neq 0}+\mathbf{1}_{x=0,\,r\geq 0}$,
$\partial_{t}F_{r}(0,x)=-2\delta_{0}(x)$. For $|x|\leq t$, $r_{0}(t,x)=0$,
while $r_{0}=-\infty$ otherwise, and
$F_{0}(t,x)=\mathop{\mathop{\text{{exp}}}}\nolimits\\{-(t^{2}-x^{2})\\}$,
$F_{k}(t,x)=0$, $k<0$, $|x|\leq t$.

_Flat_. $F_{r}(0,x)=\mathbf{1}_{r\geq 0}$, $\partial_{t}F_{r}(0,x)=0$ and
$r_{0}\equiv 0$ with
$F_{0}(t,x)=\mathop{\mathop{\text{{exp}}}}\nolimits\\{-2t^{2}\\}$ and
$F_{k}(t,x)=0$, $k<0$.

The boundary terms for the non-Abelian Toda equation are complicated to state
and will be partially addressed in Appdx. A.

####  2.2.4. Related work

It was known that the narrow wedge PNG one-point distribution is connected to
Toda $\tau$-functions via Schur measures [Oko01]. This can be partially
extended to the flat case (see Sec. 2.5.2). It is far from clear however, if,
or how, such methods could apply to general initial data. Because of this
prevailing paradigm, the fact that general initial data lead to Toda actually
comes as something of a surprise. There is a point of view that all these
integrable models lead to $\tau$-functions. But this is refuted by the key
integrable model which led to the discovery of the KPZ fixed point, TASEP,
which does not appear to fit this mold so nicely.

Very recently [CR22] studied a finite temperature version of the narrow wedge
solution, with a Fermi factor analogous to the deformation of the Airy kernel
which gives the narrow wedge solution of KPZ [ACQ11]. Note that it is far from
clear what a positive temperature deformation of PNG should look like; an
interesting suggestion is studied in [ABW22]. Using Riemann-Hilbert methods,
[CR22] show that the Fredholm determinant satisfies the same equation as ours
in the one-point narrow wedge case. They also obtain $t\searrow 0$ asymptotics
which refine what we have in the general case (see (4.34)). We note that the
works were done completely independently, and theirs was posted to the arXiv
after we had announced ours in several talks.

The scalar 2D Toda lattice also comes up in generating functions for Hurwitz
numbers [Oko00] (see also [GPH15]), “moments” of conformal maps of simply
connected domains [WZ00], as well as in the intriguingly related situation of
the multilayer heat equation [OW16] (see also [O’C12], where the partition
function of a Brownian directed polymer model is characterized in terms of a
diffusion process associated with the quantum Toda lattice). The 1D non-
Abelian case appeared earlier, governing spin correlations in the
inhomogeneous XY model [PC78]. A mean field analysis of PNG in [BN+98]
somewhat surprisingly leads to a directed version of the Toda lattice.

###  2.3. Fredholm determinant formula

The proof of Thm. 2.1 and Cor. 2.2 is based on the following explicit Fredholm
determinant formula for the multipoint distribution of the PNG height
function, given in terms of the kernel $K_{r}$ from (2.13):

######  Theorem 2.3.

Let $h_{0}\in\operatorname{\mathscr{U\\!C}}$ be deterministic and fix any
$x_{1},\dotsc,x_{n}\in\mathbb{R}$ and $r_{1},\dotsc,r_{n}\in\mathbb{Z}$ such
that $r_{i}\geq r_{0}(t,x_{i})$ (see (2.14)) for each $i=1,\dotsc,n$. The
finite dimensional distributions (2.7) are given by the Fredholm determinant

$F(t,x_{1},\ldots,x_{n},r_{1},\ldots,r_{n})=\mathop{\mathop{\text{{det}}}}\nolimits(I-K_{r})_{\oplus_{n}\ell^{2}(\mathbb{Z}_{>0})}.$
(2.28)

Note that the determinant in (2.28) is the Fredholm determinant of an operator
acting on the $n$-fold direct sum of $\ell^{2}(\mathbb{Z}_{>0})$, while (2.19)
is that of an $n\times n$ matrix.

The way we originally derived this formula was as a limit of an analogous
formula for parallel TASEP with general initial data obtained in [MR22]. In
the regime when each particle in parallel TASEP attempts to move with
probability $p$ close to $1$, the height function of that model can be thought
of as a discrete time version of PNG [GW95] . For parallel TASEP there is a
biorthogonal ensemble representation, derived in [BFS08], from which a
Fredholm determinant formula for its multipoint distributions can be derived
based on a generalization of the method introduced in [MQR21] in the context
of continuous time TASEP [MR22]. We will pursue this route to (2.28) in an
upcoming paper, where we will also show that in fact parallel TASEP converges
to PNG, after appropriate rescaling, as $p\to 1$. A potential alternate
approach could be to start from the formula in Thm. 2 of [JR22], which is an
analogue of Schutz’s formula for TASEP in the PNG context. In principle, one
could hope to pass from this to a Fredholm determinant formula. But it has
only been accomplished for a few special initial conditions (see [JR22]). In
this paper, we will prove Thm. 2.3 by directly checking that the right hand
side of (2.28) solves the Kolmogorov backward equation for PNG.

######  Remark 2.4.

We feel that the proof of (2.28) using the backward equation has considerable
explanatory power. Although it turns out to be somewhat technical, the key
fact that makes it work, Prop. 4.8, is transparent and basic, explaining why
hit kernels will produce transition probabilities for such models, and, we
hope, opening the way to stochastic analysis of the KPZ fixed point itself. In
[NQR20a], the backward equation was also proved for the TASEP Fredholm
determinant formulas. However, the key mechanism is not apparent there. The
identification of this mechanism is one of the most important contributions of
this article. It is also important to note that a proof like this, via the
backward equation, can only be achieved once one has a formula for general
initial data.

######  Remark 2.5.

The finite propagation speed of the PNG dynamics implies that if
$|x_{1}-x_{2}|>2t$ then $h(t,x_{1})$ and $h(t,x_{2})$ are independent. This
fact can actually be seen from the Fredholm determinant formula (2.28):
restricting to the two-point case for simplicity, in such a situation we have
$P^{\operatorname{\text{{hit}}}(h_{0})}_{x_{2}-t,x_{1}+t}=0$, so
$(K_{r})_{2,1}=0$, and thus the kernel is upper-triangular, which means that
its Fredholm determinant equals the product of the Fredholm determinants of
its diagonal entries, i.e.
$\mathop{\mathop{\text{{det}}}}\nolimits(I-(K_{r})_{1,1})_{\ell^{2}(\mathbb{Z}_{>0})}\mathop{\mathop{\text{{det}}}}\nolimits(I-(K_{r})_{2,2})_{\ell^{2}(\mathbb{Z}_{>0})}$.
Note that this means, in particular, that there is no dependence on the
$(K_{r})_{1,2}$ entry of the kernel, which is the only one depending on
$h_{0}$ outside of $[x_{1}-t,x_{1}+t]\cup[x_{2}-t,x_{2}+t]$ (which cannot
affect the distribution of $h(t,x_{1})$ and $h(t,x_{2})$). The same argument
works when there are several clusters of $x_{i}$’s which are separated by
distance at least $2t$.

The non-Abelian Toda equations for PNG follow from (2.28) and the structure of
the kernel $K_{r}$. A key relation to this effect which is satisfied by the
kernel is

$\partial_{\eta}K_{r}(u,v)=K_{r-1}(u+1,v)-K_{r}(u+1,v),\quad\partial_{\zeta}K_{r}(u,v)=K_{r-1}(u,v+1)-K_{r}(u,v+1),$
(2.29)

provided $r_{i}\geq r_{0}(t,x_{i})$ for each $i$. The two identities together
yield $\partial_{\eta}\partial_{\zeta}K_{r}=K_{r+1}-2K_{r}+K_{r-1}$, which
effectively provides a linear evolution for $K_{r}$ from which the PNG
multipoint distributions can be recovered through the Fredholm determinant,
thus presenting the model as a stochastic integrable system. The identities in
(2.29) follow directly (and for all $r$) from the definition (2.12)/(2.13) if
one ignores the dependence of the scattering transform
$\mathscr{T}^{h_{0}}_{x_{i}-t,x_{j}+t}$ on the $x_{i}$’s and $t$. Showing that
this factor does not contribute to $\partial_{\eta}K_{r}$ and
$\partial_{\zeta}K_{r}$ requires a separate argument, see Sec. 3.3.

The formula (2.28) for the PNG finite dimensional distributions can be
expressed equivalently in terms of the kernel $K^{\text{{ext}}}$ from (2.13),
which is how formulas of this type have often been written in the literature.
This way of writing the formula is convenient for some computations. To state
it, for a fixed vector $r\in\mathbb{Z}^{n}$ and indices $x_{1}<\dotsm<x_{n}$
let

$\chi_{r}(x_{j},u)=\mathbf{1}_{u>r_{j}}\qquad\text{and}\qquad\bar{\chi}_{r}(x_{j},u)=\mathbf{1}_{u\leq
r_{j}},$ (2.30)

which are regarded as multiplication operators acting on the space
$\ell^{2}(\\{x_{1},\dotsc,x_{m}\\}\times\mathbb{Z})$. We will also use this
notation when $n=1$ and $r\in\mathbb{Z}$, writing
$\chi_{r}(u)=1-\bar{\chi}_{r}(u)=\mathbf{1}_{u>r}$. Then we have, for
$r_{i}\geq r_{0}(t,x_{i})$, $i=1,\dotsc,n$,

$F(t,x_{1},\ldots,x_{n},r_{1},\ldots,r_{n})=\mathop{\mathop{\text{{det}}}}\nolimits(I-\chi_{r}K^{\text{{ext}}}\chi_{r})_{\ell^{2}(\\{x_{1},\dotsc,x_{n}\\}\times\mathbb{Z}}.$
(2.31)

On the other hand, the basic operators which make up the kernels appearing in
(2.11) and (2.12) can be expressed as integral operators with kernels having
an explicit contour integral formula:

$e^{2t\nabla+x\Delta}(u_{1},u_{2})=e^{-2x}\frac{1}{2\pi{\rm
i}}\oint_{\gamma_{0}}\frac{\mathrm{d}z}{z^{u_{2}-u_{1}+1}}e^{t(z-z^{-1})+x(z+z^{-1})},$
(2.32)

where $\gamma_{0}$ is any simple, positively oriented contour around the
origin. When $t>|x|$ this kernel can be expressed in terms of the Bessel
function of the first kind, $J_{n}(x)=\frac{1}{2\pi{\rm
i}}\oint_{\gamma_{0}}\mathrm{d}z\,e^{x(z-z^{-1})/2}/z^{n+1}$:

$e^{2t\nabla+x\Delta}(u_{1},u_{2})=e^{-2x}\left(\tfrac{t-x}{t+x}\right)^{(u_{2}-u_{1})/2}\\!J_{u_{2}-u_{1}}\bigl{(}2\sqrt{t^{2}-x^{2}}\bigr{)}.$
(2.33)

When $t=0$ there is another Bessel function expression for the kernel of
$e^{x\Delta}$ for $x\geq 0$ (i.e., for the semigroup of the walk
$\mathsf{N}$):

$e^{x\Delta}(u_{1},u_{2})=e^{-2x}I_{|u_{2}-u_{1}|}(2x),$ (2.34)

where $I_{n}(2x)=\frac{1}{2\pi{\rm
i}}\oint_{\gamma_{0}}\mathrm{d}z\,e^{x(z+z^{-1})}/z^{n+1}$ is the modified
Bessel function of the first kind. Formula (2.32) can be proved by
differentiation in $t$ and $x$, while (2.33) and (2.34) follow from the
standard contour integral formulas for the Bessel functions [Nis].

###  2.4. KPZ universality

The _strong KPZ universality conjecture_ states that for any model in the
class there is an analogue of the height function and if
$\varepsilon^{1/2}h_{0}(\varepsilon^{-1}x)\longrightarrow\mathfrak{h}_{0}(x)$
as $\varepsilon\to 0$ then there are (non-universal) $c$ and $C_{\varepsilon}$
such that

$\varepsilon^{1/2}h(c\hskip
0.50003pt\varepsilon^{-3/2}t,\varepsilon^{-1}x)-C_{\varepsilon}t\xrightarrow[\varepsilon\to
0]{}\mathfrak{h}(t,x;\mathfrak{h}_{0}),$ (2.35)

where $\mathfrak{h}(t,x;\mathfrak{h}_{0})$ is the KPZ fixed point starting
from $\mathfrak{h}_{0}$ [MQR21]. This can alternatively be interpreted as the
definition of the KPZ class. KPZ universality has been achieved in only a few
cases, usually by showing that Fredholm determinant formulas for the
transition probabilities converge to those of the KPZ fixed point. There are
two steps: 1. Pointwise convergence of the operator kernels; 2. Upgrade to
trace class convergence, or analogous bounds. The latter tends to be tedious
and difficult, and is sometimes skipped in the literature. In the case of PNG,
the pointwise convergence of the kernels can be checked by observation. It
will be upgraded to trace class in a future article. But for PNG, there is
anyway a more direct route using the variational formula (2.3). Implicit in
the article [DV21] is the statement that under the 1:2:3 scaling, (2.35),
$\mathscr{A}_{t}(x,y)$ converges to the rescaled Airy sheet
$t^{1/3}\mathscr{A}(t^{-2/3}x,t^{-2/3}y)$ and therefore that if one has
uniform convergence on compact sets of the rescaled initial data to a
compactly supported $\mathfrak{h}_{0}$, then at time $t>0$ one has

$\varepsilon^{1/2}h(\varepsilon^{-3/2}t,\varepsilon^{-1}x)-2\varepsilon^{-1}t\xrightarrow[\varepsilon\to
0]{}\mathop{\mathop{\text{{sup}}}}\displaylimits_{y\in\mathbb{R}}\big{\\{}t^{1/3}\mathscr{A}(t^{-2/3}x,t^{-2/3}y)+\mathfrak{h}_{0}(y)\\}$

in the same sense. The right hand side is the variational description of the
KPZ fixed point $\mathfrak{h}(t,x;\mathfrak{h}_{0})$ (see [MQR21, DOV18,
NQR20]).

For the KPZ fixed point $\mathfrak{h}(t,x)$, we showed in [QR22] that

$\mathfrak{q}=\partial_{r}^{2}\mathop{\mathop{\text{{log}}}}\nolimits\mathbb{P}(\mathfrak{h}(t,x)\leq
r)$

satisfies the scalar Kadomtsev–Petviashvili (KP-II) equation

$\partial_{t}\mathfrak{q}+\tfrac{1}{2}\partial_{r}\mathfrak{q}^{2}+\tfrac{1}{12}\partial_{r}^{3}\mathfrak{q}+\tfrac{1}{4}\partial_{r}^{-1}\partial_{x}^{2}\mathfrak{q}=0.$
(2.36)

An equation, with the flavor of (2.18)/(2.19), was derived in that case also
for the multipoint case. In order to write it down, we let
$\mathbf{K}=\mathbf{K}(t;x_{1},\dotsc,x_{n};r_{1},\dotsc,r_{n};\mathfrak{h}_{0})$
denote the _Brownian scattering operator_ introduced in [MQR21], which plays
the role of the PNG kernel $K_{r}$ for the KPZ fixed point, so that

$\mathfrak{F}(t;x_{1},\dotsc,x_{n};r_{1},\dotsc,r_{n};\mathfrak{h}_{0})\coloneqq\mathbb{P}(\mathfrak{h}(t,x_{i})\leq
r_{i},\,i=1,\dotsc,n)=\mathop{\mathop{\text{{det}}}}\nolimits(\mathbf{I}-\mathbf{K})_{\oplus_{n}L^{2}([0,\infty))}.$

Consider also the variables $\bar{r}=r_{1}+\dotsm+r_{n}$,
$\overline{x}=x_{1}+\dotsm+x_{n}$, and write as before
$\partial_{\bar{r}}=\partial_{r_{1}}+\dotsm+\partial_{r_{n}}$ and
$\partial_{\overline{x}}=\partial_{x_{1}}+\dotsm+\partial_{x_{n}}$. Finally
let

$\mathfrak{Q}=(\mathbf{I}-\mathbf{K})^{-1}\mathbf{K}(0,0),$

which is in $GL(n)$. Then the result of [QR22] is that

$\partial_{r}\mathop{\mathop{\text{{log}}}}\nolimits\mathfrak{F}=\operatorname{\text{{tr}}}(\mathfrak{Q})$

while $\mathfrak{Q}$ and its derivative
$\mathfrak{q}=\partial_{r}\mathfrak{Q}$ solve the matrix KP equation

$\partial_{t}\mathfrak{q}+\tfrac{1}{2}\hskip
0.50003pt\partial_{r}\mathfrak{q}^{2}+\tfrac{1}{12}\partial_{r}^{3}\mathfrak{q}+\tfrac{1}{4}\partial_{x}^{2}\mathfrak{Q}+\tfrac{1}{2}[\mathfrak{q},\hskip
0.50003pt\partial_{x}\mathfrak{Q}]=0,$ (2.37)

where $[A,B]=AB-BA$.

The following gives a twist on the convergence of PNG to the KPZ fixed point:
Instead of convergence of kernels, one can check convergence of equations. The
argument which we will present is far from giving a complete proof, which
would require, in addition to controlling the expansions, a uniqueness proof
at the level of the KP equations for the KPZ fixed point. This will be the
subject of future research. At any rate, the route of [DV21] is preferable for
a rigorous proof here. However, we believe that convergence of equations has
considerable explanatory value as a direct and intuitive route to the
universal fluctuations. Well-known behaviours, such as the Tracy-Widom GUE and
GOE one-point asymptotic fluctuations in KPZ models, are explained by their
appearance as special self-similar solutions of (2.36), so such a derivation
helps us understand the emergence of non-standard fluctuations in the KPZ
class. The idea is that their appearance is through such _normal form_
equations.

In the flat case $\mathfrak{h}_{0}\equiv 0$, the $n=1$ $\mathfrak{q}$ in
(2.36) does not depend on $x$ and the equation simplifies to the Korteweg-de
Vries (KdV). The derivation of the KdV equation from the Toda lattice (2.25)
is a folklore result with a long history. However, examination of the
literature actually reveals a consensus that it does not actually hold unless
one reduces the problem, often by considering waves traveling in only one
direction. For example, [BP87] states that, “it cannot be literally true” that
KdV is the limit of the Toda lattice “since KdV is first order in time and
antisymmetric” (in $r$), but instead has to be derived from first order
versions, the Langmuir, or Kac-van Moerbeke lattice. This approach goes back
to considerable work on the problem of convergence to KdV by Toda himself. In
[TW73] it is the Boussinesq equation that is derived from Toda, which is then
reduced to KdV by considering only one-way waves.

