# Three-dimensional Gaussian fluctuations of non-commutative random surface
growth with a reflecting wall

Zhengye Zhou111Texas A$\&$M University, College Station, TX, United States of
America. Email<EMAIL_ADDRESS>
###### Abstract

We consider the multi-time correlation and covariance structure of a random
surface growth with a wall introduced in [10]. It is shown that the
correlation functions associated with the model along space-like paths have
determinantal structure, which yields the convergence of height fluctuations
to that of a Gaussian free field. We also construct a continuous-time non-
commutative random walk on $U(\mathfrak{so}_{N+1})$, which matches the random
surface growth when restricting to the Gelfand-Tsetlin subalgebra of
$U(\mathfrak{so}_{N+1})$. As an application, we prove the convergence of
moments to an explicit Gaussian free field and get the covariance functions of
the associated random point process along both the space-like paths and time-
like paths. In particular, it does not match the three-dimensional Gaussian
field from spectra of overlapping stochastic Wishart matrices in [16] even
along the space-like paths.

##  1 Introduction

As an approach to study the Anisotropic Kardar–Parisi–Zhang (AKPZ) equation,
many models in the AKPZ universality class were studied over the past decades
(e.g. [5, 19, 21]). There has been lots of progress in understanding large
time asymptotics of driven interacting particle systems on the $2+1$
dimensions random growth models in the AKPZ universality class (e.g. [4, 5, 7,
8]). For example, in [8], the authors constructed a class of two-dimensional
random surface growth models, which can be interpreted as random point
processes. It was shown that along space-like paths, the point processes are
determinantal. The authors also established Gaussian fluctuations of one
specific growing random surface along space-like paths by computing the
correlation kernel and taking asymptotics. Later in [15], a continuous-time
non-commutative random walk on $U(\mathfrak{gl}_{N})$ (the universal
enveloping algebra of $\mathfrak{gl}_{N}$) was introduced, which matches the
random surface growth model introduced in [8] when restricting to the Gelfand-
Tsetlin subalgebra of $U(\mathfrak{gl}_{N})$. As an application, the
convergence to the Gaussian fluctuations along time-like paths was proved,
completing the entire three-dimensional Gaussian field. In particular, it was
also shown that this three-dimensional Gaussian field matches the one for
eigenvalues of stochastic Wigner matrices in [2].

Another model of interest is the random surface growth with a reflecting wall
constructed in [10], which can be viewed as an one-parameter family of
Plancherel measures for the infinite-dimensional orthogonal group. It is
believed that this model also belongs to the AKPZ universality class with
certain boundary condition. Shown in Figure 1, the random surface growth is
equivalent to a particle process in the quarter plane.

Figure 1: random surface growth with a wall

In [10], the determinantal formula for the correlation functions at any finite
time moment was derived. However, the authors did not provide the correlation
functions along space-like paths since the evolution of measure does not match
the general L-ensembles (see e.g. [4, 9, 11]). More specifically, this model
has $n$ particles at both $(2n-1/2\pm 1/2)$-th levels, thus the the
corresponding L-ensemble should only have virtual variables for even levels.
The Gaussian free field fluctuations for the height functions at any fixed
time moment were later proved in [13]. In this paper, we first introduce a
generalized L-ensemble, which implies the determinantal property and the
formula for the correlation function. We then apply this generalization to
extend the determinantal formula for the correlation functions to multi-time
moments. Later we introduce a continuous-time non-commutative random walk on
the universal enveloping algebra of the Lie group $\mathfrak{so}_{N+1}$
(denoted $U(\mathfrak{so}_{N+1})$), which is an analog of the non-commutative
random walks on $U(\mathfrak{gl}_{N})$ (see e.g. [1, 12, 15]). It can be
proved that this random walk matches the random surface growth. With the help
of the non-commutative random walk, we obtain the three-dimensional Gaussian
fluctuations and the covariance structures of the random surface growth. It is
natural to compare the three-dimensional Gaussian free fields from both
interacting particle systems and the eigenvalue processes of random matrices
that belong to the AKPZ universality class. For our model, an appropriate
analog is the Wishart matrices, since the eigenvalues are always non–negative,
corresponding to the fact that particles are restricted to be to the right of
the reflecting wall. However, the Gaussian fluctuation fields from this paper
and [16] are different along both the space-like paths and the time-like
paths.

Outline of the paper. In section 2, we review some facts about the algebras
and the random surface growth. In section 3, we introduce a generalized
L-ensemble to show that our model is a determinantal random point process
along space-like paths. Moreover, we compute the multi-time correlation kernel
which extends the existing formulas in [10] for the single-time kernel. Later,
we define a continuous-time non-commutative random walk on
$U(\mathfrak{so}_{N+1})$ in section 4. We also show the relation between the
random surface growth and the non-commutative random walk in section 4. In
section 5, we show that moments of the random surface converge to Gaussian
free fields along both space-like paths and time-like paths. In addition, the
explicit covariance formulas are given.

Acknowledgments. The author would like to thank Jeffrey Kuan for enlightening
discussions and providing Figure 1.

##  2 Preliminaries

###  2.1 Representation theory

We first review some useful results from representation theory (see e.g [10,
17, 18]). In this paper, for any positive integer $N$ such that
$N=2n+a-\frac{1}{2}$ with $a\in\left\\{\pm\frac{1}{2}\right\\}$, we abuse the
notation for $N$ and the corresponding pair $(n,a)$.

Let $O(N+1)$ denote the group of $(N+1)\times(N+1)$ real-valued orthogonal
matrices. The special orthogonal group $SO(N+1)$ is the subgroup of $O(N+1)$
consisting of matrices with determinant $1$. The associated Lie algebra is
denoted by $\mathfrak{so}_{N+1}$, which consists of $(N+1)\times(N+1)$ skew-
symmetric square matrices with Lie bracket the commutator. Last, let
$O(\infty)=\bigcup_{N=0}^{\infty}O(N+1)$ and
$SO(\infty)=\bigcup_{N=0}^{\infty}SO(N+1)$.