The following informal derivation (in the more general setting of non-Abelian
2D Toda and matrix KP) shows that the misconception was because the
unexpected, highly refined 1:2:3 scaling was missed. It is our hope that it
inspires further work.

####  2.4.1. Scalar KP equation

We start with the scalar equation (2.24), where the derivation is easier to
follow. The 1:2:3 rescaling (2.35) means we are interested in
$\mathfrak{F}_{\varepsilon}$ defined by

$F(t,x,r)=\mathfrak{F}_{\varepsilon}(\varepsilon^{3/2}t,\varepsilon
x,\varepsilon^{1/2}(r-ct)).$ (2.38)

For now we let the linear shift in the third argument depend on a constant
$c>0$, to be determined. For the derivation, we have to expand
$\mathfrak{F}_{\varepsilon}$ in Taylor series. We do not attempt a
justification here. Furthermore, in the $r$ variable,
$\mathfrak{F}_{\varepsilon}$ really lives on $\varepsilon^{1/2}\mathbb{Z}$,
yet we will pretend it is defined on $\mathbb{R}$ and still expand in Taylor
series. This type of argument is standard in such derivations, though, again,
considerable work would be required to justify it.

Under this scaling, (2.24) becomes

$\tfrac{1}{4}(\varepsilon^{3}\partial_{t}^{2}-2\varepsilon^{2}c\partial^{2}_{tr}+\varepsilon
c^{2}\partial^{2}_{r}-\varepsilon^{2}\partial_{x}^{2})\mathop{\mathop{\text{{log}}}}\nolimits\mathfrak{F}=\tfrac{\mathfrak{F}(\cdot+\varepsilon^{1/2})\mathfrak{F}(\cdot-\varepsilon^{1/2})}{\mathfrak{F}^{2}}-1,$
(2.39)

where for notational simplicity we removed the $\varepsilon$ from the
subscript in $\mathfrak{F}_{\varepsilon}$. Expanding (2.39) out and
multiplying by $\mathfrak{F}^{2}$, we get

$\tfrac{\varepsilon c^{2}}{4}\mathfrak{F}\hskip
1.00006pt\partial_{r}^{2}\mathfrak{F}-\tfrac{\varepsilon
c^{2}}{4}(\partial_{r}\mathfrak{F})^{2}-\tfrac{\varepsilon^{2}c}{2}\mathfrak{F}\hskip
1.00006pt\partial_{tr}^{2}\mathfrak{F}+\tfrac{\varepsilon^{2}c}{2}\partial_{t}\mathfrak{F}\hskip
1.00006pt\partial_{r}\mathfrak{F}-\tfrac{\varepsilon^{2}}{4}\mathfrak{F}\hskip
1.00006pt\partial_{x}^{2}\mathfrak{F}+\tfrac{\varepsilon^{2}}{4}(\partial_{x}\mathfrak{F})^{2}\\\
=\varepsilon\mathfrak{F}\hskip
1.00006pt\partial_{r}^{2}\mathfrak{F}-\varepsilon(\partial_{r}\mathfrak{F})^{2}+\varepsilon^{2}\big{(}\tfrac{1}{12}\mathfrak{F}\hskip
1.00006pt\partial_{r}^{4}\mathfrak{F}-\tfrac{1}{3}\partial_{r}\mathfrak{F}\hskip
1.00006pt\partial_{r}^{3}\mathfrak{F}+\tfrac{1}{4}(\partial^{2}_{r}\mathfrak{F})^{2}\big{)}+o(\varepsilon^{2}).$
(2.40)

Clearly we should choose average growth rate $c=2$ in order for the
$O(\varepsilon)$ terms to cancel. We are left with an equation at order
$\varepsilon^{2}$ which says that the associated bilinear form should vanish:

$\displaystyle\mathfrak{F}\partial_{tr}^{2}\mathfrak{F}-\partial_{t}\mathfrak{F}\hskip
1.00006pt\partial_{r}\mathfrak{F}+\tfrac{1}{4}\mathfrak{F}\hskip
1.00006pt\partial_{x}^{2}\mathfrak{F}-\tfrac{1}{4}(\partial_{x}\mathfrak{F})^{2}+\tfrac{1}{12}\mathfrak{F}\hskip
1.00006pt\partial_{r}^{4}\mathfrak{F}-\tfrac{1}{3}\partial_{r}\mathfrak{F}\hskip
1.00006pt\partial_{r}^{3}\mathfrak{F}+\tfrac{1}{4}(\partial^{2}_{r}\mathfrak{F})^{2}=0.$
(2.41)

This is the Hirota form of the scalar KP equation, and a simple computation
shows that it means that
$\mathfrak{q}=\partial_{r}^{2}\mathop{\mathop{\text{{log}}}}\nolimits\mathfrak{F}$
satisfies (2.36).

####  2.4.2. Matrix KP equation

Next we do the same thing with the non-Abelian Toda equations. Start with
(2.18) in the form

$\partial_{\zeta\eta}^{2}Q_{r}Q_{r}^{-1}-\partial_{\eta}Q_{r}Q_{r}^{-1}\partial_{\zeta}Q_{r}Q_{r}^{-1}+Q_{r}Q_{r-1}^{-1}-Q_{r+1}Q_{r}^{-1}=0.$
(2.42)

Since we already learned the average growth rate $c=2$ we write

$Q=I+\varepsilon^{1/2}\mathfrak{Q}_{\varepsilon}(\varepsilon^{3/2}t,\varepsilon
x,\varepsilon^{1/2}(r-2t)).$ (2.43)

Note the difference between $Q=R(1,1)$ in our problem and
$\mathfrak{Q}=RK(0,0)$ at the KPZ fixed point level which is reflected in this
scaling. Then (2.42) becomes $\Omega_{1}+\Omega_{2}+\Omega_{3}=0$ where

$\displaystyle\Omega_{1}=$
$\displaystyle\tfrac{1}{4}\varepsilon^{1/2}(\varepsilon^{3}\partial_{t}^{2}-4\varepsilon^{2}\partial^{2}_{tr}+4\varepsilon\partial_{r}^{2}-\varepsilon^{2}\partial_{x}^{2})\mathfrak{Q}(r)(I-\varepsilon^{1/2}\mathfrak{Q}(r)+\varepsilon\mathfrak{Q}(r)^{2}+o(\varepsilon)),$
(2.44) $\displaystyle\Omega_{2}=$
$\displaystyle-\tfrac{1}{4}\varepsilon(\varepsilon^{3/2}\partial_{t}-2\varepsilon^{1/2}\partial_{r}+\varepsilon\partial_{x})\mathfrak{Q}(r)(I-\varepsilon^{1/2}\mathfrak{Q}(r)+\varepsilon\mathfrak{Q}(r)^{2}+o(\varepsilon))$
$\displaystyle\qquad\qquad\times(\varepsilon^{3/2}\partial_{t}-2\varepsilon^{1/2}\partial_{r}-\varepsilon\partial_{x})\mathfrak{Q}(r)(I-\varepsilon^{1/2}\mathfrak{Q}(r)+\varepsilon\mathfrak{Q}(r)^{2}+o(\varepsilon)),$
$\displaystyle\Omega_{3}=$
$\displaystyle(I+\varepsilon^{1/2}\mathfrak{Q}(r))(I+\varepsilon^{1/2}\mathfrak{Q}(r-\varepsilon^{1/2}))^{-1}-(I+\varepsilon^{1/2}\mathfrak{Q}(r+\varepsilon^{1/2}))(I+\varepsilon^{1/2}\mathfrak{Q}(r))^{-1},$

and where, as before, we have removed the subscript $\varepsilon$ in
$\mathfrak{Q}_{\varepsilon}$ for convenience; in this decomposition,
$\Omega_{1}$ and $\Omega_{2}$ come from the first and second terms in (2.42),
respectively, while $\Omega_{3}$ combines the last two. Now writing
$\mathfrak{Q}$ for $\mathfrak{Q}(r)$ and $\mathfrak{Q}^{\prime}$ for
$\partial_{r}\mathfrak{Q}(r)$, we have

$\displaystyle I+\varepsilon^{1/2}\mathfrak{Q}(r+\varepsilon^{1/2})$
$\displaystyle=I+\varepsilon^{1/2}\mathfrak{Q}+\varepsilon\mathfrak{Q}^{\prime}+\tfrac{1}{2}\varepsilon^{3/2}\mathfrak{Q}^{\prime\prime}+\tfrac{1}{6}\varepsilon^{2}\mathfrak{Q}^{\prime\prime\prime}+\tfrac{1}{24}\varepsilon^{5/2}\mathfrak{Q}^{\prime\prime\prime\prime}+o(\varepsilon^{5/2}),$
(2.45) $\displaystyle(I+\varepsilon^{1/2}\mathfrak{Q}(r))^{-1}$
$\displaystyle=I-\varepsilon^{1/2}\mathfrak{Q}+\varepsilon\mathfrak{Q}^{2}-\varepsilon^{3/2}\mathfrak{Q}^{3}+\varepsilon^{2}\mathfrak{Q}^{4}-\varepsilon^{5/2}\mathfrak{Q}^{5}+o(\varepsilon^{5/2}),$
(2.46) and similarly
$\displaystyle(I+\varepsilon^{1/2}\mathfrak{Q}(r-\varepsilon^{1/2}))^{-1}$
$\displaystyle=I+\textstyle\sum_{i=1}^{5}(-1)^{i}\mathfrak{Q}(r-\varepsilon^{1/2})^{i}+o(\varepsilon^{5/2})$
(2.47)
$\displaystyle=I-\varepsilon^{1/2}\mathfrak{Q}+\varepsilon(\mathfrak{Q}^{\prime}+\mathfrak{Q}^{2})-\varepsilon^{3/2}(\tfrac{1}{2}\mathfrak{Q}^{\prime\prime}+\mathfrak{Q}\mathfrak{Q}^{\prime}+\mathfrak{Q}^{\prime}\mathfrak{Q}+\mathfrak{Q}^{3})$
(2.48)
$\displaystyle+\varepsilon^{2}(\tfrac{1}{6}\mathfrak{Q}^{\prime\prime\prime}+(\mathfrak{Q}^{\prime})^{2}+\tfrac{1}{2}\mathfrak{Q}\mathfrak{Q}^{\prime\prime}+\tfrac{1}{2}\mathfrak{Q}^{\prime\prime}\mathfrak{Q}+\mathfrak{Q}^{\prime}\mathfrak{Q}^{2}+\mathfrak{Q}\mathfrak{Q}^{\prime}\mathfrak{Q}+\mathfrak{Q}^{2}\mathfrak{Q}^{\prime}+\mathfrak{Q}^{4})$
(2.49)
$\displaystyle-\varepsilon^{5/2}\Big{(}\tfrac{1}{24}\mathfrak{Q}^{\prime\prime\prime\prime}+\tfrac{1}{6}(\mathfrak{Q}^{\prime\prime\prime}\mathfrak{Q}+\mathfrak{Q}\mathfrak{Q}^{\prime\prime\prime})+\tfrac{1}{2}(\mathfrak{Q}^{\prime}\mathfrak{Q}^{\prime\prime}+\mathfrak{Q}^{\prime\prime}\mathfrak{Q}^{\prime})+\tfrac{1}{2}(\mathfrak{Q}^{\prime\prime}\mathfrak{Q}^{2}+\mathfrak{Q}\mathfrak{Q}^{\prime\prime}\mathfrak{Q}+\mathfrak{Q}^{2}\mathfrak{Q}^{\prime\prime})$
(2.50)
$\displaystyle-((\mathfrak{Q}^{\prime})^{2}\mathfrak{Q}+\mathfrak{Q}^{\prime}\mathfrak{Q}\mathfrak{Q}^{\prime}+\mathfrak{Q}(\mathfrak{Q}^{\prime})^{2})+(\mathfrak{Q}^{\prime}\mathfrak{Q}^{3}+\mathfrak{Q}\mathfrak{Q}^{\prime}\mathfrak{Q}^{2}+\mathfrak{Q}^{2}\mathfrak{Q}^{\prime}\mathfrak{Q}+\mathfrak{Q}^{3}\mathfrak{Q}^{\prime})+\mathfrak{Q}^{5}\Big{)}+o(\varepsilon^{5/2}).$
(2.51)

Using these expansions in the definition of $\Omega_{3}$ and collecting terms
carefully, one sees that only terms of order $\varepsilon^{3/2}$ and higher
remain, and after simplification one gets

$\displaystyle\Omega_{3}=-\varepsilon^{3/2}\mathfrak{Q}^{\prime\prime}+\varepsilon^{2}((\mathfrak{Q}^{\prime})^{2}+\mathfrak{Q}\mathfrak{Q}^{\prime\prime})-\varepsilon^{5/2}\Big{(}(\mathfrak{Q}^{\prime})^{2}\mathfrak{Q}+\mathfrak{Q}^{\prime}\mathfrak{Q}\mathfrak{Q}^{\prime}+\mathfrak{Q}(\mathfrak{Q}^{\prime})^{2}+\mathfrak{Q}^{\prime\prime}\mathfrak{Q}^{2}$
(2.52) $\displaystyle\hskip
231.26378pt+\tfrac{1}{2}(\mathfrak{Q}^{\prime\prime}\mathfrak{Q}^{\prime}+\mathfrak{Q}^{\prime\prime}\mathfrak{Q})+\tfrac{1}{12}\mathfrak{Q}^{\prime\prime\prime\prime}\Big{)}+o(\varepsilon^{5/2}).$
(2.53)

Now we note that the only term of order $\varepsilon^{3/2}$ and lower in the
first three lines of (2.44) is $\partial_{r}^{2}\mathfrak{Q}$, which cancels
the first term in (2.52). We also have cancellation at order
$\varepsilon^{2}$. Thus the equation appears as the vanishing of terms of
order $\varepsilon^{5/2}$. From (2.44) we have

$\displaystyle\varepsilon^{5/2}~{}~{}\text{{coeff. of }}\Omega_{1}=$
$\displaystyle-\partial_{tr}^{2}\mathfrak{Q}-\tfrac{1}{4}\partial_{x}^{2}\mathfrak{Q}+\partial_{r}^{2}\mathfrak{Q}\mathfrak{Q}^{2},$
(2.54) $\displaystyle\varepsilon^{5/2}~{}~{}\text{{coeff. of }}\Omega_{2}=$
$\displaystyle-\tfrac{1}{2}\partial_{r}\mathfrak{Q}\partial_{x}\mathfrak{Q}+\tfrac{1}{2}\partial_{x}\mathfrak{Q}\partial_{r}\mathfrak{Q}+\partial_{r}\mathfrak{Q}\mathfrak{Q}\partial_{r}\mathfrak{Q}+(\partial_{r}\mathfrak{Q})^{2}\mathfrak{Q}.$
(2.55)

Adding them together with the $\varepsilon^{5/2}$ coefficient of $\Omega_{3}$
we get

$-\partial_{tr}^{2}\mathfrak{Q}-\tfrac{1}{4}\partial_{x}^{2}\mathfrak{Q}-\tfrac{1}{2}[\partial_{r}\mathfrak{Q},\partial_{x}\mathfrak{Q}]-\tfrac{1}{2}\partial_{r}^{2}\mathfrak{Q}\hskip
0.50003pt\partial_{r}\mathfrak{Q}-\tfrac{1}{2}\partial_{r}\mathfrak{Q}\hskip
0.50003pt\partial_{r}^{2}\mathfrak{Q}-\tfrac{1}{12}\partial_{r}^{4}\mathfrak{Q}=0,$
(2.56)

which is the matrix KP equation.

###  2.5. Narrow wedge and flat examples, discrete Painlevé II

####  2.5.1. Narrow wedge initial data

Consider PNG with narrow wedge initial condition $\mathfrak{d}_{0}$, defined
in (2.1). The scaling limit of the height function $h(t,x)$ for fixed $t=1$
was first derived in [PS02], where the limiting Airy2 process was first
described, using the transfer matrix method for the Fermi field associated to
a multilayer version of the process constructed using dynamics based on the
Robinson-Schensted-Knuth (RSK) correspondence. RSK is also behind the methods
employed in [Joh03] to study a discrete version of PNG, for which uniform
convergence on compacts to the same Airy2 process was obtained.

We begin by showing how the formulas obtained earlier in the literature can be
recovered from Thm. 2.3. Note that for narrow wedge initial data one has
$h(t,x)=-\infty$ for all $|x|>t$. This can be thought of alternatively as
restricting the model (and, in particular, the nucleations) to the domain
$\\{(t,x)\in[0,\infty)\times\mathbb{R}\\!:|x|\leq t\\}$, which is how it has
been often defined in the literature in this case (see e.g. [PS02]). The
random walk $\mathsf{N}$ can only hit
$\operatorname{\text{{hypo}}}(\mathfrak{d}_{0})$ at the origin, so
$P^{\operatorname{\text{{hit}}}(\mathfrak{d}_{0})}_{x_{i}-t,x_{j}+t}=\mathbf{1}_{x_{i}-t\leq
0\leq x_{j}+t}e^{(t-x_{i})\Delta}\bar{\chi}_{0}e^{(x_{j}-t)\Delta}$, and thus
$\mathscr{T}^{\mathfrak{d}_{0}}_{x_{i}-t,x_{j}+t}=\mathbf{1}_{x_{i}-t\leq
0\leq x_{j}+t}\hskip 1.00006pt\bar{\chi}_{0}$ and the kernel in (2.12) equals

$K^{{\text{{ext}}}}(x_{i},\cdot;x_{j},\cdot)=-\mathbf{1}_{x_{i}<x_{j}}e^{(x_{j}-x_{i})\Delta}+\mathbf{1}_{x_{i}-t\leq
0\leq x_{j}+t}e^{-2t\nabla-
x_{i}\Delta}\bar{\chi}_{0}e^{2t\nabla+x_{j}\Delta}.$ (2.57)

The indicators $\mathbf{1}_{x_{i}\leq t}$ and $\mathbf{1}_{x_{j}\geq-t}$ are
natural because the model is essentially restricted to the domain $|x|\leq t$.
In particular, if $x_{\ell}>t$ for some $\ell$ and the $x_{j}$’s are ordered,
then $x_{j}>t$ for all $\ell\leq j\leq m$ so
$K^{\text{{ext}}}(x_{i},\cdot;x_{j},\cdot)=-\mathbf{1}_{x_{i}<x_{j}}e^{(x_{j}-x_{i})\Delta}$
for $i\geq j\geq\ell$, and thus $K^{\text{{ext}}}$ can be decomposed as
$\left[\begin{array}[]{c|c}(K^{\text{{ext}}})(x_{i},\cdot;x_{j},\cdot))_{1\leq
i,j\leq\ell-1}&\star\\\ \hline\cr 0&{\sf U}\end{array}\right]$, where the
value of the $\star$ is irrelevant and ${\sf U}$ is strictly upper triangular.
Thus in this case we have
$F(t,x_{1},\dotsc,x_{n},r_{1},\dotsc,r_{n})=\mathop{\mathop{\text{{det}}}}\nolimits\bigl{(}I-\chi_{r}K^{\operatorname{\text{{PNG}}}}_{t,{\text{{ext}}}}\chi_{a}\bigr{)}_{\ell^{2}(\\{x_{1},\dotsc,x_{\ell-1}\\}\times\mathbb{Z})}$,
which equals $F(t,x_{1},\dotsc,x_{\ell},r_{1},\dotsc,r_{\ell})$. An analogous
statement holds in the case $x_{\ell}<-t$ for some $\ell$.

We focus now on (2.57) in the case $|x_{i}|<t$ for all $i$ (we throw away the
case when some $x_{i}=\pm t$ in order to simplify the presentation; at any
rate this case is physically uninteresting because by definition of the
process we know that $h(t,\pm t)=0$ almost surely). Assume first that
$x_{i}\geq x_{j}$. Using (2.33), the kernel (2.57) turns into

$K^{\text{{ext}}}(x_{i},u_{i};x_{j},u_{j})=e^{2(x_{i}-x_{j})}\sum_{\ell\leq
0}\left(\tfrac{t+x_{i}}{t-x_{i}}\right)^{\frac{u_{i}-\ell}{2}}\\!\\!J_{u_{i}-\ell}\bigl{(}{\scriptstyle{2\sqrt{t^{2}-x_{i}^{2}}}}\bigr{)}\left(\tfrac{t-x_{j}}{t+x_{j}}\right)^{\frac{u_{j}-\ell}{2}}J_{u_{j}-\ell}\bigl{(}{\scriptstyle{2\sqrt{t^{2}-x_{j}^{2}}}}\bigr{)}.$
(2.58)

If instead we have $x_{i}<x_{j}$, we write
$e^{(x_{j}-x_{i})\Delta}=e^{-2t\nabla-x_{i}\Delta}e^{2t\nabla+x_{j}\Delta}$,
and obtain (2.58) with a minus sign in front. This shows that
$K^{\text{{ext}}}(x_{j},u_{j};x_{i},u_{i})=e^{2(x_{i}-x_{j})}B^{*}_{t}(u_{i},-x_{i};u_{j},-x_{j})$,
where $B_{t}$ is the _extended discrete Bessel kernel_ derived in [PS02, Eq.
3.52]333After correcting a minor typo in that paper: $\ell+1/2$ inside the
Heaviside function in (3.52) there should be $\ell-1/2$.. Having the adjoint
of $B_{t}$ does not change the value of the Fredholm determinant, nor does the
conjugation of the kernel by $e^{2(x_{i}-x_{j})}$, while flipping the sign of
the $x_{i}$’s corresponds to reflecting the model about the $x$-axis, which
again does not change anything because the dynamics and the initial condition
are symmetric. Hence, after setting $t=1$, this recovers the formula derived
in [PS02].

Consider now the one-point case. Using the above formula we have

$K^{{\text{{ext}}}}(x,u;x,v)=\left(\tfrac{t+x}{t-x}\right)^{(u-v)/2}B_{s}(u,v),\qquad
B_{s}(u,v)=\sum_{\ell\leq
0}J_{u-\ell}\bigl{(}2s\bigr{)}J_{v-\ell}\bigl{(}2s\bigr{)}$

with $s=\sqrt{t^{2}-x^{2}}$. $B_{s}$ is the usual _discrete Bessel kernel_
[BOO00, Joh01], which is _integrable_ in the sense of [Its+90]:

$B_{s}(u,v)=\frac{s}{u-v}(J_{u-1}(2s)J_{v}(2s)-J_{u}(2s)J_{v-1}(2s))\quad\text{{for}}~{}u\neq
v$

and
$B_{s}(u,u)=s(\frac{\mathrm{d}}{\mathrm{d}u}J_{u-1}(2s)J_{u}(2s)-J_{u-1}(2s)\frac{\mathrm{d}}{\mathrm{d}u}J_{u}(2s))$.
From [BO00], removing the conjugation $((t+x)/(t-x))^{(u-v)/2}$ from
$K^{{\text{{ext}}}}$, we can rewrite

$F_{r}(s)\coloneqq
F(t,x,r)=\mathop{\mathop{\text{{det}}}}\nolimits(I-\chi_{r}B_{s}\chi_{r})_{\ell^{2}(\mathbb{Z})}=e^{-s^{2}}D_{r-1}(\varphi_{s}),$
(2.59)

where $D_{n}(\varphi_{s})$ is the $n\times n$ Toeplitz determinant associated
with the weight $\varphi_{s}(\zeta)=e^{s(\zeta+\zeta^{-1})}$ (setting
$D_{-1}(\varphi_{s})=1$); more explicitly, we have (for $r\geq 1$)

$F_{r}(s)=e^{-s^{2}}\mathop{\mathop{\text{{det}}}}\nolimits(I_{i-j}(2s))_{i,j=0,\ldots,r-1},$
(2.60)

where the $I_{n}$ are modified Bessel functions of the first kind (see
(2.34)). This formula goes back to [Ges90], and was the starting point of the
analysis in [BDJ99].

The Toeplitz determinants $D_{n}(\varphi_{s})$ are intimately related to the
orthogonal polynomials on the unit circle associated with the weight
$\varphi_{s}$. In the Szegő three term recurrence for such orthogonal
polynomials on the circle with respect to a general weight [Sze75], only one
coefficient $\alpha_{n}$ depends on the choice of weight. Thus the
$\alpha_{n}$’s, known as the _Verblunsky coefficients_ , “encode” the weight,
and can be thought of as the analogue of reflection coefficients in this
context. In the case of our weight $\varphi_{s}$ they are real, and related to
$F_{r}$ through

$\alpha_{r}^{2}=1-F_{r}F_{r+2}/{F_{r-1}^{2}}.$ (2.61)

They are known (see e.g. [VA18, Sec. 3]) to satisfy the discrete Painlevé II
equation

$-s(1-\alpha_{r}^{2})(\alpha_{r+1}+\alpha_{r-1})=(r+1)\alpha_{r}$ (2.62)

(see also [Bor03]) as well as the Ablowitz-Ladik lattice

$\partial_{s}\alpha_{r}=(1-\alpha_{r}^{2})(\alpha_{r+1}-\alpha_{r-1})$ (2.63)

(see also [AM01]). The Ablowitz-Ladik hierarchy is a reduction of the 2D Toda
hierarchy [Tak18]. However, it is not clear if (2.63) can be obtained directly
from our equation with the given symmetry,

$\tfrac{1}{4}(\partial_{s}^{2}+\tfrac{1}{s}\partial_{s})\mathop{\mathop{\text{{log}}}}\nolimits
F_{r}=\tfrac{F_{r+1}F_{r-1}}{F_{r}^{2}}-1.$ (2.64)

We mention also that in Thm. 7.12 of [BDS16] it is shown, using the Riemann-
Hilbert problem for the Toeplitz determinant $D_{n}(\varphi_{s})$, that the
functions
$g_{m}=-\mathop{\mathop{\text{{log}}}}\nolimits(\tfrac{F_{-2m+1}}{F_{-2m}}(1-\alpha_{-2m-1}))$,
defined for $m\leq 0$, satisfy the Toda lattice (2.25).

####  2.5.2. Flat initial data

Consider now PNG with flat initial condition $h_{0}\equiv 0$. For fixed $t=1$,
this case was analyzed in [BFS08]. Clearly it is enough to consider $r_{i}\geq
0$ for each $i$. In view of (2.31), this means that we are interested in the
kernel evaluated at values of $u_{i}>0$ for each $i$. In that region, the
scattering kernel $e^{-2t\nabla-
x_{i}\Delta}\mathscr{T}^{0}_{x_{i}-t,x_{j}+t}e^{2t\nabla+x_{j}\Delta}=e^{-2t\nabla+t\Delta}P^{\operatorname{\text{{hit}}}(0)}_{x_{i}-t,x_{j}+t}e^{2t\nabla-t\Delta}$
equals

$\mathbf{1}_{2t\geq
x_{i}-x_{j}}\big{(}e^{-2t\nabla+t\Delta}\bar{\chi}_{0}e^{2t\nabla+(x_{j}-x_{i}+t)\Delta}+e^{-2t\nabla+t\Delta}\chi_{0}P^{\operatorname{\text{{hit}}}(0)}_{x_{i}-t,x_{j}+t}e^{2t\nabla-t\Delta}\big{)}.$
(2.65)

A simple computation using (2.32) shows that the first term inside the
parenthesis can be represented by the contour integral
$\frac{e^{2(x_{i}-x_{j})}}{(2\pi{\rm i})^{2}}\oint_{\hskip
0.70004pt\gamma_{0}}\mathrm{d}z\oint_{\hskip
0.70004pt\gamma_{0}}\mathrm{d}w\frac{z^{u_{i}-1}}{w^{u_{j}}(w-z)}e^{2t(w-z)+(x_{j}-x_{i})(w+w^{-1})}$
with the contours chosen so that $|w|>|z|$. But in the case of interest,
$u_{i}>0$, the integrand is analytic in $z$ and hence this term vanishes. The
second term on the right hand side of (2.65) is computed using the reflection
principle for $\mathsf{N}$: assuming $x_{j}-x_{i}+2t\geq 0$ and $u>0$, and
writing $\underline{x}=x_{i}-t$, $\overline{x}=x_{j}+t$,

$P^{\operatorname{\text{{hit}}}(0)}_{\underline{x},\overline{x}}(u,v)=e^{(\overline{x}-\underline{x})\Delta}(u,-v)\mathbf{1}_{v>0}+e^{(\overline{x}-\underline{x})\Delta}(u,v)\mathbf{1}_{v\leq
0}.$

$P^{\operatorname{\text{{hit}}}(0)}_{\underline{x},\overline{x}}e^{2t\nabla-t\Delta}(u,v)$
equals the sum of two integrals,
$\frac{e^{2(\overline{x}-\underline{x})}}{(2\pi{\rm i})^{2}}\oint_{\hskip
0.70004pt\gamma_{0}}\mathrm{d}z\oint_{\hskip
0.70004pt\gamma_{0}^{\prime}}\mathrm{d}w\,\frac{z^{u}}{w^{v}(1-wz)}e^{(\overline{x}-\underline{x})(z+\frac{1}{z})-\frac{2t}{w}}$
with contours chosen so that $|wz|<1$, and
$\frac{e^{2(\overline{x}-\underline{x})}}{(2\pi{\rm i})^{2}}\oint_{\hskip
0.70004pt\gamma_{0}}\mathrm{d}z\oint_{\hskip
0.70004pt\gamma_{0}}^{\prime}\mathrm{d}w\frac{z^{u-1}}{w^{v}(w-z)}e^{(\overline{x}-\underline{x})(z+\frac{1}{z})-\frac{2t}{w}}$
with contours chosen so that $|w|>|z|$. The two integrals can be brought
together if we choose contours with $|z|<1$ and $|z|<|w|<1/|z|$, and
simplifying we get

$\textstyle
P^{\operatorname{\text{{hit}}}(0)}_{\underline{x},\overline{x}}e^{2t\nabla-t\Delta}(u,v)=\frac{e^{2(\overline{x}-\underline{x})}}{(2\pi{\rm
i})^{2}}\oint_{\hskip
0.70004pt\gamma_{0}}\mathrm{d}z\,\frac{1-z^{2}}{z^{1-u}}e^{(\overline{x}-\underline{x})(z+z^{-1})}\oint_{\hskip
0.70004pt\gamma_{0}^{\prime}}\mathrm{d}w\,\frac{1}{w^{v}(w-z)(1-zw)}e^{-2tw^{-1}}.$
(2.66)

Expanding the $w$ contour to a contour $\gamma_{0}^{\prime\prime}$ with
$|w|>1/|z|$ and picking up the residue at $w=z^{-1}$, the $w$ integral can be
written as $\frac{1}{2\pi{\rm
i}}\oint_{\gamma_{0}^{\prime\prime}}\frac{\mathrm{d}w}{w^{v}(w-z)(1-zw)}e^{-2tw^{-1}}-\frac{z^{v}}{z^{2}-1}e^{-2tz}$,
and changing variables $w\longmapsto 1/w$, this expression becomes
$\frac{1}{2\pi{\rm
i}}\oint_{\gamma_{0}^{\prime\prime\prime}}\mathrm{d}w\,\frac{w^{v}}{(w-z)(1-zw)}e^{-2tw}-\frac{z^{v}}{z^{2}-1}e^{-2tz}$,
where the contour includes only the singularity at $0$. We are interested only
in the case $v>0$, for which the $w$ integral vanishes, and thus using this
for such $v$ in the above expression we get that (2.66) is given by
$e^{2(\overline{x}-\underline{x})}\frac{1}{2\pi{\rm
i}}\oint_{\gamma_{0}}\frac{\mathrm{d}z}{z^{u+v+1}}e^{2tz+(x_{j}-x_{i})(z+z^{-1})}$,
where in the last integral we changed $z$ to $z^{-1}$. Continuing the
computation in the same way we get, for $u_{i},u_{j}>0$,
$e^{-2t\nabla+t\Delta}\chi_{0}P^{\operatorname{\text{{hit}}}(0)}_{\underline{x},\overline{x}}e^{2t\nabla-t\Delta}(u_{i},u_{j})=e^{2(x_{i}-x_{j})}\frac{1}{2\pi{\rm
i}}\oint_{\gamma_{0}}\frac{\mathrm{d}w}{w^{u_{i}}}\frac{1}{2\pi{\rm
i}}\oint_{\gamma_{0}}\frac{\mathrm{d}z}{z^{u_{j}+1}(z-w)}e^{2t(z-w^{-1})+(x_{j}-x_{i})(z+z^{-1})}$
with contours chosen so that $|z|>|w|$. Now we enlarge the $w$ contour to a
circle of radius $r$, $r>|z|$. With this new contour the integral is clearly
bounded by a constant times $r\hskip 0.50003pte^{-2t/r}/r^{u_{i}+1}$, which
goes to $0$ as $r\to\infty$. Hence we are only left with the residue at $w=z$,
which finally yields, for $u_{i},u_{j}>0$,

$\displaystyle\textstyle K^{\text{{ext}}}(x_{i},u_{i};x_{j},u_{j})$
$\displaystyle\textstyle=-e^{2(x_{i}-x_{j})}I_{|u_{i}-u_{j}|}(2(x_{j}-x_{i}))\mathbf{1}_{x_{i}<x_{j}}$
(2.67) $\displaystyle\textstyle\qquad+\mathbf{1}_{2t\geq
x_{i}-x_{j}}e^{2(x_{i}-x_{j})}\frac{1}{2\pi{\rm
i}}\oint_{\gamma}\frac{\mathrm{d}w}{w^{u_{i}+u_{j}+1}}e^{(x_{j}-x_{i})(w+w^{-1})}e^{2t(w-w^{-1})}.$
(2.68)

When $x_{j}-x_{i}\leq 2t$, the kernel coincides with the one obtained in
[BFS08, Prop. 4]. When $x_{j}-x_{i}<-2t$ this does not coincide exactly with
the expression in that result, which has an absolute value in the indicator
function, but in terms of the determinant it does not make any difference.

In the one-point case the flat kernel is independent of the spatial variable
$x$, and simplifies to

$\textstyle K^{\text{{ext}}}(u,v)=\frac{1}{2\pi{\rm
i}}\oint_{\gamma}\frac{\mathrm{d}w}{w^{u+v+1}}e^{2t(w-w^{-1})}=J_{u+v}(4t).$

The kernel is Hankel, and in fact the associated Fredholm determinant is known
to be equal to a Toeplitz-plus-Hankel determinant [BR01],

$F_{r}(s)\coloneqq
F(s,0,r)=e^{-s^{2}}\mathop{\mathop{\text{{det}}}}\nolimits(I_{i-j}(2s)-I_{i+j+2}(2s))_{i,j=0,\ldots,r-1}$
(2.69)

where $I_{n}$ are modified Bessel functions of the first kind (see (2.34)). By
analogy with the narrow wedge case, one expects a connection with discrete
Painlevé. Such a connection has appeared implicitly in the literature and we
include it here for completeness; we thank Jinho Baik for explaining it to us.

From Eqn. 4.18 in [BR01a], the determinant on the right hand side of (2.69)
can be written as $\prod_{j\geq r}N_{2j+2}(s)^{-1}(1-\pi_{2j+2}(0;s))$ where
$\pi_{k}(\zeta;s)$ is the monic orthogonal polynomial of degree $k$ on the
unit circle with respect to the weight $\varphi_{s}(\zeta)$ defined below
(2.59) and $N_{k}(s)$ is its norm, defined in [BR01a, Eq. 4.10]. Writing
$b_{k}(s)=-\pi_{k}(0;s)$ and $a_{k}(s)=-N_{k-1}(s)^{-1}$, we get from this
that

${F_{r}(s)}/{F_{r+1}(s)}=-a_{2r+3}(s)(1+b_{2r+2}(s)).$

The relation to Painlevé is provided by Eqn. 1.6 and Thm. 5.1 of [Bai03],
which show that the $b_{k}$’s satisfy the discrete Painlevé II equation (2.62)
with $\alpha_{r}=b_{r-1}$ while the $a_{k}$’s can be recovered from the
$b_{k}$’s through $a_{k}=(1-b_{k}^{2})a_{k+1}$. Using arguments similar to
those in [BDS16, Sec. 7.3], it should also be possible to derive the Toda
lattice (2.25) for flat PNG from these formulas.

##  3\. Derivation of the non-Abelian Toda equation

Our goal in this section is to prove Thm. 2.2. The proof is divided in three
parts. In Sec. 3.1, we abstract away details particular to PNG and prove the
non-Abelian Toda equations for a general class of matrix kernels $K_{r}$
satisfying certain structural conditions. In Sec. 3.2 we explain how the
general result is used to derive the non-Abelian Toda equations for a
simplified version of the PNG kernel where the dependence on the interval
$[x_{i}-t,x_{j}+t]$ in the scattering transform
$\mathscr{T}^{h_{0}}_{x_{i}-t,x_{j}+t}$ in (2.12) is dropped. Finally in Sec.
3.3 we show that dropping this dependence does not matter in the region above
$r_{0}(t,x)$.