Recall that if $O\in SO(N+1)$, then the spectrum of $O$ is of the form
$\\{z_{1},z_{1}^{-1},\ldots,z_{n},z_{n}^{-1}\\}$ if $N+1=2n$, while when
$N+1=2n+1$, the spectrum of $O$ is of the form
$\\{z_{1},z_{1}^{-1},\ldots,z_{n},z_{n}^{-1},1\\}$, where $z_{i}$ are roots of
unity in both cases. Let $\Omega$ be the set of all
$\omega=(\alpha,\beta,\delta)$ such that

$\alpha=(\alpha_{1}\geq\alpha_{2}\geq\ldots\geq 0)\in\mathbb{R}^{\infty},\
\beta=(\beta_{1}\geq\beta_{2}\geq\ldots\geq 0)\in\mathbb{R}^{\infty},\
\delta\in\mathbb{R}$

and

$\sum_{i=1}^{\infty}(\alpha_{i}+\beta_{i})\leq\delta.$

For any $\omega=(\alpha,\beta,\delta)\in\Omega$, we define a function

$\chi^{\omega}(O)=\prod_{j=1}^{n}E^{\omega}\left(\frac{z_{j}+z_{j}^{-1}}{2}\right),$
(2.1)

where

$E^{\omega}(x)=e^{\left(\delta-\sum_{i=1}^{\infty}(\alpha_{i}+\beta_{i})\right)(x-1)}\prod_{i=1}^{\infty}\frac{1-\beta_{i}(1-x)+\beta_{i}^{2}(1-x)/2}{1-\alpha_{i}(1-x)+\alpha_{i}^{2}(1-x)/2},$
(2.2)

if we set $x=(z+z^{-1})/2$. Then each $\omega\in\Omega$ identifies an extreme
character $\chi^{\omega}$ of $O(\infty)$.

When $n\in\mathbb{N}$, it is a classical result that the set of all
irreducible representations of $SO(2n+1)$ over $\mathbb{C}$ is parameterized
by partitions of length $\leq n$, which is a sequence of nonincreasing
nonnegative integers $\lambda=(\lambda_{1}\geq\ldots\geq\lambda_{n}\geq 0)$.
For each partition $\lambda$, denote the corresponding character of the
irreducible representation of $SO(2n+1)$ by $\chi^{\lambda}_{SO(2n+1)}$, whose
dimension is denoted $\text{dim}_{SO(2n+1)}\lambda$. Similarly, the set of all
irreducible representations of $SO(2n)$ over $\mathbb{C}$ is parameterized by
sequences of integers $\lambda=(\lambda_{1},\ldots,\lambda_{n})$ satisfying
$\lambda_{1}\geq\ldots\geq\lambda_{n-1}\geq|\lambda_{n}|$. Again, for each
$\lambda$, we denote the corresponding character of the irreducible
representation of $SO(2n)$ by $\chi^{\lambda}_{SO(2n)}$ with dimension
$\text{dim}_{SO(2n)}\lambda$.

Let $\mathbb{J}_{n}$ denotes the set of all partitions of length $\leq n$ and
$\mathbb{J}_{n,a}$ with $a\in\\{\pm 1/2\\}$ be two copies of $\mathbb{J}_{n}$.
In what follows, we slightly abuse notations to interchange the use of $a=\pm
1/2$ and its sign, for example, we will write $\mathbb{J}_{n,\pm 1/2}$ as
$\mathbb{J}_{n,\pm}$.

For any $\omega\in\Omega$, the restriction of $\chi^{\omega}$ defined in (2.1)
to any $SO(M)$ defines two measures $P^{\omega}_{n,a}$ on $\mathbb{J}_{n,a}$
such that

$\chi^{\omega}|_{SO(2n+1)}=\sum_{\lambda\in\mathbb{J}_{n}}P^{\omega}_{n,\frac{1}{2}}(\lambda)\frac{\chi^{\lambda}_{SO(2n+1)}}{\text{dim}_{SO(2n+1)}\lambda},$
(2.3)
$\chi^{\omega}|_{SO(2n)}=\sum_{\lambda\in\mathbb{J}_{n}}P^{\omega}_{n,-\frac{1}{2}}(\lambda)\frac{\chi^{\lambda}_{SO(2n)}+\chi^{\lambda^{*}}_{SO(2n)}}{\text{dim}_{SO(2n)}\lambda},$
(2.4)

where $\lambda^{*}=(\lambda_{1},\ldots,\lambda_{n-1},-\lambda_{n})$.

Let $J_{k}^{(a,b)}$ denote the $k$-th Jacobi polynomial with parameters $a,b$
(see e.g. [20]), define the constant $c_{k}$ to be

$c_{k}=\left\\{\begin{array}[]{cc}\frac{1\cdot 3\cdot\ldots\cdot(2k-1)}{2\cdot
4\cdot\ldots\cdot 2k}&\text{ if }k>0,\\\ 1&\text{ if }k=0,\end{array}\right.$

and let $\mathcal{J}_{k}^{(a,b)}=J_{k}^{(a,b)}/c_{k}$.

Then for any $\lambda\in\mathbb{J}_{n,a}$, and $\omega\in\Omega$, the measure
$P^{\omega}_{n,a}$ provided in [10] is as follows:

$P^{\omega}_{n,a}(\lambda)=C_{n,a}\times\text{det}\left[\psi_{n-j}^{n,a}\left(\lambda_{i}-i+n\right)\right]_{1\leq
i,j\leq n}\times\text{dim}_{SO(2n+1/2+a)}\lambda,$ (2.5)

where

$\psi_{n-l}^{n,a}(k)=\frac{W^{(a,-1/2)}(k)}{\pi}\int_{-1}^{1}E^{\omega}(x)(x-1)^{n-l}\mathcal{J}_{k}^{(a,-1/2)}(x)(1-x)^{a}(1+x)^{-1/2}dx,$
$W^{(a,b)}(k)=\left\\{\begin{array}[]{ll}2,&\text{ if }k>0,\ a=b=-1/2,\\\
1,&\text{ if }k=0,\ a=b=-1/2,\\\ 1,&\text{ if }k\geq 0,\ a=1/2,\
b=-1/2,\end{array}\right.$

and $C_{n,a}$ is the normalization term for $P^{\omega}_{n,a}$. In particular,
$P^{(0,0,0)}_{n,a}$ is the delta measure at the point $(1,2,\cdots,n)$.