###  3.1. General kernel

######  Theorem 3.1.

Fix two integers $\underaccent{\bar}{r}<\bar{r}-1$ and some open domain
$O\subseteq\mathbb{R}^{2}$. Let $K_{\eta,\zeta,r}$ be a family of kernels
defined for $r\in\\{\underaccent{\bar}{r},\dotsc,\bar{r}\\}$ and
$(\eta,\zeta)\in O$, which acts on $\oplus_{n}\ell^{2}(\mathbb{Z}_{>0})$, the
$n$-fold direct sum of $\ell^{2}(\mathbb{Z}_{>0})$. Write
$K_{r}=K_{\eta,\zeta,r}$ and suppose that its matrix kernel $K_{r}(u,v)$
satisfies, for all $u,v>0$,

  1. (1)

$K_{r+1}(u,v)=K_{r}(u+1,v+1)$,
$r\in\\{\underaccent{\bar}{r},\dotsc,\bar{r}-1\\}$,

  2. (2)

$\partial_{\eta}K_{r}(u,v)=K_{r-1}(u+1,v)-K_{r}(u+1,v)$,
$r\in\\{\underaccent{\bar}{r}+1,\dotsc,\bar{r}\\}$,

  3. (3)

$\partial_{\zeta}K_{r}(u,v)=K_{r-1}(u,v+1)-K_{r}(u,v+1)$,
$r\in\\{\underaccent{\bar}{r}+1,\dotsc,\bar{r}\\}$.

Assume that, for all $r\in\\{\underaccent{\bar}{r},\dotsc,\bar{r}\\}$ and
fixed $(\eta,\zeta)\in O$, $K_{r}$ is trace class as an operator acting on
$\oplus_{n}\ell^{2}(\mathbb{Z}_{>0})$, and that for such $r$ the derivatives
in (2) and (3) above exist in trace norm. Assume furthermore that $I-K_{r}$ is
invertible for all $r\in\\{\underaccent{\bar}{r},\dotsc,\bar{r}\\}$ and define
the invertible $n\times n$ matrix

$\displaystyle Q_{r}=(I-K_{r})^{-1}(1,1).$ (3.1)

Then for $r\in\\{\underaccent{\bar}{r}+1,\dotsc,\bar{r}-1\\}$ and
$(\eta,\zeta)\in O$, $Q_{r}$ satisfies the non-Abelian 2D Toda equation

$\partial_{\zeta}(\partial_{\eta}Q_{r}Q_{r}^{-1})+Q_{r}Q_{r-1}^{-1}-Q_{r+1}Q_{r}^{-1}=0.$
(3.2)

We stress that, although we used the same notation, $K_{r}$ and $Q_{r}$ in
this result are general and not necessarily related to PNG (in particular, the
index $r$ in these objects denotes an integer rather than element of
$\mathbb{Z}^{n}$).

Before starting the proof, we need to introduce some notations. For
$r\in\\{\underaccent{\bar}{r},\dotsc,\bar{r}\\}$ let

$R_{r}=(I-K_{r})^{-1},\qquad\text{{so}}\quad Q_{r}=R_{r}(1,1),$

and define, for $f\in\ell^{2}(\mathbb{Z}_{>0})$,

$\sigma f(u)=f(u+1),\qquad\sigma^{*}f(u)=f(u-1)\mathbf{1}_{u>1},$

which have explicit kernels given by $\sigma(u,v)=\mathbf{1}_{v=u+1}$ and
$\sigma^{*}(u,v)=\mathbf{1}_{v=u-1>0}$.

######  Remark 3.2.

In [BP87] it is shown, in this language, that if one has a scalar kernel
$K_{r}(t)$ satisfying $K_{r+1}=\sigma K_{r}=K_{r}\sigma^{*}$ and
$\partial_{t}K_{r}=K_{r+1}-K_{r-1}$, then
$p_{r}=\mathop{\mathop{\text{{det}}}}\nolimits(I+K_{r})$ satisfies the one-
sided bilinear Toda lattice equation
$\dot{p}_{r+1}p_{r}-p_{r+1}\dot{p}_{r}=p_{r+2}p_{r-1}-p_{r+1}p_{r}$, which can
be transformed into the classic Toda lattice (2.25) (see the introduction of
[BP87]).

We will use the same notation $\sigma$ for the diagonal matrix kernel with
$\sigma$ in each diagonal entry, which acts on
$f\in\oplus_{n}\ell^{2}(\mathbb{Z}_{>0})$ as above. Then (1) in the
assumptions of the theorem becomes

$K_{r+1}=\sigma K_{r}\sigma^{*},$ (3.3)

while

$\sigma\sigma^{*}=I\qquad\text{and}\qquad P\coloneqq
I-\sigma^{*}\sigma\quad\text{{is rank $n$}}.$

$P$ is a diagonal matrix kernel with diagonal entries $\Pi$ which have an
explicit kernel given by

$\Pi(u,v)=\mathbf{1}_{u=v=1}.$ (3.4)

On the other hand, (2) and (3) in the assumptions of the theorem can be
expressed in terms of $\sigma$ and $\sigma^{*}$ as

$\partial_{\eta}K=\sigma(K_{r-1}-K_{r}),\qquad\text{and}\qquad\partial_{\zeta}K_{r}=(K_{r-1}-K_{r})\sigma^{*}.$
(3.5)

By taking the $\eta$ derivative of the first or the $\zeta$ derivative of the
second, we also have

$\partial_{\eta}\partial_{\zeta}K_{r}=K_{r+1}-2K_{r}+K_{r-1}.$ (3.6)

The proof of Thm. 3.1 is based on two identities which we collect in the next
result:

######  Lemma 3.3.

  1. (i)

$(I+PR_{r}K_{r}P)^{-1}=(I-P)+(I-K_{r})(I+\sigma^{*}R_{r+1}\sigma K_{r})P$.

  2. (ii)

$(I+PR_{r}K_{r}P)(I+PR_{r-1}K_{r-1}P)^{-1}=I+PR_{r}(K_{r}-K_{r-1})(I+\sigma^{*}R_{r}\sigma
K_{r-1})P$.

###### Proof.

The first step of the proof is to show the following two simple identities:

$\displaystyle(I+PR_{r}K_{r})^{-1}$
$\displaystyle=(I-K_{r})(I+\sigma^{*}R_{r+1}\sigma K_{r}),$ (3.7)
$\displaystyle(I+PR_{r}K_{r}P)^{-1}$
$\displaystyle=(I-P)+(I+PR_{r}K_{r})^{-1}P.$ (3.8)

For the first one, use the fact that $R_{r}K_{r}=K_{r}R_{r}=R_{r}-I$ to write
$I+PR_{r}K_{r}=I-P+PR_{r}=(I-\sigma^{*}\sigma K_{r})R_{r}$, and then multiply
by $(I-K_{r})(I+\sigma^{*}R_{r+1}\sigma K_{r})$ to get

$\displaystyle(I+PR_{r}K_{r})(I-K_{r})(I+\sigma^{*}R_{r+1}\sigma
K_{r})=(I-\sigma^{*}\sigma K_{r})(I+\sigma^{*}R_{r+1}\sigma K_{r})$ (3.9)
$\displaystyle\hskip 57.81621pt=I-\sigma^{*}\sigma
K_{r}+\sigma^{*}R_{r+1}\sigma K_{r}-\sigma^{*}\sigma
K_{r}\sigma^{*}R_{r+1}\sigma K_{r}=I,$ (3.10)

where in the last equality we used $\sigma K_{r}\sigma^{*}=K_{r+1}$ together
with $R_{r}K_{r}=K_{r}R_{r}=R_{r}-I$ again.

To get (3.8), decompose $I+PR_{r}K_{r}P$ as $(I+PR_{r}K_{r})-PR_{r}K_{r}(I-P)$
and then multiply by $(I-P)+(I+PR_{r}K_{r})^{-1}P$ on the right. The result
can be simplified using the fact that $P$ is a projection and we get

$I-PR_{r}K_{r}(I-P)(I+PR_{r}K_{r})^{-1}P=I,$

the last equality following from the easily checked fact that
$(I+PR_{r}K_{r})^{-1}P=P(I+R_{r}K_{r}P)^{-1}$.

The identity in (i) now follows directly from (3.7) and (3.8). For (ii), write
the left hand side as

$I+P(R_{r}K_{r}-R_{r-1}K_{r-1})P(I+PR_{r-1}K_{r-1}P)^{-1}$

and then use (i) to rewrite this as

$I+P(R_{r}K_{r}-R_{r-1}K_{r-1})P(I-K_{r-1})(I+\sigma^{*}R_{r}\sigma
K_{r-1})P,$ (3.11)

where we used again the fact that $P$ is a projection. Now use (3.3) to write

$\displaystyle(I-P)(I-K_{r-1})(I+\sigma^{*}R_{r}\sigma K_{r-1})P$
$\displaystyle=\sigma^{*}\sigma(I-K_{r-1})(I+\sigma^{*}R_{r}\sigma K_{r-1})P$
(3.12) $\displaystyle=(\sigma^{*}\sigma(I-K_{r-1})+\sigma^{*}\sigma
K_{r-1})P=\sigma^{*}\sigma P=0,$ (3.13)

and use this to rewrite (3.11) further as

$I+P(R_{r}K_{r}-R_{r-1}K_{r-1})(I-K_{r-1})(I+\sigma^{*}R_{r}\sigma K_{r-1})P.$
(3.14)

Since $R_{r-1}K_{r-1}=K_{r-1}R_{r-1}$ and $R_{r}K_{r}=R_{r}-I$ we have

$(R_{r}K_{r}-R_{r-1}K_{r-1})(I-K_{r-1})=R_{r}K_{r}(I-K_{r-1})-K_{r-1}=R_{r}(K_{r}-K_{r-1})$

and using this in (3.14) shows that it is equal to the desired right hand
side. ∎

###### Proof of Thm. 3.1.

Let

$U_{r}=Q_{r}Q_{r-1}^{-1},\qquad V_{r}=-\partial_{\eta}Q_{r}Q_{r}^{-1}.$ (3.15)

Then (3.2) can be rewritten as

$\partial_{\zeta}V_{r}+U_{r+1}-U_{r}=0.$ (3.16)

Our goal then will be to prove this last equation.

In order to achieve this, it is convenient to lift (3.16) to an equivalent
equation for operators acting on $\oplus_{n}\ell^{2}(\mathbb{Z}_{>0})$. To
this end, we introduce the operators

$\mathcal{X}_{r}=PR_{r}K_{r}P,\qquad\mathcal{Y}_{r}=I+\mathcal{X}_{r}=I+PR_{r}K_{r}P.$

From the previous lemma, $\mathcal{Y}_{r}$ is invertible. On the other hand,
since $K_{r}$ is differentiable in $\eta$ and $\zeta$ in trace norm, its
inverse $R_{r}$ is also differentiable, and since $P$ is finite rank then
$\mathcal{X}_{r}$, and thus also $\mathcal{Y}_{r}$ and $\mathcal{Y}_{r}^{-1}$,
are differentiable in $\eta$ and $\zeta$ in trace norm. In particular, we may
introduce the following analogs of $U_{r}$ and $V_{r}$ in terms of the
operator $\mathcal{Y}_{r}$:

$\displaystyle\mathcal{U}_{r}=\mathcal{Y}_{r}\mathcal{Y}_{r-1}^{-1},\qquad\mathcal{V}_{r}$
$\displaystyle=-\partial_{\eta}\mathcal{Y}_{r}\mathcal{Y}_{r}^{-1}.$ (3.17)

We claim that (3.16) is equivalent to the same equation at the level of
$\mathcal{U}_{r}$ and $\mathcal{V}_{r}$:

$\partial_{\zeta}\mathcal{V}_{r}+\mathcal{U}_{r+1}-\mathcal{U}_{r}=0.$ (3.18)

To see this note first that, in view of (3.4), $P$ is a projection onto an
$n$-dimensional subspace of $\oplus_{n}\ell^{2}(\mathbb{Z}_{>0})$, call it
$Z$. Then we may think of $\mathcal{Y}_{r}$ as acting on $Z\oplus Z^{\perp}$
as a block matrix operator
$\mathcal{Y}_{r}=\left[\begin{smallmatrix}I_{Z}+\mathcal{X}_{r}&0\\\
0&I_{Z^{\perp}}\end{smallmatrix}\right]$, and we have
$\mathcal{Y}_{r}^{-1}=\left[\begin{smallmatrix}(I_{Z}+\mathcal{X}_{r})^{-1}&0\\\
0&I_{Z^{\perp}}\end{smallmatrix}\right]$. In this way we thus have

$\mathcal{U}_{r}=\begin{bmatrix}(I_{Z}+\mathcal{X}_{r})(I_{Z}+\mathcal{X}_{r-1})^{-1}&0\\\
0&I_{Z^{\perp}}\end{bmatrix},\qquad\mathcal{V}_{r}=\begin{bmatrix}\partial_{\eta}\mathcal{X}_{r}(I_{Z}+\mathcal{X}_{r-1})^{-1}&0\\\
0&I_{Z^{\perp}}\end{bmatrix}.$

In particular, (3.18) is completely trivial for the second row in this block
representation. For the first row we simply note that $Z$ is isomorphic to
$\mathbb{R}^{n}$ and the matrix of the linear transformation
$I_{Z}+\mathcal{X}_{r}$ on $\mathbb{R}^{n}$ is $Q_{r}$, so the equation is
equivalent to (3.16).

Hence our goal is to show (3.18). Using $R_{r}K_{r}=K_{r}R_{r}=R_{r}-I$ we
have

$\displaystyle\partial_{\eta}\hskip
0.50003pt\mathcal{Y}_{r}=P\partial_{\eta}R_{r}P=PR_{r}\partial_{\eta}K_{r}R_{r}P,\qquad\partial_{\zeta}\hskip
0.50003pt\mathcal{Y}_{r}=P\partial_{\zeta}R_{r}P=PR_{r}\partial_{\zeta}K_{r}R_{r}P,$
(3.19) $\displaystyle\partial_{\zeta}\partial_{\eta}\hskip
0.50003pt\mathcal{Y}_{r}=P\partial_{\zeta}\partial_{\eta}R_{r}P=PR_{r}(\partial_{\zeta}\partial_{\eta}K_{r}+\partial_{\zeta}K_{r}R_{r}\partial_{\eta}K_{r}+\partial_{\eta}K_{r}R_{r}\partial_{\zeta}K_{r})R_{r}P,$
(3.20)

so

$\displaystyle\partial_{\zeta}\mathcal{V}_{r}$
$\displaystyle=-\partial_{\zeta}\partial_{\eta}\mathcal{Y}_{r}\mathcal{Y}_{r}^{-1}+\partial_{\eta}\mathcal{Y}_{r}\mathcal{Y}_{r}^{-1}\partial_{\zeta}\mathcal{Y}_{r}\mathcal{Y}_{r}^{-1}$
(3.21)
$\displaystyle=-PR_{r}\Big{[}\partial_{\zeta}\partial_{\eta}K_{r}+\partial_{\zeta}K_{r}R_{r}\partial_{\eta}K_{r}+\partial_{\eta}K_{r}R_{r}\partial_{\zeta}K_{r}-\partial_{\eta}K_{r}R_{r}P\mathcal{Y}_{r}^{-1}PR_{r}\partial_{\zeta}K_{r}\Big{]}R_{r}P\mathcal{Y}_{r}^{-1}.$
(3.22)

Now from (i) of Lem. 3.3 we have
$R_{r}P\mathcal{Y}_{r}^{-1}=R_{r}P(I-K_{r})(I+\sigma^{*}R_{r+1}\sigma
K_{r})P$, while

$\displaystyle R_{r}P(I-K_{r})(I+\sigma^{*}R_{r+1}\sigma K_{r})$
$\displaystyle=I+\sigma^{*}R_{r+1}\sigma
K_{r}-R_{r}\sigma^{*}\sigma(I-K_{r})(I+\sigma^{*}R_{r+1}\sigma K_{r})$ (3.23)
$\displaystyle=I+\sigma^{*}R_{r+1}\sigma
K_{r}-R_{r}(\sigma^{*}\sigma(I-K_{r})+\sigma^{*}(I-K_{r+1})R_{r+1}\sigma
K_{r})$ (3.24) $\displaystyle=I+\sigma^{*}R_{r+1}\sigma
K_{r}-R_{r}\sigma^{*}\sigma=I+\sigma^{*}R_{r+1}\sigma K_{r}-R_{r}(I-P),$
(3.25)

where in the second equality we simplified using (3.3), and therefore
$R_{r}P\mathcal{Y}_{r}^{-1}=(I+\sigma^{*}R_{r+1}\sigma K_{r})P$. Using this in
(3.21) gives

$\partial_{\zeta}\mathcal{V}_{r}=-PR_{r}\Omega(I+\sigma^{*}R_{r+1}\sigma
K_{r})P,$ (3.26)

where $\Omega$ equals (using again the formula for
$R_{r}P\mathcal{Y}_{r}^{-1}$)

$\displaystyle\partial_{\zeta}\partial_{\eta}K_{r}+\partial_{\zeta}K_{r}R_{r}\partial_{\eta}K_{r}+\partial_{\eta}K_{r}R_{r}\partial_{\zeta}K_{r}-\partial_{\eta}K_{r}R_{r}P\mathcal{Y}_{r}^{-1}PR_{r}\partial_{\zeta}K_{r}$
(3.27)
$\displaystyle\;\;=(K_{r+1}-2K_{r}+K_{r-1})+(K_{r}-K_{r-1})\sigma^{*}R_{r}\sigma(K_{r}-K_{r-1})$
(3.28)
$\displaystyle\quad+\sigma(K_{r}-K_{r-1})R_{r}(K_{r}-K_{r-1})\sigma^{*}-\sigma(K_{r}-K_{r-1})(I+\sigma^{*}R_{r+1}\sigma
K_{r})PR_{r}(K_{r}-K_{r-1})\sigma^{*}$ (3.29)
$\displaystyle\;\;=(K_{r+1}-2K_{r}+K_{r-1})+(K_{r}-K_{r-1})\sigma^{*}R_{r}\sigma(K_{r}-K_{r-1})$
(3.30) $\displaystyle\quad-\sigma(K_{r}-K_{r-1})\sigma^{*}R_{r+1}\sigma
K_{r}PR_{r}(K_{r}-K_{r-1})\sigma^{*}+\sigma(K_{r}-K_{r-1})\sigma^{*}\sigma
R_{r}(K_{r}-K_{r-1})\sigma^{*}$ (3.31)
$\displaystyle\;\;=(K_{r+1}\\!-\\!2K_{r}\\!+\\!K_{r-1})+(K_{r}\\!-\\!K_{r-1})\sigma^{*}R_{r}\sigma(K_{r}\\!-\\!K_{r-1})+(K_{r+1}\\!-\\!K_{r})R_{r+1}(K_{r+1}\\!-\\!K_{r}),$
(3.32)

and where we used (3.3) again together with the identity $R_{r+1}\sigma
K_{r}P-\sigma=R_{r+1}\sigma
K_{r}(I-\sigma^{*}\sigma)-\sigma=-R_{r+1}\sigma(I-K_{r})$. On the other hand,
(ii) of Lem. 3.3 gives

$\mathcal{U}_{r+1}-\mathcal{U}_{r}=PR_{r+1}(K_{r+1}-K_{r})(I+\sigma^{*}R_{r+1}\sigma
K_{r})P-PR_{r}(K_{r}-K_{r-1})(I+\sigma^{*}R_{r}\sigma K_{r-1})P,$

and using $R_{r}(K_{r+1}-K_{r})R_{r+1}=R_{r+1}-R_{r}$ we have
$R_{r+1}(K_{r+1}-K_{r})=R_{r}(K_{r+1}-K_{r})(I+R_{r+1}(K_{r+1}-K_{r}))$. Then

$\mathcal{U}_{r+1}-\mathcal{U}_{r}=PR_{r}(K_{r+1}-K_{r})(I+\sigma^{*}R_{r+1}\sigma
K_{r})PPR_{r}(K_{r}-K_{r-1})(I+\sigma^{*}R_{r}\sigma K_{r-1})P\\\
\qquad+PR_{r}(K_{r+1}-K_{r})R_{r+1}(K_{r+1}-K_{r})(I+\sigma^{*}R_{r+1}\sigma
K_{r})P.$ (3.33)

Comparing with (3.26), the last term on the right hand cancels with the piece
coming from the third term in (3.27), so we get

$\displaystyle\partial_{\zeta}\mathcal{V}_{r}+\mathcal{U}_{r+1}-\mathcal{U}_{r}$
$\displaystyle=PR_{r}(K_{r}-K_{r-1})(I+\sigma^{*}R_{r+1}\sigma K_{r})P$ (3.34)
$\displaystyle\quad\qquad-
PR_{r}(K_{r}-K_{r-1})\sigma^{*}R_{r}\sigma(K_{r}-K_{r-1})(I+\sigma^{*}R_{r+1}\sigma
K_{r})P$ (3.35) $\displaystyle\quad\qquad-
PR_{r}(K_{r}-K_{r-1})(I+\sigma^{*}R_{r}\sigma K_{r-1})P$ (3.36)
$\displaystyle=PR_{r}(K_{r}-K_{r-1})\sigma^{*}(R_{r+1}\sigma K_{r}-R_{r}\sigma
K_{r-1})P$ (3.37) $\displaystyle\quad\qquad-
PR_{r}(K_{r}-K_{r-1})\sigma^{*}R_{r}\sigma(K_{r}-K_{r-1})(I+\sigma^{*}R_{r+1}\sigma
K_{r})P$ (3.38)
$\displaystyle=PR_{r}(K_{r}-K_{r-1})\sigma^{*}[R_{r+1}-R_{r}-R_{r}K_{r+1}R_{r+1}+R_{r}K_{r}R_{r+1}]\sigma
K_{r}P.$ (3.39)

The term in brackets vanishes as $R_{r}K_{r}=K_{r}R_{r}=R_{r}-I$, yielding
(3.18) and completing the proof. ∎

###  3.2. PNG kernel

Recall our definition $(K_{r})_{ij}(u,v)=K^{\text{{ext}}}(u+r_{i},v+r_{j})$.
In Sec. 2 we used $r$ to denote the vector
$(r_{1},\dotsc,r_{n})\in\mathbb{Z}^{n}$. Here it will be more convenient to
regard $r_{1},\dotsc,r_{n}$ as fixed parameters and $r$ as an auxiliary
variable taking values in $\mathbb{Z}$, and redefine

$(K_{r})_{ij}(u,v)=K^{\text{{ext}}}(u+r_{i}+r,v+r_{j}+r).$ (3.40)

This notation is consistent with our usage of $K_{r\pm 1}$ in Sec. 2, and then
(2.18) coincides with (3.16) with $r=0$. Therefore, to show that (2.18) holds
at the given choice of parameters $(r_{1},\dotsc,r_{n})$ satisfying
$r_{i}>r_{0}(t,x_{i})$ for each $i$, it is enough to show that $K_{r}$
satisfies the assumptions of Thm. 3.1 with $\underaccent{\bar}{r}=-1$ and
$\bar{r}=1$.

That $K_{r}$ satisfies (1) of Thm. 3.1 is straightforward by definition, so we
turn to assumptions (2) and (3) of that result. We fix $i$ and $j$ and focus
on the $i,j$ entry of $K_{r}$, which we denote by $K^{ij}_{r}$. We need to
check that (2) and (3) hold for
$r\in\\{\underaccent{\bar}{r}+1,\dotsc,\bar{r}\\}$, which in our setting means
$r\in\\{0,1\\}$. From (2.12) we have that $\partial_{t}K^{ij}_{r}(u,v)$ equals
$K^{ij}_{r}(u-1,v)-K^{ij}_{r}(u+1,v)+K^{ij}_{r}(u,v-1)-K^{ij}_{r}(u,v+1)$ plus
a term coming from differentiating the scattering operator, and
$\partial_{\overline{x}}K^{ij}_{r}(u,v)$ equals
$-K^{ij}_{r}(u-1,v)-K^{ij}_{r}(u+1,v)+K^{ij}_{r}(u,v-1)+K^{ij}_{r}(u,v+1)$
plus a similar term. Using this we get, making now the scattering terms
explicit,

$\displaystyle\partial_{\eta}K^{ij}_{r}(u,v)$
$\displaystyle=K^{ij}_{r}(u,v-1)-K^{ij}_{r}(u+1,v)+W^{(\eta)}_{t,x_{i},x_{j}}(u+r_{i}+r,v+r_{j}+r),$
(3.41) $\displaystyle\partial_{\zeta}K^{ij}_{r}(u,v)$
$\displaystyle=K^{ij}_{r}(u,v+1)-K^{ij}_{r}(u-1,v)+W^{(\zeta)}_{t,x_{i},x_{j}}(u+r_{i}+r,v+r_{j}+r)$

with

$W^{(\zeta)}_{t,x,x^{\prime}}=e^{-2t\nabla-x\Delta}\Big{(}\partial_{a}\mathscr{T}^{h_{0}}_{a,b}\big{|}_{\begin{subarray}{c}a=x-t\\\
b=x^{\prime}+t\end{subarray}}\Big{)}e^{2t\nabla+x^{\prime}\Delta},\quad
W^{(\eta)}_{t,x,x^{\prime}}=e^{-2t\nabla-x\Delta}\Big{(}\partial_{b}\mathscr{T}^{h_{0}}_{a,b}\big{|}_{\begin{subarray}{c}a=x-t\\\
b=x^{\prime}+t\end{subarray}}\Big{)}e^{2t\nabla+x^{\prime}\Delta}.$ (3.42)

Hence (2) and (3) will follow (since $K_{r}(u,v-1)=K_{r-1}(u,v)$,
$K_{r}(u-1,v)=K_{r-1}(u,v+1)$) if we show that the
$W^{(\eta)}_{t,x_{i},x_{j}}$ and $W^{(\zeta)}_{t,x_{i},x_{j}}$ terms on the
right hand side of (3.41) vanish for $u,v>0$ or, what is the same (since we
are assuming $r_{i}>r_{0}(t,x_{i})$, $r_{j}>r_{0}(t,x_{j})$, and
$r\in\\{0,1\\}$), that

$W^{(\eta)}_{t,x_{i},x_{j}}(u,v)=W^{(\zeta)}_{t,x_{i},x_{j}}(u,v)=0\quad\forall
u>r_{0}(t,x_{i})+1,\,v>r_{0}(t,x_{j})+1.$ (3.43)

We do this in the next subsection.

$K_{r}$ itself is not trace class in general, but there is a multiplication
operator $\vartheta$ (independent of $t$, the $x_{i}$’s and the $r_{i}$’s)
such that the conjugated kernel $\vartheta K_{r}\vartheta^{-1}$ is trace
class. This is proved in Prop. B.2, and since the above argument remains valid
if we replace $K_{r}$ by this conjugation, and since both (2.18) and (2.19)
also do not change under this conjugation, we may perform this replacement to
prove Thm. 2.1. The necessary differentiability in trace norm of the
conjugated kernel appears in Appdx. B.3.

It only remains to show that $I-K_{r}$ is invertible for $r\in\\{-1,0,1\\}$.
In fact, this is true for all $r\geq-1$, which is equivalent (by (3.40)) to
showing that $K^{\text{{ext}}}$ is invertible whenever $r_{i}\geq
r_{0}(t,x_{i})$ for each $i$. We prove this by appealing to Thm. 2.3 and a
simple probabilistic argument. Consider the event that there are no
nucleations inside the region
$C_{t,x_{1},x_{m}}=\\{(s,y)\\!:s\in[0,t],\,x_{1}-s\leq y\leq x_{m}+s\\}$,
which obviously has positive probability. By definition of the PNG dynamics we
necessarily have in that case that $h(t,x_{i})=r_{0}(t,x_{i})$ for each $i$,
and then since we are taking $r_{i}\geq r_{0}(t,x_{i})$ we have

$\mathbb{P}_{h_{0}}(h(t,x_{i})\leq r_{i})\geq\mathbb{P}_{h_{0}}(\text{{no
nucl. in }}C_{t,x_{1},x_{m}})>0.$

But by (2.31) the left hand side equals the Fredholm determinant of the
identity minus the (trace class, after conjugation) kernel $K^{\text{{ext}}}$,
so $I-K^{\text{{ext}}}$ is invertible.

To finish the proof of the theorem we need to prove (2.19). By (2.28) we have
$F(t,x_{1},\ldots,x_{n},r_{1}+1,\ldots,r_{n}+1)=\mathop{\mathop{\text{{det}}}}\nolimits(I-K_{r+1})_{\oplus_{n}\ell^{2}(\mathbb{Z}_{>0})}$,
where we keep thinking of $r$ as a scalar variable. The kernel $K_{r}$
satisfies the hypotheses of Thm. 2.1, so in particular the formalism employed
there also applies here. In particular, by (3.3) and the cyclic property of
the Fredholm determinant we have

$\displaystyle F(t,x_{1},\ldots,x_{n},r_{1}+1,\ldots,r_{n}+1)$
$\displaystyle=\mathop{\mathop{\text{{det}}}}\nolimits(I-\sigma
K_{r}\sigma^{*})_{\oplus_{n}\ell^{2}(\mathbb{Z}_{>0})}=\mathop{\mathop{\text{{det}}}}\nolimits(I-(I-P)K_{r})_{\oplus_{n}\ell^{2}(\mathbb{Z}_{>0})}$
(3.44)
$\displaystyle=\mathop{\mathop{\text{{det}}}}\nolimits(I+PR_{r}K_{r})_{\oplus_{n}\ell^{2}(\mathbb{Z}_{>0})}\mathop{\mathop{\text{{det}}}}\nolimits(I-K_{r})_{\oplus_{n}\ell^{2}(\mathbb{Z}_{>0})}$
(3.45)
$\displaystyle=\mathop{\mathop{\text{{det}}}}\nolimits(I+PR_{r}K_{r})_{\oplus_{n}\ell^{2}(\mathbb{Z}_{>0})}F(t,x_{1},\ldots,x_{n},r_{1},\ldots,r_{n}).$
(3.46)

Now $P$ is a projection onto an $n$-dimensional subspace of
$\oplus_{n}\ell^{2}(\mathbb{Z}_{>0})$, and the matrix of the linear
transformation $P+PR_{r}K_{r}$ on $\mathbb{R}^{n}$ is given by
$\big{(}\mathbf{1}_{i=j}+(R_{r}K_{r})_{ij}(1,1)\big{)}_{i,j=1}^{n}=\big{(}(R_{r})_{ij}(1,1)\big{)}_{i,j=1}^{n}$
(since $R_{r}K_{r}=R_{r}-I$). Hence

$\mathop{\mathop{\text{{det}}}}\nolimits(I+PR_{r}K_{r})_{\oplus_{n}\ell^{2}(\mathbb{Z}_{>0})}=\mathop{\mathop{\text{{det}}}}\nolimits((I-P)+(P+PR_{r}K_{r}))_{\oplus_{n}\ell^{2}(\mathbb{Z}_{>0})}=\mathop{\mathop{\text{{det}}}}\nolimits(R_{r}(1,1)),$

the last being a determinant of an $n\times n$ matrix. Using this and the
definition $Q_{r}=R_{r}(1,1)$ in (3.45) yields (2.19).

###  3.3. The forcing term

Our goal now is to prove (3.43). In fact, we will prove a more general result,
which will also play a crucial role when we prove in Sec. 4 that our Fredholm
determinant satisfies the Kolmogorov backward equation for PNG. To state it we
introduce the following notational convention: when an object $A_{a}$ depends
on a parameter $a\in\mathbb{R}$, then the notations $A_{a^{-}}$ and $A_{a+}$
will always mean

$A_{a^{+}}=\mathop{\mathop{\text{{lim}}}}\displaylimits_{s\searrow
a}A_{s}\qquad\text{and}\qquad
A_{a^{-}}=\mathop{\mathop{\text{{lim}}}}\displaylimits_{s\nearrow a}A_{s},$

assuming these limits are well defined. We also define
$P^{\operatorname{\text{{no
hit}}}(h)}_{a,b}=e^{(b-a)\Delta}-P^{\operatorname{\text{{no hit}}}(h)}_{a,b}$.
In the following lemma and its proof the above notational convention will be
used to denote by $P^{\operatorname{\text{{hit}}}/\operatorname{\text{{no
hit}}}(h)}_{a^{+},b}$ and
$P^{\operatorname{\text{{hit}}}/\operatorname{\text{{no hit}}}(h)}_{a,b^{-}}$
the transition probabilities for $\mathsf{N}$ hitting/not hitting
$\operatorname{\text{{hypo}}}(h)$ on the half-open intervals $(a,b]$ and
$[a,b)$, respectively.

######  Lemma 3.4.

Let $t\geq 0$ and $x,x^{\prime}\in\mathbb{R}$ and let
$W^{(\zeta)}_{t,x,x^{\prime}}$ and $W^{(\eta)}_{t,x,x^{\prime}}$ be defined as
in (3.42). Then

$W^{(\zeta)}_{t,x,x^{\prime}}(u,v)=0\quad\forall\hskip
0.50003ptu>h_{0}(x-t)+1,\qquad
W^{(\eta)}_{t,x,x^{\prime}}(u,v)=0\quad\forall\hskip
0.50003ptv>h_{0}(x^{\prime}+t)+1.$ (3.47)

Moreover,

$\displaystyle W^{(\zeta)}_{t,x,x^{\prime}}(h_{0}(x-t)+1,v)$
$\displaystyle=-\tfrac{1}{2}P^{\operatorname{\text{{no
hit}}}(h_{0})}_{(x-t)^{+},x^{\prime}+t}e^{2t+2t\nabla-t\Delta}(h_{0}(x-t),v),$
(3.48) $\displaystyle W^{(\eta)}_{t,x,x^{\prime}}(u,h_{0}(x^{\prime}+t)+1)$
$\displaystyle=-\tfrac{1}{2}e^{2t-2t\nabla-t\Delta}P^{\operatorname{\text{{no
hit}}}(h_{0})}_{x-t,(x^{\prime}+t)^{-}}(u,h_{0}(x^{\prime}+t)).$

Since $r_{0}(t,x)>h_{0}(x\pm t)$ for all $x$, (3.47) clearly implies (3.43),
finishing the proof of Thm. 2.1. Note that in this use of the lemma in the
proof Thm. 2.1 we have only employed the condition $r_{k}>h_{0}(t,x_{k})$; the
stronger condition $r_{k}>r_{0}(t,x_{k})$ assumed in the theorem is required
to ensure that $K_{r}$ is invertible where needed. The identities in (3.48),
on the other hand, will be used in Sec. 4.6.