For $N+1=2n+1$ or $N+1=2n$ respectively, we number the rows and columns of
$(N+1)\times(N+1)$ matrices by the indices $\\{-n,\ldots,-1,0,1,\ldots,n\\}$
and $\\{-n,\ldots,-1,1,\ldots,n\\}$.

The Lie algebra $\mathfrak{so}_{N+1}$ is spanned by the generators

$F_{ij}=E_{ij}-E_{-j,-i},\ \ \ -n\leq i,j\leq n,i\neq-j.$

The commutation relation in $\mathfrak{so}_{N+1}$ is

$[F_{ij},F_{kl}]=\delta_{kj}F_{il}-\delta_{il}F_{kj}-\delta_{i,-k}F_{-j,l}+\delta_{-l,j}F_{k,-i}.$

The coproduct
$\Delta:U(\mathfrak{so}_{N+1})\xrightarrow[]{}U(\mathfrak{so}_{N+1})\otimes
U(\mathfrak{so}_{N+1})$ is given by $\Delta(F_{ij})=F_{ij}\otimes 1+1\otimes
F_{ij}$.

Let $Z(U(\mathfrak{so}_{N+1}))$ be the centre of $U(\mathfrak{so}_{N+1})$, we
introduce the generators of $Z(U(\mathfrak{so}_{N+1}))$ (see e.g. chapter 7 of
[17]). Denote by $F^{(m)}$ the matrix with entries $F_{ij}$, where $1\leq
m\leq n$ and $i,j=-m,-m+1,\cdots,m$. Denote by $\tilde{F}^{(m)}$ the submatrix
of $F^{(m)}$ obtained by removing the row and column enumerated by $-m$. Let
$\rho_{i}=-\rho_{-i}=-i+1$ if $N+1$ is even, while
$\rho_{i}=-\rho_{-i}=-i+\frac{1}{2}$ if $N+1$ is odd.

Let $\Lambda_{k}^{(m)}$, $\Phi_{k}^{(m)}$, $\tilde{\Lambda}_{k}^{(m)}$,
$\tilde{\Phi}_{k}^{(m)}$ denote the noncommutative symmetric functions defined
as the coefficients in the expansion of the following quasideterminant (see
Definition 1.10.1 in [17]):

$\begin{array}[]{l}1+\sum_{k=1}^{\infty}\Lambda_{k}^{(m)}q^{k}=\left|1+q\left(F^{(m)}+\rho_{m}\right)\right|_{mm},\\\
\sum_{k=1}^{\infty}\Phi_{k}^{(m)}q^{k-1}=-\frac{d}{dq}log\left|1-q\left(F^{(m)}+\rho_{m}\right)\right|_{mm},\\\
1+\sum_{k=1}^{\infty}\tilde{\Lambda}_{k}^{(m)}q^{k}=\left|1+q\left(-\tilde{F}^{(m)}-\rho_{m}\right)\right|_{mm},\\\
\sum_{k=1}^{\infty}\tilde{\Phi}_{k}^{(m)}q^{k-1}=-\frac{d}{dq}log\left|1-q\left(-\tilde{F}^{(m)}-\rho_{m}\right)\right|_{mm}.\\\
\end{array}$

In addition, there is a graphical construction of $\Lambda_{k}^{(m)}$,
$\Phi_{k}^{(m)}$, $\tilde{\Lambda}_{k}^{(m)}$, $\tilde{\Phi}_{k}^{(m)}$.

For a $M\times M$ matrix $A$, we consider the complete oriented graph
$\mathcal{A}$ with vertices 1,2,…,M and label the arrow from vertex $i$ to $j$
by $a_{ij}$. For each directed path of length $k$ in graph $\mathcal{A}$ which
starts from vertex $i$ and ends at vertex $j$, we define a monomial of the
form $A_{i,r_{1}}A_{r_{1},r_{2}}\cdots A_{r_{k-1,j}}$.

Let $A=F^{(m)}+\rho_{m}$, then $(-1)^{k-1}\Lambda_{k}^{(m)}$ is the sum of all
monomials labeling simple path in $\mathcal{A}$ of length $k$ going from $m$
to $m$. $\Phi_{2k}^{(m)}$ is the sum of all monomials labeling path in
$\mathcal{A}$ of length $k$ going from $m$ to $m$, the coefficient of each
monomial being the ratio of $k$ to the number of returns to $m$. Similarly,
$\tilde{\Lambda}_{k}^{(m)}$ and $\tilde{\Phi}_{k}^{(m)}$ are defined with
$A=-\tilde{F}^{(m)}-\rho_{m}$.

Now, we define
$\Lambda_{2n}^{N+1}=\prod_{m=1}^{n}\tilde{\Lambda}_{1}^{(m)}\Lambda_{1}^{(m)}$
and
$\Phi_{2k}^{N+1}=\sum_{m=1}^{n}\left(\tilde{\Phi}_{2k}^{(m)}+\Phi_{2k}^{(m)}\right)$.

######  Example 1.

  1. (1)

$\Phi_{2}^{N+1}=2\sum_{m=1}^{n}\left(\left(F_{mm}+\rho_{m}\right)^{2}+2\sum_{-m<i<m}F_{mi}F_{im}\right).$

  2. (2)

$\Phi_{4}^{N+1}=2\sum_{m=1}^{n}\Bigg{(}\left(F_{mm}+\rho_{m}\right)^{4}+2\sum_{-m<i,j<m}F_{mi}F_{im}F_{mj}F_{jm}\\\
+2\sum_{-m<i,j<m}\left(F_{mm}+\rho_{m}\right)F_{mi}\left(F_{ij}+\delta_{ij}\rho_{m}\right)F_{jm}+2\sum_{-m<i,j<m}F_{mi}\left(F_{ij}+\delta_{ij}\rho_{m}\right)F_{jm}\left(F_{mm}+\rho_{m}\right)\\\
+4\sum_{-m<i,j<m}F_{mi}\left(F_{ij}+\delta_{i,j}\rho_{m}\right)\left(F_{ji}+\delta_{i,j}\rho_{m}\right)F_{im}+2\sum_{-m<i<m}F_{mi}F_{i-m}F_{-mi}F_{im}\\\
+\frac{4}{3}\sum_{-m<i<m}\left(F_{mi}F_{im}\left(F_{mm}+\rho_{m}\right)^{2}+\left(F_{mm}+\rho_{m}\right)^{2}F_{mi}F_{im}+\left(F_{mm}+\rho_{m}\right)F_{mi}F_{im}\left(F_{mm}+\rho_{m}\right)\right)\Bigg{)}.$

In Chapter 7 of [17], the explicit generators for $Z(U(\mathfrak{so}_{N+1}))$
were identified using the Harish-Chandra isomorphism with the ring of shifted
symmetric polynomials.