######  Remark 3.5.

In the one-point case there is a transparent (though partial) argument that
the extra derivatives coming from the dependence of
$\mathscr{T}^{h_{0}}_{x-t,x+t}$ on $t$ and $x$ do not contribute to the
computation (i.e. (3.43)). Fix $t\geq 0$, $x\in\mathbb{R}$. For $a\leq b$, let
$K^{a,b}_{r}(u,v)=e^{-2t\nabla-x\Delta}\mathscr{T}^{h_{0}}_{a,b}e^{2t\nabla+x\Delta}(u+r,v+r)$,
the kernel from (2.13) in the one-point case except that in the scattering
transform appearing in (2.12) we remove the dependence on $x-t$ and $x+t$ and
replace it by arbitrary parameters $a,b$. If we let

$F_{r}(t,x)=F_{r,x-t,x+t}(t,x),\qquad
F_{r,a,b}(t,x)=\mathop{\mathop{\text{{det}}}}\nolimits(I-K^{a,b}_{r})_{\ell^{2}(\mathbb{Z}_{>0})}$

for $r\geq r_{0}(t,x)$, then a simple adaptation of the arguments from the
last subsection tells us that
$\tfrac{1}{4}(\partial_{t}^{2}-\partial_{x}^{2})\mathop{\mathop{\text{{log}}}}\nolimits
F_{r}-\frac{F_{r+1}F_{r-1}}{F_{r}^{2}}$ is given by

$-\big{(}\partial_{x+t}\partial_{a}\mathop{\mathop{\text{{log}}}}\nolimits(F_{r,a,b})-\partial_{x-t}\partial_{b}\mathop{\mathop{\text{{log}}}}\nolimits(F_{r,a,b})-\partial_{a}\partial_{b}\mathop{\mathop{\text{{log}}}}\nolimits(F_{r,a,b})\big{)}\big{|}_{a=x-t,b=x+t}.$
(3.49)

To prove that the 2D Toda equation (2.24) holds, we need to show that this
_forcing term_ vanishes for $r>r_{0}(t,x)$. The argument we present next will
show that the left derivative
$\partial_{b^{-}}\mathop{\mathop{\text{{log}}}}\nolimits(F_{r,a,b})|_{b=x+t}$
vanishes; the same argument shows
$\partial_{a^{+}}\mathop{\mathop{\text{{log}}}}\nolimits(F_{r,a,b})|_{a=x-t}=0$.
Introduce truncated initial data
$h_{0}^{[a,b]}(x)=h_{0}(x)\mathbf{1}_{x\in[a,b]}-\infty\cdot\mathbf{1}_{x\notin[a,b]}$.
If $x-t\leq a\leq b\leq x+t$ then
$P^{\operatorname{\text{{hit}}}(h_{0}^{[a,b]})}_{x-t,x+t}=e^{(a-x+t)\Delta}P^{\operatorname{\text{{hit}}}(h_{0})}_{a,b}e^{(x+t-b)\Delta}$
because $\mathsf{N}$ cannot hit $\operatorname{\text{{hypo}}}(h_{0}^{[a,b]})$
outside $[a,b]$. Then we may write
$\mathscr{T}^{h_{0}}_{a,b}=e^{(x-t)\Delta}P^{\operatorname{\text{{hit}}}(h_{0}^{[a,b]})}_{x-t,x+t}e^{-(x+t)\Delta}=\mathscr{T}^{h_{0}^{[a,b]}}_{x-t,x+t}$.
In other words, when $[a,b]\subseteq[x-t,x+t]$, the scattering transform for
$h_{0}$ in the interval $[a,b]$ is the same as for the truncated initial data
$h_{0}^{[a,b]}$ in the interval $[x-t,x+t]$, and thus

$F_{r,a,b}(t,x)=F_{r}(t,x;h_{0}^{[a,b]})=\mathbb{P}_{h_{0}^{[a,b]}}\bigl{(}h(t,x)\leq
r\bigr{)}=\mathbb{P}_{\mathfrak{d}_{x}^{-r}}\bigl{(}h(t,\cdot)\leq-
h_{0}^{[a,b]}\bigr{)}.$ (3.50)

The third equality comes from the skew time reversal invariance of PNG (2.6),
with $\mathfrak{d}_{x}^{-r}$ a shifted narrow wedge given by
$\mathfrak{d}_{x}^{-r}(y)=-r$ if $x=y$ and $-\infty$ otherwise. In view of
this we may compute
$\partial_{b^{-}}F_{r,a,b}\big{|}_{b=x+t}=\mathop{\mathop{\text{{lim}}}}\displaylimits_{\delta\to
0}\delta^{-1}\left(\mathbb{P}_{\mathfrak{d}_{x}^{-r}}\bigl{(}h(t,\cdot)\leq-
h_{0}^{[a,x+t]}\bigr{)}-\mathbb{P}_{\mathfrak{d}_{x}^{-r}}\bigl{(}h(t,\cdot)\leq-
h_{0}^{[a,x+t-\delta]}\bigr{)}\right)$ as

$-\mathop{\mathop{\text{{lim}}}}\displaylimits_{\delta\to
0}\delta^{-1}\mathbb{P}_{\mathfrak{d}_{x}^{-r}}\bigl{(}\forall
y\in[a,x+t-\delta],~{}h(t,y)\leq-h_{0}(y);~{}\exists
z\in(x+t-\delta,x+t],h(t,z)>-h_{0}(z)\bigr{)}.$

The key is that the derivative is being computed at the edge of the forward
light cone, and that, in view of the initial condition, $h(t,x+t)=-r$. Suppose
first that $x+t$ is not a jump point for $-h_{0}$ so that, in particular,
$h_{0}$ is constant, and equal to $h_{0}(x+t)$, on $[x+t-\delta,x+t]$ if
$\delta$ is small enough. Then on the event inside the probability we have
$h(t,x+t-\delta)\leq-h_{0}(x+t)$ while $h(t,x+t)=-r\leq-h_{0}(x+t)$. Hence,
for the event to occur, $h(t,y)$ has to jump up _and then down_ in the
interval $[x+t-\delta,x+t]$, which has probability $\mathcal{O}(\delta^{2})$.
The same holds if $h_{0}$ has a down jump at $x+t$, because $h_{0}$ is upper
semi-continuous, so it is still constant on $[x+t-\delta,x+t]$. The only
relevant possibility then is that $h_{0}$ has an up jump at $x+t$. In this
case, and up to terms of order $\delta^{2}$, the event inside the probability
will occur if $h(t,y)$ stays below $-h_{0}$ on $[a,x+t-\delta]$ and
$-r=h(t,x)>-h_{0}(x+t)$, but we are assuming $r\geq h_{0}(x+t)$. This explains
why the forcing terms (3.49) vanish.

We turn now to the full proof of Lem. 3.4. The first step is to prove the
following preliminary result:

######  Lemma 3.6.

For $a<b$ and $h\in\operatorname{\mathscr{U\\!C}}$,

$\displaystyle\partial_{a}P^{\operatorname{\text{{hit}}}(h)}_{a,b}(u,v)$
$\displaystyle=\Delta
P^{\operatorname{\text{{hit}}}(h)}_{a^{+},b}(u,v)+\tfrac{1}{2}\mathbf{1}_{u=h(a)+1}P^{\operatorname{\text{{no
hit}}}(h)}_{a^{+},b}(h(a),v)\quad\forall\hskip 0.50003ptu>h(a),$ (3.51a)
$\displaystyle\partial_{b}P^{\operatorname{\text{{hit}}}(h)}_{a,b}(u,v)$
$\displaystyle=P^{\operatorname{\text{{hit}}}(h)}_{a,b^{-}}\Delta(u,v)+\tfrac{1}{2}P^{\operatorname{\text{{no
hit}}}(h)}_{a,b^{-}}(u,h(b))\mathbf{1}_{v=h(b)+1}\quad\;\forall\hskip
0.50003ptv>h(b).$ (3.51b)

Note that the second term on the right hand side of each identity only appears
when $h$ has a jump at the edges of $[a,b]$ (more precisely, a down jump at
$a$ for (3.51a) and an up jump at $b$ for (3.51b)), because if $h(b^{-})=h(b)$
then $P^{\operatorname{\text{{no hit}}}(h)}_{a,b^{-}}(u,h(b))=0$, and
similarly for the other edge.Note also that the derivatives may become
singular without the restrictions on $u$ and $v$. For example, take $h(x)$ to
be $-\infty$ for $x<1$ and $0$ for $x\geq 1$. Then
$P^{\operatorname{\text{{hit}}}h}_{0,b}(0,0)$ is not even continuous at $b=1$.

###### Proof.

We will only prove (3.51b), (3.51a) is completely analogous (and can also be
derived by considering the adjoint of the kernel in the second line and
reversing the direction of the random walk). Decomposing
$P^{\operatorname{\text{{hit}}}(h)}_{a,b+\varepsilon}$ as
$P^{\operatorname{\text{{hit}}}(h)}_{a,b-\varepsilon}e^{2\varepsilon\Delta}+P^{\operatorname{\text{{no
hit}}}(h)}_{a,b-\varepsilon}P^{\operatorname{\text{{hit}}}(h)}_{b-\varepsilon,b+\varepsilon}$,
where $P^{\operatorname{\text{{no
hit}}}(h)}_{a,b}=e^{(b-a)\Delta}-P^{\operatorname{\text{{hit}}}(h)}_{a,b}$,
one checks directly that

$\partial_{b}P^{\operatorname{\text{{hit}}}(h)}_{a,b}=\mathop{\mathop{\text{{lim}}}}\displaylimits_{\varepsilon\searrow
0}(2\varepsilon)^{-1}\big{(}P^{\operatorname{\text{{hit}}}(h)}_{a,b+\varepsilon}-P^{\operatorname{\text{{hit}}}(h)}_{a,b-\varepsilon}\big{)}=P^{\operatorname{\text{{hit}}}(h)}_{a,b^{-}}\Delta+P^{\operatorname{\text{{no
hit}}}(h)}_{a,b^{-}}\mathop{\mathop{\text{{lim}}}}\displaylimits_{\varepsilon\searrow
0}(2\varepsilon)^{-1}P^{\operatorname{\text{{hit}}}(h)}_{b-\varepsilon,b+\varepsilon},$
(3.52)

where $P^{\operatorname{\text{{hit}}}(h)}_{a,b^{-}}$ is the transition
probability for $\mathsf{N}$ without hitting $\operatorname{\text{{hypo}}}(h)$
on the half-open interval $[a,b)$.

Consider the limit on the right hand side of (3.52). Since
$P^{\operatorname{\text{{no hit}}}(h)}_{a,b^{-}}(u,\eta)=0$ if $\eta\leq
h(b^{-})$, we only need to consider
$(2\varepsilon)^{-1}P^{\operatorname{\text{{hit}}}(h)}_{b-\varepsilon,b+\varepsilon}(\eta,v)$
for $\eta>h(b^{-})$. By upper semi-continuity we may assume that $\varepsilon$
is small enough so that $h(s)\leq h(b)$ for all
$s\in[b-\varepsilon,b+\varepsilon]$. If $\eta>h(b)$ then
$P^{\operatorname{\text{{hit}}}(h)}_{b-\varepsilon,b+\varepsilon}(\eta,v)=\mathcal{O}(\varepsilon^{2})$,
because to go from $\eta$ to $v$ hitting $\operatorname{\text{{hypo}}}(h)$
involves $\mathsf{N}$ jumping at least twice in the interval
$[b-\varepsilon,b+\varepsilon]$ (first down to go below level $h(b)$, and then
up to get to $v>h(b)$). Now suppose $h(b^{-})<\eta\leq h(b)$. If $v-\eta>1$
then
$P^{\operatorname{\text{{hit}}}(h)}_{b-\varepsilon,b+\varepsilon}(\eta,v)=\mathcal{O}(\varepsilon^{2})$.
The only other alternative is $\eta=h(b)$, $v=h(b)+1$, in which case
$\mathsf{N}$ can go from $\eta$ to $v$ hitting
$\operatorname{\text{{hypo}}}(h)$ in $[b-\varepsilon,b+\varepsilon]$ by
staying put up to time $b$ and then jumping up by one inside the interval
$[b,b+\varepsilon]$. The conclusion is that for $\eta>h(b^{-})$ and $v>h(b)$,

$P^{\operatorname{\text{{hit}}}(h)}_{b-\varepsilon,b+\varepsilon}(\eta,v)=e^{\varepsilon\Delta}(0,1)\mathbf{1}_{\eta=h(b)>h(b^{-}),v=h(b)+1}+\mathcal{O}(\varepsilon^{2}).$

Using this in (3.52) and computing the limit we deduce that, for $v>h(b)$,

$\partial_{b}P^{\operatorname{\text{{hit}}}(h)}_{a,b}(u,v)=P^{\operatorname{\text{{hit}}}(h)}_{a,b^{-}}\Delta(u,v)+\tfrac{1}{2}P^{\operatorname{\text{{no
hit}}}(h)}_{a,b^{-}}(u,h(b))\mathbf{1}_{v=h(b)+1}\mathbf{1}_{h(b^{-})<h(b)}.$

The last indicator function can be removed (because
$P^{\operatorname{\text{{no hit}}}(h)}_{a,b^{-}}(u,h(b))=0$ if
$h(b)=h(b^{-})$) and we get the claimed identity. ∎

We also state the following simple result, which we will use often:

######  Lemma 3.7.

The kernel $e^{2t\nabla-t\Delta}$ is lower triangular, i.e.
$e^{2t\nabla-t\Delta}(u,v)=0$ for all $v-u\geq 1$.

###### Proof.

From (2.32) we have $e^{2t\nabla-t\Delta}(u,v)=e^{2t}\frac{1}{2\pi{\rm
i}}\oint_{\gamma_{r}}\frac{\mathrm{d}z}{z^{v-u+1}}e^{-2tz^{-1}}$, which
vanishes if $v-u\geq 1$ since then there is no residue at infinity. ∎

###### Proof of Lem. 3.4.

We will only consider the $\eta$ the derivatives, corresponding to the right
edge of the interval $[x-t,x^{\prime}+t]$; the $\zeta$ derivatives follow in
the same way. Given $a,b\in\mathbb{R}$, write

$W=e^{-2t\nabla-x\Delta}\mathscr{T}^{h_{0}}_{a,b}e^{2t\nabla+x^{\prime}\Delta},$
(3.53)

so that what we are trying to compute is
$\partial_{b}W(u,v)|_{a=x-t,b=x^{\prime}+t}$ for $v>h_{0}(x^{\prime}+t)$.

Start by assuming that $x-t<x^{\prime}+t$, which means that in $W$ we are
taking $a<b$. By definition of $\mathscr{T}^{h_{0}}_{a,b}$ we have
$\partial_{b}\mathscr{T}^{h_{0}}_{a,b}=e^{a\Delta}\big{(}-P^{\operatorname{\text{{hit}}}(h_{0})}_{a,b}\Delta+\partial_{b}P^{\operatorname{\text{{hit}}}(h_{0})}_{a,b}\big{)}e^{-b\Delta}$,
so

$\partial_{b}W=e^{-2t\nabla+(a-x)\Delta}\big{(}-P^{\operatorname{\text{{hit}}}(h_{0})}_{a,b}\Delta+\partial_{b}P^{\operatorname{\text{{hit}}}(h_{0})}_{a,b}\big{)}e^{2t\nabla+(x^{\prime}-b)\Delta}.$
(3.54)

Evaluating at $b=x^{\prime}+t$, the last factor on the right becomes
$e^{2t\nabla-t\Delta}$, and then by Lem. 3.7, if we want to compute
$\partial_{b}W\big{|}_{b=x^{\prime}+t}(u,v)$ for $v>h_{0}(x^{\prime}+t)$ then
we only need to consider the factor
$\partial_{b}P^{\operatorname{\text{{hit}}}(h_{0})}_{a,b}\big{|}_{b=x^{\prime}+t}(u,\eta)$
for $\eta>h_{0}(x^{\prime}+t)$. In this case Lem. 3.6 shows that
$\partial_{b}P^{\operatorname{\text{{hit}}}(h_{0})}_{a,b}\big{|}_{b=x^{\prime}+t}(u,\eta)=P^{\operatorname{\text{{hit}}}(h_{0})}_{a,(x^{\prime}+t)^{-}}\Delta(u,\eta)+\frac{1}{2}P^{\operatorname{\text{{no
hit}}}(h_{0})}_{a,(x^{\prime}+t)^{-}}\Delta(u,h_{0}(x^{\prime}+t))\mathbf{1}_{\eta=h_{0}(x^{\prime}+t)+1}$,
so going back to (3.54) and using
$P^{\operatorname{\text{{hit}}}(h_{0})}_{x-t,x^{\prime}+t}-P^{\operatorname{\text{{hit}}}(h_{0})}_{x-t,(x^{\prime}+t)^{-}}=-P^{\operatorname{\text{{no
hit}}}(h_{0})}_{x-t,(x^{\prime}+t)^{-}}$ we deduce that

$\partial_{b}W\big{|}_{\begin{subarray}{c}a=x-t\\\
b=x^{\prime}+t\end{subarray}}(u,v)=-e^{-2t\nabla-t\Delta}P^{\operatorname{\text{{no
hit}}}(h_{0})}_{x-t,(x^{\prime}+t)^{-}}\bar{\chi}_{h_{0}(x^{\prime}+t)}\Delta
e^{2t\nabla-t\Delta}(u,v)\\\
+\tfrac{1}{2}e^{-2t\nabla-t\Delta}P^{\operatorname{\text{{no
hit}}}(h_{0})}_{x-t,(x^{\prime}+t)^{-}}(u,h_{0}(x^{\prime}+t))e^{2t\nabla-t\Delta}(h_{0}(x^{\prime}+t)+1,v).$
(3.55)

From Lem. 3.7, the last factor on the second line vanishes if
$v>h_{0}(x^{\prime}+t)+1$, so by our assumption on $v$ this term only
contributes to the last expression in the case $v=h_{0}(x^{\prime}+t)+1$. Now
we focus on the first term on the right hand side, and consider the factor
$\bar{\chi}_{h_{0}(x^{\prime}+t)}\Delta e^{2t\nabla-t\Delta}(\xi,v)$ ($\xi$ is
an intermediate variable in the kernel compositions), which involves
$\bar{\chi}_{h_{0}(x^{\prime}+t)}e^{2t\nabla-t\Delta}(\xi+i,v)$ for
$i\in\\{-1,0,1\\}$. Using Lem. 3.7 again together with the assumption
$v>h_{0}(x^{\prime}+t)$, all these terms vanish except in the case $i=1$,
$\xi+1=v=h_{0}(x^{\prime}+t)+1$, and we conclude that
$\bar{\chi}_{h_{0}(x^{\prime}+t)}\Delta
e^{2t\nabla-t\Delta}(\xi,v)=\mathbf{1}_{\xi+1=v=h_{0}(x^{\prime}+t)+1}e^{2t\nabla-t\Delta}(h_{0}(x^{\prime}+t)+1,v)$,
and hence that

$\begin{multlined}e^{-2t\nabla-t\Delta}P^{\operatorname{\text{{no
hit}}}(h_{0})}_{x-t,(x^{\prime}+t)^{-}}\bar{\chi}_{h_{0}(x^{\prime}+t)}\Delta
e^{2t\nabla-t\Delta}(u,v)\\\ =e^{-2t\nabla-t\Delta}P^{\operatorname{\text{{no
hit}}}(h_{0})}_{x-t,(x^{\prime}+t)^{-}}(u,h_{0}(x^{\prime}+t))e^{2t\nabla-t\Delta}(h_{0}(x^{\prime}+t)+1,v)\mathbf{1}_{v=h_{0}(x^{\prime}+t)+1}.\end{multlined}e^{-2t\nabla-t\Delta}P^{\operatorname{\text{{no
hit}}}(h_{0})}_{x-t,(x^{\prime}+t)^{-}}\bar{\chi}_{h_{0}(x^{\prime}+t)}\Delta
e^{2t\nabla-t\Delta}(u,v)\\\ =e^{-2t\nabla-t\Delta}P^{\operatorname{\text{{no
hit}}}(h_{0})}_{x-t,(x^{\prime}+t)^{-}}(u,h_{0}(x^{\prime}+t))e^{2t\nabla-t\Delta}(h_{0}(x^{\prime}+t)+1,v)\mathbf{1}_{v=h_{0}(x^{\prime}+t)+1}.$
(3.56)

Using this, together with the identity
$e^{2t\nabla-t\Delta}(\eta,\eta)=e^{2t}$ coming from (2.32), in (3.55) yields

$\partial_{b}W\big{|}_{\begin{subarray}{c}a=x-t\\\
b=x^{\prime}+t\end{subarray}}(u,v)=-\tfrac{1}{2}e^{-2t\nabla-t\Delta}P^{\operatorname{\text{{no
hit}}}(h_{0})}_{x-t,(x^{\prime}+t)^{-}}(u,h_{0}(x^{\prime}+t))e^{2t}\mathbf{1}_{v=h_{0}(x^{\prime}+t)},$
(3.57)

which proves the second identity in each of (3.47) and (3.48) under the
assumption $x+t<x^{\prime}+t$.

Now if $x-t>x^{\prime}+t$, then $\mathscr{T}^{h_{0}}_{x-t,b}=0$ for $b$ in a
neighborhood of $x^{\prime}+t$, and the identities are trivial. What is left
is to consider the case $x-t=x^{\prime}+t$, for which we need to compute

$e^{-2t\nabla-t\Delta}\big{(}\partial_{b}P^{\operatorname{\text{{hit}}}(h_{0})}_{a,b}\big{|}_{a=b=x^{\prime}+t}\big{)}e^{2t\nabla-t\Delta}=\mathop{\mathop{\text{{lim}}}}\displaylimits_{\varepsilon\searrow
0}(2\varepsilon)^{-1}e^{-2t\nabla-t\Delta}(P^{\operatorname{\text{{hit}}}(h_{0})}_{x^{\prime}+t,x^{\prime}+t+\varepsilon}-P^{\operatorname{\text{{hit}}}(h_{0})}_{x^{\prime}+t,x^{\prime}+t-\varepsilon})e^{2t\nabla-t\Delta}.$

We have
$P^{\operatorname{\text{{hit}}}(h_{0})}_{x^{\prime}+t,x^{\prime}+t-\varepsilon}=0$
(because $\varepsilon>0$) while proceeding as above we have, for
$\xi>h_{0}(x^{\prime}+t)$,
$P^{\operatorname{\text{{hit}}}(h_{0})}_{x^{\prime}+t,x^{\prime}+t+\varepsilon}(\eta,\xi)=\varepsilon\mathbf{1}_{\eta=h_{0}(x^{\prime}+t),\,\xi=h_{0}(x^{\prime}+t)+1}+\mathcal{O}(\varepsilon^{2})$
and hence furthermore that the above limit equals
$\frac{1}{2}e^{-2t\nabla-t\Delta}(u,h_{0}(x^{\prime}+t))e^{2t\nabla-t\Delta}(h_{0}(x^{\prime}+t)+1,v)$
when evaluated at $u>h_{0}(x-t)$, $v>h_{0}(x^{\prime}+t)$. Using this in
(3.54) and proceeding as above show that
$\partial_{b}W\big{|}_{\begin{subarray}{c}a=x-t\\\
b=x^{\prime}+t\end{subarray}}(u,v)$ equals

$-e^{-2t\nabla-t\Delta}\bar{\chi}_{h_{0}(x^{\prime}+t)}\Delta
e^{2t\nabla-t\Delta}(u,v)+\tfrac{1}{2}e^{2t-2t\nabla-t\Delta}(u,h_{0}(x^{\prime}+t))\mathbf{1}_{v=h_{0}(x^{\prime}+t)+1}.$

Exactly the same argument we used to get to (3.56) and (3.57) shows that the
first term equals
$e^{2t-2t\nabla-t\Delta}(u,h_{0}(x^{\prime}+t))\mathbf{1}_{v=h_{0}(x^{\prime}+t)+1}$,
so the above expression equals
$-\frac{1}{2}e^{2t-2t\nabla-t\Delta}(u,h_{0}(x^{\prime}+t))\mathbf{1}_{v=h_{0}(x^{\prime}+t)+1}$,
which yields the $\eta$ derivatives in (3.47) and (3.48) in the remaining case
$x-t=x^{\prime}+t$. ∎

##  4\. PNG as a Markov process

We will need to develop some basic Markov machinery for PNG in order to show
that the Fredholm determinant expressions (2.28) uniquely solve the backward
(Kolmogorov) equations. To our surprise, there does not seem to be any
previous reference, except [Ben+81] where the equation is written down and
invariant measures computed from a physics viewpoint.

Let us briefly describe the plan. First of all, PNG is constructed directly as
a Markov process, with a generator $\hat{\mathscr{L}}$ coming from general
Markov process theory. Continuous time simple random walks are shown to be
invariant modulo the height shift and the adjoint $\hat{\mathscr{L}}^{*}$ is
computed. It is exactly the generator of the backward dynamics for PNG. Note
that one can also think of this as the forward process “upside down”, i.e.
$h\mapsto-h$.

The transition probabilities should satisfy the backward equation, but there
is a technical issue which permeates the proof: $\mathbf{1}_{h(x_{i})\leq
r_{i},i=1,\ldots,n}$ is _not_ a continuous function on
$\operatorname{\mathscr{U\\!C}}$ (for example, if
$h_{\varepsilon}(x)=\mathop{\mathop{\text{{log}}}}\nolimits\mathbf{1}_{x\geq\varepsilon}$
then $h_{\varepsilon}\longrightarrow h_{0}$ in
$\operatorname{\mathscr{U\\!C}}$, but
$\mathbf{1}_{h_{\varepsilon}(0)\leq-1}=1$ for $\varepsilon>0$ and $0$ for
$\varepsilon=0$). But the exact formulas are only for
$\mathbb{P}_{h}(h(t,x_{i})\leq r_{i},i=1,\ldots,n)$. In fact, the backward
equations only hold in a weak sense and one cannot just appeal to general
theory. For the Fredholm determinants the key behind the backward equations is
the identity (4.37), which shows that the time evolution of the scattering
kernels corresponds to the action of the generator.

Once one knows that the determinantal formulas satisfy the backward equation
in an appropriate sense, uniqueness is provided by the existence of a large
enough class of solutions of the adjoint equation (see Prop. 4.7). This is now
automatic, since the Fredholm determinant formulas can be “flipped on their
head” to produce solutions of the adjoint equation, which is just the backward
equation for the solution upside down.

###  4.1. Semigroup

The state space $\operatorname{\mathscr{U\\!C}}$ consists of upper semi-
continuous height functions $h:\mathbb{R}\to\mathbb{Z}\cup\\{-\infty\\}$. A
function is upper semi-continuous if and only if its hypograph
$\operatorname{\text{{hypo}}}(f)=\\{(x,y):y\leq f(x)\\}$ is closed. We
compactify at $-\infty$ by introducing a metric on
$\mathbb{Z}\cup\\{-\infty\\}$ so that $[-\infty,0]$ has finite measure. Then
$f_{n}\longrightarrow f$ if there exist444In [MFQR13] this is mistakenly
written as “for all $M\geq 1$”. $M_{\ell}\to\infty$ such that
$\operatorname{\text{{hypo}}}(f_{n})\cap[-M_{\ell},M_{\ell}]\longrightarrow\operatorname{\text{{hypo}}}(f)\cap[-M_{\ell},M_{\ell}]$
in the Hausdorff topology. Equivalently, $f_{n}\longrightarrow f$ if for each
$x$, $\mathop{\mathop{\text{{lim\hskip
1.0ptsup}}}}\displaylimits_{n\to\infty}f_{n}(x_{n})\leq f(x)$ whenever
$x_{n}\to\infty$ and there exists $x_{n}\to x$ with
$\mathop{\mathop{\text{{lim\hskip
1.0ptinf}}}}\displaylimits_{n\to\infty}f_{n}(x_{n})\geq f(x)$.

The first order of business is to construct the Markov semigroup acting on
continuous functions on $\operatorname{\mathscr{U\\!C}}$. Note that because of
the finite propagation speed, unlike models such as the exclusion process,
there is an elementary construction.

Start with a space-time set $\mathcal{N}$ of nucleation points which is
locally finite. Let $\mathcal{N}_{A}$ denote the restriction to a subset $A$
of space-time. Define a map $\mathcal{T}_{s,t,x}$ taking a
$\operatorname{\mathscr{U\\!C}}$ function $h$ on $[x-(t-s),x+t-s]$ “at time
$s$” and $\mathcal{N}_{A}$, $A={\\{(u,y):|y-x|\leq t-u,s\leq u\leq t\\}}$ into
$h(t,x)$ as follows. Let the points of $\mathcal{N}_{A}$ be denoted
$(u_{1},y_{1}),\ldots,(u_{M},y_{N})$ with ordered times $u_{1}<\cdots<u_{M}$.
$M\leq N$ since several points may have the same $u_{i}$. On the time interval
$(s,u_{1})$ we evolve $h$ to
$h((u_{1})_{-},z)=\mathop{\mathop{\text{{sup}}}}\displaylimits_{|\tilde{z}-z|\leq
u_{1}-s}h(\tilde{z})$. At time $u_{1}$ there are nucleation points
$y_{1},\ldots,y_{m_{1}}$ and we add $1$ to $h$ at each to get
$h(u_{1},z)=h((u_{1})_{-},z)+\sum_{i=1}^{m_{1}}\mathbf{1}_{z=y_{i}}$. Now on
the time interval $(u_{1},u_{2})$ we evolve $h$ to
$h((u_{2})_{-},z)=\mathop{\mathop{\text{{sup}}}}\displaylimits_{|\tilde{z}-z|\leq
u_{2}-u_{1}}h(u_{1},\tilde{z})$ and then, at time $u_{2}$ there are nucleation
points $y_{m_{1}+1},\ldots,y_{m_{2}}$ and we add $1$ to $h$ at each to get
$h(u_{2},z)=h((u_{2})_{-},z)+\sum_{i=m_{1}+1}^{m_{2}}\mathbf{1}_{z=y_{i}}$,
etc. Since there are finitely many points in $\mathcal{N}_{A}$, the process is
well-defined. Since $\mathcal{T}_{s,t,x}$ knows already to only use the points
of $\mathcal{N}$ in $A$ and the part of $h$ in $[x-(t-s),x+t-s]$, we can just
think of it as a map
$(h,\mathcal{N})\mapsto\mathcal{T}_{s,t,x}(h,\mathcal{N})$. It has the
property that for $\mathfrak{d}_{y}$ defined in (2.1),

$\mathcal{T}_{s,t,x}(h,\mathcal{N})=\mathop{\mathop{\text{{max}}}}\displaylimits_{|y-x|\leq
t-s}\\{\mathcal{T}_{0,t,x}(\mathfrak{d}_{y},\mathcal{N})+h(y)\\},$ (4.1)

as well as the semigroup property that for $s<u<t$,

$\mathcal{T}_{s,t,x}(h,\mathcal{N})=\mathcal{T}_{u,t,x}(\mathcal{T}_{s,u,\cdot}(h,\mathcal{N}),\mathcal{N}).$
(4.2)

In other words, given the background nucleations, PNG is actually a well-
defined, _deterministic_ Markov semigroup on $\operatorname{\mathscr{U\\!C}}$.
We then choose this background $\mathcal{N}$ as a rate $2$ space-time Poisson
point process, and this constructs our PNG model as

$h(t,x)=h(t,x;h_{0})=\mathcal{T}_{0,t,x}(h_{0},\mathcal{N}).$ (4.3)

$\operatorname{\mathscr{U\\!C}}$ is a complete, separable, locally compact
metric space. If we define, for $f\in\mathscr{C}_{0}:=$ continuous functions
on $\operatorname{\mathscr{U\\!C}}$ which vanish at infinity,

$\mathscr{P}_{t}f(h)=\mathbb{E}[f(\mathcal{T}_{0,t,\cdot}(h,\mathcal{N}))],$
(4.4)

then $\\{\mathscr{P}_{t},t\geq 0\\}$ is a semigroup. To see this, note first
that $\mathscr{P}_{0}$ is the identity. By (4.2),

$\mathscr{P}_{s+t}f(h)=\mathbb{E}[f(\mathcal{T}_{s,s+t,\cdot}(\mathcal{T}_{0,s,\cdot}(h,\mathcal{N}_{0,s}),\mathcal{N}_{s,s+t})],$
(4.5)

where the subscripts on the $\mathcal{N}$ indicate which time interval is
being used. Since the Poisson processes on the two time intervals are
independent, we can take the expectation over $\mathcal{N}_{s,s+t}$ first to
get

$\mathbb{E}[\mathscr{P}_{t}f(\mathcal{T}_{0,s,\cdot}(h,\mathcal{N}_{0,s})]=\mathscr{P}_{s}\mathscr{P}_{t}f(h).$
(4.6)

$\mathscr{P}_{t}f$ are also strongly continuous. Recall that such a semigroup
is Markov if, in addition, $\mathscr{P}_{t}1=1$ and $\mathscr{P}_{t}f\geq 0$
for all nonnegative $f\in\mathscr{C}_{0}$. Both are immediate from the
definition. Note that we could alternatively (and equivalently) have shown
directly that $h(t)$ defined by (4.3) is a Feller Markov process. The
generator $\hat{\mathscr{L}}$ is defined for
$f\in\mathscr{D}(\hat{\mathscr{L}})$ by

$\hat{\mathscr{L}}f=\mathop{\mathop{\text{{lim}}}}\displaylimits_{t\searrow
0}\tfrac{1}{t}(\mathscr{P}_{t}f-f),$ (4.7)

where $\mathscr{D}(\hat{\mathscr{L}})$ is just the set of those
$f\in\mathscr{C}_{0}$ for which the limit exists (in $\mathscr{C}_{0}$, with
the sup topology). The Hille-Yosida theorem [EK86, Thm. 2.6] tells us that
$\mathscr{D}(\hat{\mathscr{L}})$ is dense in $\mathscr{C}_{0}$ and that for
$f\in\mathscr{C}_{0}$ and $t>0$,
$\mathscr{P}_{t}\in\mathscr{D}(\hat{\mathscr{L}})$ with

$\partial_{t}\mathscr{P}_{t}f=\hat{\mathscr{L}}\mathscr{P}_{t}f.$ (4.8)

There is also a version of PNG on $[-L,L)$ with periodic boundary conditions.
We call the corresponding generator $\hat{\mathscr{L}}_{L}$. Clearly,
everything we have described has an analogue in that case.

######  Remark 4.1.

Our situation is unusual. Usually one starts with a generator and attempts to
build from it a semigroup/Markov process. In the case of PNG, the
semigroup/Markov process is explicit, so we can just read off the generator.

###  4.2. Generator

The above description of the generator is not explicit and in order to do
calculations, one has to introduce coordinates. A
$\operatorname{\mathscr{U\\!C}}$ function essentially consists of unit steps
up and down, together with an absolute height at some point, say $h(0)$. We
consider cases where the locations of the steps are an at most countably
infinite vector of points $y_{i}\in\mathbb{R}$, $i\in\mathbb{Z}$, which for
simplicity we will assume to be ordered by their label, $y_{i}\leq y_{i+1}$.
To each step $y_{i}$ corresponds a $\pm 1$ variable $\sigma_{i}$; we set
$\sigma_{i}=+1$ if the step is up and $\sigma_{i}=-1$ if the step is down
(i.e. $\mathop{\mathop{\text{{lim}}}}\displaylimits_{\varepsilon\searrow
0}h(y_{i}-\sigma_{i}\varepsilon)=h(y_{i})-1$). We interpret a pair
$(y_{i},\sigma_{i})$ as a particle with spin $\sigma_{i}$ located at $y_{i}$.
Note that this allows for steps of an arbitrary size because we are allowing
for jump locations to coincide.

It is certainly not the case that all $\operatorname{\mathscr{U\\!C}}$
functions fit this description. For example, $\mathfrak{d}_{0}$ “wants” to
correspond to $h(0)=0$ and $(y_{i},\sigma_{i})=(0^{-},+1)$ for $i<0$ and
$(0^{+},-1)$ for $i\geq 0$, but this is not allowed by the above description.
Another example is the indicator function of the Cantor set. It is in
$\operatorname{\mathscr{U\\!C}}$, but it certainly cannot be written in this
manner. Furthermore, the valid $\operatorname{\mathscr{U\\!C}}$ function
$h(x)=-\infty$, $x<0$, $h(x)=0$, $x\geq 0$ corresponds, for example, to
$(y_{i},\sigma_{i})=(0,+1)$, $i<0$ and $(\infty,\pm 1)$, $i\geq 0$. And a
configuration like $(y_{i},\sigma_{i})=(i,+1)$, $i<0$ and $(1,+1)$, $i\geq 0$,
corresponds to a _non_ -$\operatorname{\mathscr{U\\!C}}$ function taking the
illegal value $+\infty$ on $[1,\infty)$.

On the other hand, these cases are extreme. If one looks within one of the
invariant measures for the process (see Prop. 4.2), the point process $y_{i}$
is locally finite and simple (no coincidences) with probability one. If we
restrict to locally finite configurations, the map
$h(x)\longleftrightarrow(h(0),(y_{i},\sigma_{i})_{i\in\mathbb{Z}})$ is one-to-
one up to shifts in numbering if we make the convention that unneeded
particles are placed at either $+\infty$ with a plus sign, or $-\infty$ with a
minus sign. Furthermore, it is not hard to see that the dynamics stays within
this class.

So we will work with generators on this restricted subspace

$\hat{\mathcal{X}}=\mathbb{Z}\times\mathcal{X},\quad\mathcal{X}=\big{\\{}(\vec{y},\vec{\sigma})\in((\mathbb{R}\cup\\{-\infty,\infty\\})\times\\{-1,1\\})^{\mathbb{Z}}\\!:\vec{y}\text{{
is locally finite}},\,y_{i}\leq y_{i+1}\,\forall i,\\\ y_{i}=y_{i+1}\text{{
only if }}(\sigma_{i},\sigma_{i+1})=(+1,-1)\big{\\}},$ (4.9)

and we say $h\in\hat{\mathcal{X}}$ whenever the pair $(\vec{y},\vec{\sigma})$
corresponding to $h$ is in $\mathcal{X}$. At the end, we extend the results to
$\operatorname{\mathscr{U\\!C}}$ by continuity from above. It is reminiscent
of entrance laws in diffusion theory. Here one is misled because the narrow
wedges $\mathfrak{d}_{y}$ appear to lead to the simplest evolution, but they
do not fit nicely within the explicit computations with the generator.

The dynamics of the vector $(y_{i},\sigma_{i})_{i\in\mathbb{Z}}\in\mathcal{X}$
is also Markovian, and we let $\mathscr{L}$ denote its generator. The second
Markov process has a slightly reduced description, as we lose the absolute
height, but we will see that it is essentially enough for our purposes. The
state space for the $(y_{i},\sigma_{i})_{i\in\mathbb{Z}}$ process is
$\mathcal{X}$ with the topology inherited from the
$\operatorname{\mathscr{U\\!C}}$ topology on the associated height functions
(with, say, $h(0)=0$).