######  Theorem 2.1.

[17] When $N=2n$, $Z\left(U\left(\mathfrak{so}_{N+1}\right)\right)$ is
generated by $1$ and $\left\\{\Phi_{2k}^{N+1}\right\\}_{k=1}^{n}$. When
$N=2n-1$, $Z(U(\mathfrak{so}_{N+1}))$ is generated by $1$,
$\left\\{\Phi_{2k}^{N+1}\right\\}_{k=1}^{n-1}$ together with
$\sqrt{\Lambda_{2n}^{N+1}}$. Images of $\Phi_{2k}^{N+1}$ and
$\Lambda_{2n}^{N+1}$ under the Harish-Chandra isomorphism are

$\begin{array}[]{lr}\frac{\Phi_{2k}^{N+1}}{2}\xrightarrow{}l_{1}^{2k}+\cdots+l_{n}^{2k},&\Lambda_{2n}^{N+1}\xrightarrow[]{}(-1)^{n}l_{1}^{2}\cdots
l_{n}^{2},\end{array}$

where $l_{i}=\lambda_{i}+\rho_{i}$.

Thus, any $Y\in Z(U(\mathfrak{so}_{N+1}))$ acts as a multiplication by a
scalar $p_{Y}(\lambda_{1},\ldots,\lambda_{n})$ in $V_{\lambda}$. When $N=2n$,
$p_{Y}(\lambda_{1},\ldots,\lambda_{n})$ is symmetric in the variables
$l_{1}^{2},\ldots,l_{n}^{2}$. When $N=2n-1$,
$p_{Y}(\lambda_{1},\ldots,\lambda_{n})$ is the sum of a symmetric polynomial
in $l_{1}^{2},\ldots,l_{n}^{2}$ and $l_{1}\cdots l_{n}$ times a symmetric
polynomial in $l_{1}^{2},\ldots,l_{n}^{2}$.

###  2.2 Random surface growth with a wall

In this subsection, we provide more details about the model.

Consider the two-dimensional lattice $\mathbb{Z}_{\geq
0}\times\mathbb{Z}_{+}$. On each horizontal level
$\mathbb{Z}_{+}\times\\{N\\}$, there are
$n=\left\lfloor\frac{N+1}{2}\right\rfloor$ particles.

Denote the horizontal coordinates of all particles with vertical coordinate
$N$ by $y_{1}^{N}>y_{2}^{N}>\cdots>y^{N}_{n}$. The reflecting wall forces
$y^{N}_{n}\geq a+\frac{1}{2}$. We have particle configurations

$\\{y_{k}^{N}\in\mathbb{Z}_{\geq 0}|k=1,2,\ldots,n;N=1,2,\ldots\\}.$

Additionally, the particles satisfy the interlacing property
$y_{k+1}^{N+1}<y_{k}^{N}<y_{k}^{N+1}$ for all meaningful values of $k$ and
$N$.

The densely packed initial condition is given by the configuration where
$y_{k}^{N}=N-2k+1$ for all $k$ and $N$, which means all the particles are as
much to the left as possible. Each particle has two exponential clocks of rate
$\frac{1}{2}$, one is responsible for left jump and another for right jump.
All clocks are independent. When the clock for particle $y_{k}^{N}$ rings, the
particle tries to jump by $1$ in the corresponding direction. Right jumps are
blocked if $y_{k}^{N}+1=y_{k-1}^{N-1}$ and left jumps are blocked if
$y_{k}^{N}-1=y_{k}^{N-1}$. If the particle is against the wall (i.e.
$y^{N}_{n}=0$) and the left jump clock rings, the particle is reflected and
tries to jump to the right.

When $y_{k}^{N}$ tries to jump to the right and the jump is not blocked, we
find the largest non-negative $r$ such that $y_{k}^{N+i}=y_{k}^{N}+i$ for
$0\leq i\leq r$, and all particles $\\{y_{k}^{N}\\}_{i=0}^{r}$ jump to right
by $1$ simultaneously. If $y_{k}^{N}$ tries to jump to the left and is not
blocked, we find the largest $l$ such that $y_{k+j}^{N+j}=y_{k}^{N}-j$ for
$0\leq j\leq l$, and all particles $\\{y_{k+j}^{N+j}\\}_{j=0}^{l}$ jump to
left by $1$.

In words, in order to maintain the interlacing property, the jump of one
particle is blocked by particles below it, but if the jump is not blocked, the
particle pushes particles above it to jump together. Additionally, jumps are
reflected against the wall.

We then recall several equivalent ways to define the particle configurations
of our model.

It is worth mentioning that the particle configurations of the above point
process are obtained as an equivalent interpretation of paths of partitions
[10].

When $\lambda\in\mathbb{J}_{n,-}$ and $\mu\in\mathbb{J}_{n,+}$, we write
$\lambda\prec\mu$ if
$0\leq\lambda_{n}\leq\mu_{n}\leq\ldots\leq\lambda_{1}\leq\mu_{1}$. When
$\lambda\in\mathbb{J}_{n,+}$ and $\mu\in\mathbb{J}_{n+1,-}$, we write
$\lambda\prec\mu$ if
$0\leq\mu_{n+1}\leq\lambda_{n}\leq\mu_{n}\leq\ldots\leq\lambda_{1}\leq\mu_{1}$.
Next, set $\mathbb{J}=\bigcup_{n\geq
1}\left(\mathbb{J}_{n,+}\cup\mathbb{J}_{n,-}\right)$. A path in $\mathbb{J}$
is a sequence
$\mathbf{\lambda}=\left(\lambda^{(1),-}\prec\lambda^{(1),+}\prec\lambda^{(2),-}\prec\ldots\right)$
such that $\lambda^{(i),\pm}\in\mathbb{J}_{i,\pm}$. Last, let $\mathbb{J}^{N}$
to be the set of finite paths in $\mathbb{J}$ of length $N=2n-1/2+a$.