To describe the generator $\mathscr{L}$, we first need to describe a core for
its domain. This will consist of _local_ functions $F$, which are regular in
some sense. _Local_ means there is a finite interval $(a,b)$ so that $F$
depends only on $(y_{i},\sigma_{i})$ with $y_{i}\in(a,b)$. We are assuming
that the configuration is locally finite, so there is a finite number $n$ of
$y_{i}$’s in the interval. We may as well relabel them $y_{1}\leq
y_{2}\leq\cdots\leq y_{n}$, and let $\sigma_{1},\ldots,\sigma_{n}$ be their
corresponding spins. Let $E^{a,b}_{n}$ be the set of such configurations. For
$n=0$ we set $E^{a,b}_{0}=\\{\emptyset\\}$. We define
$E^{a,b}=\bigcup_{n=0}^{\infty}E^{a,b}_{n}$ to be the set of all
configurations of finitely many particles in $(a,b)$. We will use
interchangeably the two notations
$F(\dotsc,(y_{i},\sigma_{i}),(y_{i+1},\sigma_{i+1}),\dotsc)=F(\vec{y},\vec{\sigma})$,
where $\vec{y}=(\dotsc,y_{i},y_{i+1},\dotsc)$ and
$\vec{\sigma}=(\dotsc,\sigma_{i},\sigma_{i+1},\dotsc)$. Denote by $F_{n}$ the
restriction555We always mean that $F$ only depends on variables in the
interval $(a,b)$, so this just means we are considering the restriction of $F$
to the situation where there are $n$ particles in that interval.  of $F$ to
$E^{a,b}_{n}$.

An important consequence of the $\operatorname{\mathscr{U\\!C}}$ topology is
that configurations with $y_{i}=y_{i+1}$ and $\sigma_{i}=-1$,
$\sigma_{i+1}=+1$, are identified with the configuration where the two points
are removed from the list. Thus continuity of $F$ with respect to the
$\operatorname{\mathscr{U\\!C}}$ topology corresponds to _matching conditions_
between its restrictions,

$\mathop{\mathop{\text{{lim}}}}\displaylimits_{y_{i+1}-y_{i}\searrow
0}F_{n}(\ldots,(y_{i},-1),(y_{i+1},+1),\ldots)=F_{n-2}(\ldots,(y_{i-1},\sigma_{i-1}),(y_{i+2},\sigma_{i+2}),\ldots).$
(4.10)

The dynamics of the up/down steps consists of two pieces, a deterministic
part, with generator $\mathscr{L}_{\operatorname{\text{{det}}}}$, and a
stochastic part, with generator $\mathscr{L}_{\operatorname{\text{{cr}}}}$,
for _creation_ ,

$\mathscr{L}=\mathscr{L}_{\operatorname{\text{{det}}}}+\mathscr{L}_{\operatorname{\text{{cr}}}}.$
(4.11)

This generator acts on local functions $F$, smooth in the interior of each
$E_{n}^{a,b}$. The deterministic part
$\mathscr{L}_{\operatorname{\text{{det}}}}$ has each up-step moving to the
left with unit speed, and each down-step moving to the right with unit speed,
so that at interior points,

$\mathscr{L}_{\operatorname{\text{{det}}}}F_{n}(\vec{y},\vec{\sigma})=-\sum_{i=1}^{n}\sigma_{i}\partial_{y_{i}}F_{n}(\vec{y},\vec{\sigma}).$
(4.12)

The stochastic part of the dynamics has new $(+1,-1)$ pairs created randomly
uniformly with rate $2$,

$\mathscr{L}_{\operatorname{\text{{cr}}}}F_{n}(\vec{y},\vec{\sigma})=2\int_{-\infty}^{\infty}\bigl{(}F_{n+2}(C_{z}(\vec{y},\vec{\sigma}))-F_{n}(\vec{y},\vec{\sigma})\bigr{)}\mathrm{d}z,$
(4.13)

where the configuration $C_{z}(\vec{y},\vec{\sigma})$ is obtained by inserting
$(z,+1)$ and $(z,-1)$ into $(\vec{y},\vec{\sigma})$ to get an element of
$E^{a,b}_{n+2}$. Since $F$ is local the integral is really only over $(a,b)$.
Since we only consider $F\in\mathscr{C}_{0}$ as candidates for the domain of
the generator, $F_{n}$ are necessarily all bounded and (4.13) always makes
sense. The domain of $\mathscr{L}$ therefore consists of functions for which
the right hand side of (4.12) is in $\mathscr{C}_{0}$ and our core consists of
all local functions in the domain.

The analogous generator in the periodic case, on $[-L,L)$, without the
heights, will be called $\mathscr{L}_{L}$. The periodic case differs from the
whole line case in that many configurations $(y_{i},\sigma_{i})$ do not
correspond to valid height functions.

In order to write the full generator $\hat{\mathscr{L}}$ of the PNG height
function in this language we would need to take into account the dynamics of
$h(0)$. To this end one would need to count the number of up/down steps which
have crossed the origin to the right/left up to time $t$. Adding this part of
the dynamics to get the full generator amounts to adding an additional term on
the right hand side of (4.11). It would not be so hard to write the resulting
$\hat{\mathscr{L}}$, but we will never need its explicit form in the
computations.

###  4.3. Invariant measures

The following measures $\\{\mu_{\rho}\\}_{\rho\in(0,\infty)}$ are invariant
for the $(y_{i},\sigma_{i})_{i\in\mathbb{Z}}$ process on the line: Fix
$\rho\in(0,\infty)$ and let $\Xi^{+}$ and $\Xi^{-}$ be independent Poisson
point processes on $\mathbb{R}$ with intensities $\rho$ and $\rho^{-1}$,
respectively (so, for instance,
$\mathbb{P}(|\Xi^{+}\cap[a,b]|=n)=\frac{(\rho(b-a))^{n}}{n!}e^{-\rho(b-a)}$
for any $a<b$). Superpose $\Xi^{+}$ and $\Xi^{-}$, order the resulting points
$(y_{i})_{i\in\mathbb{Z}}$, and assign $\sigma_{i}=+1$ to points coming from
$\Xi^{+}$ and $\sigma_{i}=-1$ to points coming from $\Xi^{-}$. This results in
a configuration $(\vec{y},\vec{\sigma})$ which, in simple words, has $+1$
points and $-1$ points as independent Poisson point processes with intensities
$\rho$ and $\rho^{-1}$, respectively. $\mu_{\rho}$ denotes the law of this
configuration.

We also have the analogous measures $\mu_{L,\rho}$ which are just the
restrictions of $\mu_{\rho}$ to $[-L,L)$. Note that while it is not always
possible to build a periodic height function out of a configuration
$(y_{i},\sigma_{i})$ on $[-L,L)$, if one starts PNG with a periodic height
function one can run the process of up and down steps and build a periodic
height function out of the result at a later time.

######  Proposition 4.2.

For any $\rho>0$, the measure $\mu_{\rho}$ is invariant for the
$(\vec{y},\vec{\sigma})$ system.

In order to prove the result we need to show that
$\int\mathscr{L}F\mathrm{d}\mu_{\rho}=0$. We will actually perform a more
general computation, which will be useful later on, and from which the
proposition will follow.

Given an integer $m$, let $\mu^{a,b,m}_{\rho}$ denote the measure $\mu_{\rho}$
conditioned on the event that $\sum_{i\in\mathbb{Z}:\hskip
0.70004ptx_{i}\in[a,b]}\sigma_{i}=m$. In terms of the associated height
function $h$, this just means conditioning on $h(b)-h(a)=m$.

######  Proposition 4.3.

Fix $a<b$ in $\mathbb{R}$, $u\in\mathbb{Z}$, $\mu^{a,b,m}_{\rho}$ as above,
and suppose $F$ is a function which depends only on the $(y_{i},\sigma_{i})$
where $y_{i}\in(a,b)$ and on the value $u=h(a)$, such that each $F_{n}$ is
continuously differentiable in $y_{i}$ in the interior of $E_{n}^{a,b}$. Then

$\int\mathscr{L}F(\vec{y},\vec{\sigma};u)\,\mathrm{d}\mu^{a,b,m}_{\rho}(\vec{y},\vec{\sigma})=\frac{\hskip
1.00006ptd_{a,b,m-1}}{d_{a,b,m}}\int(F(\vec{y},\vec{\sigma};u)+F(\vec{y},\vec{\sigma};u+1))\,\mathrm{d}\mu_{\rho}^{a,b,m-1}\\\
-\frac{d_{a,b,m+1}}{d_{a,b,m}}\int(F(\vec{y},\vec{\sigma};u)+F(\vec{y},\vec{\sigma};u-1))\,\mathrm{d}\mu_{\rho}^{a,b,m+1},$
(4.14)

with $d_{a,b,m}=e^{(b-a)\Delta}(0,m)$ (see (2.34)). Furthermore, in the
periodic case with period $L$, for any $F\in\mathscr{D}(\mathscr{L}_{L})$,
$\int\mathscr{L}_{L}F\mathrm{d}\mu_{L,\rho}=0$.

###### Proof.

Throughout the proof it will be convenient to think of a configuration
$(\vec{y},\vec{\sigma})$ chosen from $\mu_{\rho}$ as follows: sample a single
Poisson point process $\Xi$ with intensity $r$, order the points $y_{i}$,
$i\in\mathbb{Z}$, and then for each $i$ flip a coin to give $\sigma_{i}=1$
with probability $p$ and $\sigma_{i}=-1$ with probability $1-p$. For this to
match the above description we need to choose $r$ and $p$ so that $rp=\rho$
and $r(1-p)=\rho^{-1}$. The parameters $r$ and $p$ are dependent, but we will
keep them in because it clarifies the computation.

Since $F$ is supported on $(a,b)$, we only need to focus on the particles of
$\Xi$ inside that interval. The number $n$ of those particles has distribution
Poisson$[r(b-a)]$ and their positions $y_{1}\leq\cdots\leq y_{n}$ are the
ordering of independent uniformly distributed variables on $[a,b]$. The
conditioning in the definition of $\mu^{a,b,m}_{\rho}$ means that there are
$k$ pluses and $\ell$ minuses with $n=k+\ell$, $m=k-\ell$. From this it
follows after a computation, writing $D_{k+\ell}$ for the domain $\\{a\leq
y_{1}<\cdots<y_{k+\ell}\leq b\\}$, that

$\int
F(\vec{y},\vec{\sigma};u)\,\mathrm{d}\mu^{a,b,m}_{\rho}(\vec{y},\vec{\sigma})=\frac{1}{d_{a,b,m}}\sum_{\begin{subarray}{c}k,\ell\geq
0:\\\
k-\ell=m\end{subarray}}\upsilon_{k,\ell}\sum_{\begin{subarray}{c}\vec{\sigma}\in\\{-1,1\\}^{k+\ell}:\\\
|\\{i:\hskip
0.70004pt\sigma_{i}=1\\}|=k\end{subarray}}\int_{D_{k+\ell}}F_{k+\ell}(\vec{y},\vec{\sigma};u)\,\mathrm{d}{y_{1}}\dotsm\mathrm{d}{y_{k+\ell}}$
(4.15)

where $\upsilon_{k,\ell}=e^{-r(b-a)}r^{k+\ell}p^{k}(1-p)^{\ell}$. To simplify
notation below, we write $F_{k+\ell}$ for
$F_{k+\ell}(\vec{y},\vec{\sigma};u)$, omit the
$\mathrm{d}{y_{1}}\cdots\mathrm{d}{y_{k+\ell}}$, and write the $k,\ell$ and
$\vec{\sigma}$ sums as $\sum_{k,\ell,\vec{\sigma}}$. Using this we have

$\int\mathscr{L}_{\operatorname{\text{{cr}}}}F(\vec{y},\vec{\sigma};u)\,\mathrm{d}\mu^{a,b,m}_{\rho}(\vec{y},\vec{\sigma})=\frac{2}{d_{a,b,m}}\sum_{k,\ell,\vec{\sigma}}\upsilon_{k,\ell}\int_{a}^{b}\mathrm{d}z\int_{D_{k+\ell}}\bigl{(}F_{k+\ell+2}(C_{z}(\vec{y},\vec{\sigma});u)-F_{k+\ell}(\vec{y},\vec{\sigma};u)\bigr{)}$
(4.16)

for the stochastic part of the dynamics (4.13), while for the deterministic
part (4.12) we have

$\int\mathscr{L}_{\operatorname{\text{{det}}}}F(\vec{y},\vec{\sigma};u)\,\mathrm{d}\mu^{a,b,m}_{\rho}(\vec{y},\vec{\sigma})=-\frac{1}{d_{a,b,m}}\sum\nolimits_{k,\ell,\vec{\sigma}}\upsilon_{k,\ell}\sum_{i=1}^{k+\ell}\sigma_{i}\int_{D_{k+\ell}}\partial_{y_{i}}F_{k+\ell}.$
(4.17)
$\sum\nolimits_{k,\ell,\vec{\sigma}}\upsilon_{k,\ell}\Big{(}-\sum_{i=1}^{k+\ell-1}\sigma_{i}\int_{D_{k+\ell}}F_{k+\ell}\cdot\delta_{y_{i}=y_{i+1}}+\sum_{i=2}^{k+\ell}\sigma_{i}\int_{D_{k+\ell}}F_{k+\ell}\cdot\delta_{y_{i}=y_{i-1}}\Big{)}$
(4.18)

and one from the two boundary integrals

$\sum\nolimits_{k,\ell,\vec{\sigma}}\upsilon_{k,\ell}\Big{(}\sigma_{k+\ell}\int_{D_{k+\ell}}F_{k+\ell}\cdot\delta_{y_{k+\ell}=b}-\sigma_{1}\int_{D_{k+\ell}}F_{k+\ell}\cdot\delta_{y_{1}=a}\Big{)}.$
(4.19)

Changing variables $i\longmapsto i-1$ in the second sum inside the brackets on
the right hand side of (4.18), all terms with $\sigma_{i}=\sigma_{i+1}$ cancel
with those same ones coming from the first sum, and hence the brackets equal
$\sum_{i=1}^{k+\ell-1}(\sigma_{i+1}-\sigma_{i})\mathbf{1}_{\sigma_{i}\neq\sigma_{i+1}}\int_{D_{k+\ell}}F_{k+\ell}\cdot\delta_{y_{i}=y_{i+1}}$,
which equals

$-2\sum_{\begin{subarray}{c}1\leq i<k+\ell-1:\\\
\sigma_{i}=-\sigma_{i+1}=1\end{subarray}}\int_{D_{k+\ell}}F_{k+\ell}\cdot\delta_{y_{i}=y_{i+1}}+2\sum_{\begin{subarray}{c}1\leq
i<k+\ell-1:\\\
-\sigma_{i}=\sigma_{i+1}=1\end{subarray}}\int_{D_{k+\ell}}F^{\wedge
i}_{k+\ell-2}\cdot\delta_{y_{i}=y_{i+1}}$ (4.20)

by the continuity condition (4.10), with $F^{\wedge i}_{k+\ell-2}$ referring
to $F_{k+\ell-2}$ evaluated at $(\vec{y},\vec{\sigma})$ with the $i$-th and
$(i+1)$-th entries of $\vec{y}$ and $\vec{\sigma}$ removed. Now we focus on
the second sum in (4.20). In the $i$-th term, the $y_{i}$ and $y_{i+1}$
integrals only involve the delta function, so they yield a factor
$y_{i+2}-y_{i-1}$. If we now perform the summation over $\vec{\sigma}$ on this
term, the sums over $\sigma_{i}$ and $\sigma_{i+1}$ disappear (the integrand
does not depend on these variables, and only $\sigma_{i}=-1$, $\sigma_{i+1}=1$
survives) and then, after relabeling variables, it is not hard to see that the
result can be rewritten as
$(b-a)\sum\nolimits_{\vec{\sigma}}^{k-1,\ell-1}\int_{D_{k+\ell-2}}F_{k+\ell-2}$.
Using this above shows that (4.18) is given by

$-2\sum\nolimits_{k,\ell,\vec{\sigma}}\upsilon_{k,\ell}\sum_{\begin{subarray}{c}1\leq
i<k+\ell-1:\\\
\sigma_{i}=-\sigma_{i+1}=1\end{subarray}}\sigma_{i}\int_{D_{k+\ell}}F_{k+\ell}\cdot\delta_{y_{i}=y_{i+1}}+2r^{2}p(1-p)(b-a)\sum\nolimits_{k,\ell}\upsilon_{k,\ell}\int_{D_{k+\ell}}F_{k+\ell},$
(4.21)

where in the second term we have changed variables
$(k,\ell)\longmapsto(k+1,\ell+1)$ in the sum, giving the $r^{2}p(1-p)$
prefactor. We conclude that
$\int\\!\mathscr{L}_{\operatorname{\text{{det}}}}F(\cdot,\cdot;u)\,\mathrm{d}\mu^{a,b,m}_{\rho}$
equals this expression plus (4.19).

Now consider the right hand side of (4.16), and split it into two terms. If we
decompose the first term according to the position $z$ of the newly created
pair among the $y_{i}$’s to rewrite it as $\sum_{1\leq i<k+\ell+1:\hskip
0.70004pt\sigma_{i}=-\sigma_{i+1}=1}\int_{D_{k+\ell+2}}F_{k+\ell+2}\cdot\delta_{y_{i}=y_{i+1}}$,
and then change variables $(k,\ell)\longmapsto(k-1,\ell-1)$ in the second
term, we see (using $r^{2}p(1-p)=1$) that
$\int\\!\mathscr{L}_{\operatorname{\text{{cr}}}}F(\cdot,\cdot;u)\,\mathrm{d}\mu^{a,b,m}_{\rho}$
equals minus the above expression. From this and the above computation we get

$\int\\!\mathscr{L}F(\vec{y},\vec{\sigma};u)\,\mathrm{d}\mu^{a,b,m}_{\rho}(\vec{y},\vec{\sigma})=\eqref{eq:Bdet}.$
(4.22)

Although we did not introduce all the necessary notation for the periodic
case, it should be clear from the computations that there we would have
$\int\mathscr{L}_{L}F(\vec{y},\vec{\sigma};u)\,\mathrm{d}\mu_{L,\rho}(\vec{y},\vec{\sigma})=0$.

It only remains to rewrite (4.19). Consider the sum corresponding to the first
term in the brackets on the right hand side of (4.19). Here
$F_{k+\ell}\cdot\delta_{y_{k+\ell}=b}$ is being integrated against the measure
$\upsilon_{k,\ell}\mathbf{1}_{a\leq y_{1}<\dotsm<y_{k+\ell}\leq b}$ over
$k,\ell\geq 0$ with $k-\ell=m$ and over choices of
$\vec{\sigma}\in\\{-1,1\\}^{k+\ell}$ with $k$ of the jumps being up. This
measure simply encodes the jump times and steps of the random walk
$\mathsf{N}$ (with up jumps at rate $\rho$ and down jumps at rate $\rho^{-1}$)
conditioned to go from $0$ at time $a$ to $m$ at time $b$. In the case
$\sigma_{k+\ell}=1$, the delta function in the integrand corresponds to the
last jump being up, and taking place at time $b$. We may think of this as the
walk being conditioned to go to $m-1$ instead of $m$, with $k-1$ up jumps
(note here $k\geq 1$ because otherwise $\sigma_{k+\ell}$ cannot take the value