Set $\mathfrak{X}=\mathbb{Z}_{\geq
0}\times\mathbb{Z}_{>0}\times\\{\pm\frac{1}{2}\\}$ and
$\mathfrak{Y}=\mathbb{Z}_{\geq 0}\times\mathbb{Z}_{>0}$ . $\mathfrak{Y}$ is
equivalent to $\mathfrak{X}$ via the bijection sending
$(x,n,a)\in\mathfrak{X}$ to
$(2x+a+\frac{1}{2},2n+a-\frac{1}{2})\in\mathfrak{Y}$.

To any path
$\mathbf{\lambda}=\left(\lambda^{(1),-}\prec\lambda^{(1),+}\prec\lambda^{(2),-}\prec\ldots\right)\in\mathbb{J}$,
we associate point configurations in $\mathfrak{X}$ and $\mathfrak{Y}$ as
follows:

$\mathcal{L}_{\mathfrak{X}}(\mathbf{\lambda})=\left\\{\left(x_{k}^{n,a},n,a\right):\
1\leq k\leq n,\ a\in\left\\{\pm 1/2\right\\},\ n\geq 1\right\\},$
$\mathcal{L}_{\mathfrak{Y}}(\mathbf{\lambda})=\left\\{\left(y_{k}^{2n-1/2+a},2n-1/2+a\right):\
1\leq k\leq n,\ a\in\left\\{\pm 1/2\right\\},\ n\geq 1\right\\},$

where $x_{k}^{n,a}=\lambda_{k}^{n,a}+n-k$ and
$y^{2n-1/2+a}_{k}=2(\lambda_{k}^{n,a}+n-k)+a+1/2$. In what follows, we
interchange the use of $x_{k}^{n,a}$ and $y^{2n-1/2+a}_{k}$ for ease of
notation.

We also adopt the notations and definitions from [10] as follows.

Define $T_{n,a}^{\varphi_{t}}$ on $\mathbb{J}_{n}\times\mathbb{J}_{n}$ by

$T_{n,a}^{\varphi_{t}}(\mu,\lambda)=\text{det}\left[I_{a}^{\varphi_{t}}(\mu_{i}-i+n,\lambda_{j}-j+n)\right]_{1\leq
i,j\leq
n}\frac{\text{dim}_{SO(2n+1/2+a)}\lambda}{\text{dim}_{SO(2n+1/2+a)}\mu},$

where $I_{a}^{\varphi_{t}}$ is given by

$I_{a}^{\varphi_{t}}(l,i)=\frac{W^{\left(a,-\frac{1}{2}\right)}(i)}{\pi}\int_{-1}^{1}\mathcal{J}_{i}^{(a,-\frac{1}{2})}(x)\mathcal{J}_{l}^{(a,-\frac{1}{2})}(x)\varphi_{t}(x)(1-x)^{a}(1+x)^{-1/2}dx.$

Define the matrix $T^{n,+}_{n,-}$ on $\mathbb{J}_{n,+}\times\mathbb{J}_{n,-}$
by

$T^{n,+}_{n,-}(\mu,\lambda)=\text{det}\left[\phi^{+}_{-}(\mu_{i}-i+n,\lambda_{j}-j+n)\right]_{1\leq
i,j\leq n}\frac{\text{dim}_{SO(2n)}\lambda}{\text{dim}_{SO(2n+1)}\mu},$

where

$\phi^{+}_{-}(x,y)=\left\\{\begin{array}[]{cc}1&x\geq y=0,\\\ 2&x\geq y>0,\\\
0&x<y.\end{array}\right.$

Define the matrix $T^{n,-}_{n-1,+}$ on
$\mathbb{J}_{n,-}\times\mathbb{J}_{n-1,+}$ by

$T^{n,-}_{n-1,+}(\mu,\lambda)=\text{det}\left[\phi^{-}_{+}(\mu_{i}-i+n,\lambda_{j}-j+n-1)\right]_{1\leq
i,j\leq n}\frac{\text{dim}_{SO(2n-1)}\lambda}{\text{dim}_{SO(2n)}\mu},$

where

$\phi^{-}_{+}(x,y)=\left\\{\begin{array}[]{cc}1&x>y,\\\ 0&x\leq
y.\end{array}\right.$

For each measure $P^{\omega}_{n,a}$ in (2.5), we define the measure

$P^{N,\omega}=P^{\omega}_{n,a}\left(\lambda^{(n),a}\right)T^{n,a}_{n-1/2+a,-a}\left(\lambda^{(n),a},\lambda^{(n-1/2+a),-a}\right)\cdots
T^{2,-}_{1,+}\left(\lambda^{(2),-1/2},\lambda^{(1),1/2}\right)T^{1,+}_{1,-}\left(\lambda^{(1),1/2},\lambda^{(1),-1/2}\right)$
(2.6)

on $\mathbb{J}^{N}$. When $\omega=(0,0,0)$, $P^{N,\omega}$ is the delta
measure at the densely packed initial condition, i.e. $x^{m,a}_{k}=m-k$ for
all $1\leq k\leq m$ and $1\leq m\leq n$.

Define a stochastic matrix $A^{\varphi_{t}}_{N}$ on
$\mathbb{J}^{N}\times\mathbb{J}^{N}$ by

$A_{N}^{\varphi_{t}}(\mathbf{\lambda},\mathbf{\mu})=\frac{T^{\varphi_{t}}_{n,a}\left(\lambda^{(n),a},\mu^{(n),a})T^{n,a}_{n-1/2+a,-a}(\mu^{(n),a},\mu^{(n-1/2+a),-a}\right)}{T^{n,a}_{(n-1/2+a),-a}T^{\varphi_{t}}_{(n-1/2+a),-a}\left(\lambda^{(n),a},\mu^{(n-1/2+a),-a}\right)}\times\cdots\\\
\cdots\frac{T^{\varphi_{t}}_{1,+}\left(\lambda^{(1),1/2},\mu^{(1),1/2}\right)T^{1,+}_{1,-}\left(\mu^{(1),1/2},\mu^{(1),-1/2}\right)}{T^{1,+}_{1,-}T^{\varphi_{t}}_{1,-}\left(\lambda^{(1),1/2},\mu^{(1),-1/2}\right)}T^{\varphi_{t}}_{1,-}\left(\lambda^{(1),-1/2},\mu^{(1),-1/2}\right).$

If we let $Q_{N}$ be the generator of the random surface growth with a wall
restricted to $\mathbb{J}^{N}$, then the continuous-time Markov chain can be
characterized by matrix $A^{\varphi_{t}}_{N}$.

######  Theorem 2.2.

(Theroem 3.12 in [10]) Let $\varphi_{t}(x)=e^{t(x-1)}$, then

$e^{tQ_{N}}P^{N,\omega}=A^{\varphi_{t}}_{N}P^{N,\omega}.$

##  3 Correlations along space-like paths

In this section, we compute the multi-time correlation functions for the
Markov process on $\mathbb{J}^{N}$.

######  Proposition 3.1.

Consider the evolution of the measure $P^{N,\omega}$ on $\mathbb{J}^{N}$
defined in (2.6) under the Markov chain $A^{\varphi_{t}}_{N}$, and denote by
$\left(x^{1,-1/2}(t),\ldots,x^{n,a}(t)\right)$ the result after time $t\geq
0$. Then for any

$0=t^{N}_{0}\leq\cdots\leq t^{N}_{c(N)}=t^{N-1}_{0}\leq\cdots\leq
t^{N-1}_{c(N-1)}=\\\ t^{N-2}_{0}\leq\cdots\leq
t^{2}_{c(2)}=t^{1}_{0}\leq\cdots\leq t^{1}_{c(1)},$

where $c(i)$ are arbitrary non-negative integers, the joint distribution of

$x^{n,a}\left(t^{N}_{0}\right),\ldots
x^{n,a}\left(t^{N}_{c(N)}\right),x^{n-1/2+a,-a}\left(t^{N-1}_{0}\right),\ldots\\\
\ldots,x^{n-1/2+a,-a}\left(t^{N-1}_{c(N-1)}\right),x^{n-3/2-a,a}\left(t^{N-2}_{0}\right),\ldots,x^{1,1/2}\left(t^{2}_{c(2)}\right),x^{1,-1/2}\left(t^{1}_{0}\right),\ldots,x^{1,-1/2}\left(t^{1}_{c(1)}\right)$

coincides with the stochastic evolution of $P^{N,\omega}$ under transition
matrices

$T^{\varphi_{t^{N}_{1}-t^{N}_{0}}}_{n,a},\ldots,T^{\varphi_{t^{N}_{c(N)}-t^{N}_{c(N)-1}}}_{n,a},T^{n,a}_{n-1/2+a,-a},T^{\varphi_{t^{N-1}_{1}-t^{N-1}_{0}}}_{n-1/2+a,-a},\ldots,\\\
\ldots
T^{\varphi_{t^{N-1}_{c(N-1)}-t^{N-1}_{c(N-1)-1}}}_{n-1/2+a,-a},T^{n-1/2+a,-a}_{n-3/2-a,a},T^{\varphi_{t^{N-2}_{1}-t^{N-2}_{0}}}_{n-3/2-a,a},\ldots,T^{\varphi_{t^{1}_{c(1)}-t^{1}_{c(1)-1}}}_{1,+}.$

###### Proof.

See Proposition 2.5 of [8] and Proposition 3.6 of [10]. ∎

######  Definition 3.1.

For any $M\geq 1$, pick $M$ points

$\varkappa_{j}=(n_{j},a_{j},t_{j},s_{j})\in\mathbb{Z}_{>0}\times\\{\pm\frac{1}{2}\\}\times\mathbb{R}_{\geq
0}\times\mathbb{Z}_{\geq 0},$

Given the Markov process $x(t)$, The $M$-th correlation function $\rho_{M}$ at
$(\varkappa_{1},\ldots,\varkappa_{M})$ is defined as

$\rho(\varkappa_{1},\ldots,\varkappa_{M})=\text{Prob}\Big{(}\text{For each
}1\leq j\leq M,\text{ there exists a }k_{j},1\leq k_{j}\leq n_{j},\text{ such
that }x_{k_{j}}^{n_{j},a_{j}}(t_{j})=s_{j}\Big{)}.$

Next we introduce a partial order on $(n,a,t)$ via
$(n_{1},a_{1},t_{1})\prec(n_{2},a_{2},t_{2})$ if $2n_{1}-1/2+a_{1}\leq
2n_{2}-1/2+a_{2}$, $t_{1}\geq t_{2}$ and
$(n_{1},a_{1},t_{1})\neq(n_{2},a_{2},t_{2})$.

######  Theorem 3.1.

Consider the Markov process with the initial distribution $P^{N,\omega}$ given
by (2.6), let $\varphi_{t}(x)=e^{t(x-1)}$. Assume any two distinct pairs in
$\left\\{\varkappa_{j}=(n_{j},a_{j},t_{j},s_{j})|1\leq j\leq M\right\\}$ are
comparable with respect to $\prec$. Then

$\rho(\varkappa_{1},\cdots,\varkappa_{M})=\text{det}\left[K(\varkappa_{i},\varkappa_{j})\right]_{i,j=1}^{M},$
(3.1)

where $K(\varkappa_{i},\varkappa_{j})$ is the correlation kernel given by

$K(\varkappa_{i},\varkappa_{j})=\\\
\frac{W^{\left(a_{1},-\frac{1}{2}\right)}(s_{1})}{\pi}\frac{1}{2\pi
i}\int_{-1}^{1}\oint_{C}\mathcal{J}_{s_{1}}^{\left(a_{1},-\frac{1}{2}\right)}(y){J}_{s_{2}}^{\left(a_{2},-\frac{1}{2}\right)}(u)\frac{E^{\omega}(y)}{E^{\omega}(u)}\frac{e^{t_{1}(y-1)}}{e^{t_{2}(u-1)}}\frac{(y-1)^{n_{1}}}{(u-1)^{n_{2}}}\frac{(1-y)^{a_{1}}(1+y)^{-1/2}}{y-u}dudy\\\
+\mathbb{1}_{(n_{1},a_{1},t_{1})\succ(n_{2},a_{2},t_{2})}\frac{W^{\left(a_{1},-\frac{1}{2}\right)}(s_{1})}{\pi}\int_{-1}^{1}\mathcal{J}_{s_{1}}^{\left(a_{1},-\frac{1}{2}\right)}(y){J}_{s_{2}}^{\left(a_{2},-\frac{1}{2}\right)}(y)\frac{e^{t_{1}(y-1)}}{e^{t_{2}(y-1)}}(y-1)^{n_{1}-n_{2}}(1-y)^{a_{1}}(1+y)^{-1/2}dy.$

The $u$-contour $C$ is a positively oriented simple loop that encircles the
interval $[-1,1]$ but does not encircle any zeroes of $E^{\omega}$.

######  Remark 3.1.

The correlation kernel in Theorem 3.1 degenerates to the kernel in Theorem 4.1
of [10] when $t_{1}=t_{2}$, but with slightly different notations: In [10],
the time was implicitly written in the parameter $\omega$, where they
introduced a parameter
$\gamma=\delta-\sum_{i=1}^{\infty}(\alpha_{i}+\beta_{i})$ in the definition of
function $E^{\omega}$ and let it to be the time parameter. In this paper, the
function $E^{\omega}$ defined in 2.2 is fixed as time changes.

###  3.1 Determinantal structure of the correlation functions

The initial conditions for the Markov process are the distributions
$P^{N,\omega}$. Proposition 3.1 implies that the joint distribution of

$x^{n,a}\left(t^{N}_{0}\right),\ldots
x^{n,a}\left(t^{N}_{c(N)}\right),x^{n-1/2+a,-a}\left(t^{N-1}_{0}\right),\ldots\\\
\ldots,x^{n-1/2+a,-a}\left(t^{N-1}_{c(N-1)}\right),x^{n-3/2-a,a}\left(t^{N-2}_{0}\right),\ldots,x^{1,1/2}\left(t^{2}_{c(2)}\right),x^{1,-1/2}\left(t^{1}_{0}\right),\ldots,x^{1,-1/2}\left(t^{1}_{c(1)}\right)$

takes the form

$\text{const}\times\prod_{m=1}^{n}\Bigg{[}\text{det}\left[\phi_{+}^{-}\left(x_{l}^{m,-}\left(t^{2m-1}_{c(2m-1)}\right),x_{k}^{m-1,+}\left(t^{2m-2}_{0}\right)\right)\right]_{1\leq
k,l\leq m}\\\
\times\prod_{b=1}^{c(2m-1)}\text{det}\left[\mathcal{T}_{t^{2m-1}_{b-1},t^{2m-1}_{b}}^{m,-}\left(x_{l}^{m,-}\left(t^{2m-1}_{b-1}\right),x_{k}^{m,-}\left(t^{2m-1}_{b}\right)\right)\right]_{1\leq
k,l\leq m}\\\
\times\text{det}\left[\phi^{+}_{-}\left(x_{l}^{m,+}\left(t^{2m}_{c(2m)}\right),x_{k}^{m,-}\left(t^{2m-1}_{0}\right)\right)\right]_{1\leq
k,l\leq m}\\\
\times\prod_{b=1}^{c(2m)}\text{det}\left[\mathcal{T}_{t^{2m}_{b-1},t^{2m}_{b}}^{m,+}\left(x_{l}^{m,+}\left(t^{2m}_{b-1}\right),x_{k}^{m,+}\left(t^{2m}_{b}\right)\right)\right]_{1\leq
k,l\leq m}\Bigg{]}\\\
\times\text{det}\left[\psi_{n-l}^{n,a}\left(x_{k}^{n,a}\left(t^{N}_{0}\right)\right)\right]_{1\leq
k,l\leq n},$ (3.2)

where $\mathcal{T}_{t_{1},t_{2}}^{n,a}=I_{a}^{\varphi_{t_{2}-t_{1}}}$ and
$x^{m-1,+}_{m}$ is set to be $-1$, we refer to this variable as virt. If
$a=-1/2$, then the last determinant of
$\phi^{+}_{-}\left(x_{l}^{n,+}\left(t^{2n}_{c(2n)}\right),x_{k}^{n,-}\left(t^{2n-1}_{0}\right)\right)$
and the product of determinants of
$\mathcal{T}^{n,+}_{t^{2n}_{b-1},t^{2n}_{b}}$ do not occur.

With the above formula for the joint distribution, we apply an analogue of
Theorem 4.2 of [6] to compute the correlation kernel. Let $*$ denote
convolution:

$f*g(x,y)=\sum_{z\geq 0}f(x,z)g(z,y).$

For any $n$, $a$ and any two time moments $t_{i}^{2n-1/2+a}<t_{j}^{2n-1/2+a}$,
define

$\mathcal{T}^{n,a}_{t^{2n-1/2+a}_{i},t_{j}^{2n-1/2+a}}=\mathcal{T}^{n,a}_{t^{2n-1/2+a}_{i},t_{i+1}^{2n-1/2+a}}\ast\mathcal{T}^{n,a}_{t^{2n-1/2+a}_{i+1},t_{i+2}^{2n-1/2+a}}\ast\cdots\ast\mathcal{T}^{n,a}_{t^{2n-1/2+a}_{j-1},t_{j}^{2n-1/2+a}}$

and

$\mathcal{T}^{n,a}=\mathcal{T}^{n,a}_{t_{0}^{2n-1/2+a},t^{2n-1/2+a}_{c({2n-1/2+a})}}.$

For any time moments
$t^{2n_{2}-1/2+a_{2}}_{b_{2}}<t^{2n_{1}-1/2+a_{1}}_{b_{1}}$ with $n_{1}\leq
n_{2}$, we denote the convolution over all transitions between them by
$\phi^{t^{2n_{2}-1/2+a_{2}}_{b_{2}},t^{2n_{1}-1/2+a_{1}}_{b_{1}}}$:

$\phi^{t^{2n_{2}-1/2+a_{2}}_{b_{2}},t^{2n_{1}-1/2+a_{1}}_{b_{1}}}=\mathcal{T}^{n_{2},a_{2}}_{t^{2n_{2}-1/2+a_{2}}_{b_{2}},t^{2n_{2}-1/2+a_{2}}_{c(2n_{2}-1/2+a_{2})}}\ast\phi_{-a_{2}}^{a_{2}}\ast\mathcal{T}^{n_{2}-1/2+a_{2},-a_{2}}\ast\cdots\\\
\cdots\ast\phi_{a_{1}}^{-a_{1}}\ast\mathcal{T}^{n_{1},a_{1}}_{t^{2n_{1}-1/2+a_{1}}_{0},t^{2n_{1}-1/2+a_{1}}_{b_{1}}}.$

For example, when $a_{1}=-a_{2}=1/2$,

$\phi^{t^{2n_{2}-1/2+a_{2}}_{b_{2}},t^{2n_{1}-1/2+a_{1}}_{b_{1}}}=\mathcal{T}^{n_{2},-}_{t^{2n_{2}-1}_{b_{2}},t^{2n_{2}-1}_{c(2n_{2}-1)}}\ast\phi_{+}^{-}\ast\mathcal{T}^{n_{2}-1,+}\ast\cdots\ast\phi_{+}^{-}\ast\mathcal{T}^{n_{1},+}_{t^{2n_{1}}_{0},t^{2n_{1}}_{b_{1}}}.$

When $t^{2n_{1}-1/2+a_{1}}_{b_{1}}\leq t^{2n_{2}-1/2+a_{2}}_{b_{2}}$, let

$\phi^{t^{2n_{2}-1/2+a_{2}}_{b_{2}},t^{2n_{1}-1/2+a_{1}}_{b_{1}}}=0.$

For $a=\pm\frac{1}{2}$, we define matrices $M^{n,a}=[M^{n,a}]_{k,l=1}^{n}$
with entries

$M_{k,l}^{n,a}=\psi^{n,a}_{n-l}\ast\mathcal{T}^{n,a}\ast\phi_{-a}^{a}\ast\mathcal{T}^{n-1/2+a,-a}\ast\cdots\ast\mathcal{T}^{k,-}\ast\phi_{+}^{-}(\text{virt}).$

where virt is the virtual variable for $\phi_{+}^{-}$, which is equal to $-1$.

Last, for $N=2n-1/2+a$ and $l\leq n$, define

$\psi_{k-l}^{t^{2k-1/2+a_{1}}_{b},N}=\psi^{n,a}_{n-l}\ast\phi^{t^{N}_{0},t^{2k-1/2+a_{1}}_{b}}.$

######  Lemma 3.1.

If $M^{n,a}$ is upper triangular and invertible, there exist functions
$\Phi_{k-j}^{t^{2k-1/2+a_{1}}_{b},N}(s)$ such that

  1. (1)

$\left\\{\Phi_{k-j}^{t^{2k-1/2+a_{1}}_{b},N}(s)\right\\}_{j=1}^{k}$ is a basis
of the linear span of

$\left\\{\mathcal{T}_{t^{2k-1/2+a_{1}}_{b},t^{2k-1/2+a_{1}}_{c\left(2k-1/2+a_{1}\right)}}^{k,a_{1}}\ast\cdots\ast\mathcal{T}_{t^{2j-1}_{0},t^{2j-1}_{c\left(2j-1\right)}}^{j,-}\ast\phi_{+}^{-}(s,\text{virt})\right\\}_{j=1}^{k}.$

  2. (2)

For $0\leq j_{1},j_{2}\leq k-1$,

$\sum_{s\geq
0}\Phi_{j_{1}}^{t^{2k-1/2+a_{1}}_{b},N}(s)\psi_{j_{2}}^{t^{2k-1/2+a_{1}}_{b},N}(s)=\delta_{j_{1},j_{2}}.$

Then the correlation kernel for two comparable pairs $(n_{1},a_{1},t_{1})$ and
$(n_{2},a_{2},t_{2})$ with $2n_{i}-1/2+a_{i}\leq N$ is given by

$K(n_{1},a_{1},t_{1},s_{1};n_{2},a_{2},t_{2},s_{2})=-\phi^{t^{2n_{2}-1/2+a_{2}}_{b_{2}},t^{2n_{1}-1/2+a_{1}}_{b_{1}}}(s_{1},s_{2})+\sum_{k=1}^{n_{2}}\psi_{n_{1}-k}^{t^{2n_{1}-1/2+a_{1}}_{b_{1}},N}(s_{1})\Phi_{n_{2}-k}^{t^{2n_{2}-1/2+a_{2}}_{b_{2}},N}(s_{2}),$

where $t^{2n_{i}-1/2+a_{i}}_{b_{i}}=t_{i}$, $i=1,2$.

###### Proof.

The proof is similar to the proof of Theorem 4.2 in [6] and Lemma 3.4 of [9].

First we define a matrix $L^{n,a}$ such that the measure in (3.2) is
proportional to a suitable symmetric minor of $L^{n,a}$.

$L^{n,a}=\left(\begin{array}[]{cccccccccccc}0&E_{1}&0&0&0&E_{2}&\cdots&E_{n}&0&0&0\\\
0&0&-\mathbb{T}^{1,-}&0&0&0&\cdots&0&0&0&0\\\
0&0&0&-F_{1,-}^{1,+}&0&0&\cdots&0&0&0&0\\\
0&0&0&0&-\mathbb{T}^{1,+}&0&\cdots&0&0&0&0\\\
0&0&0&0&0&-F_{1,+}^{2,-}&\cdots&0&0&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots\\\
0&0&0&0&0&0&\cdots&-F_{n-1,+}^{n,-}&0&0&0\\\
0&0&0&0&0&0&\cdots&0&-\mathbb{T}^{n,-}&0&0\\\
0&0&0&0&0&0&\cdots&0&0&-F_{n,-}^{n,+}&0\\\
0&0&0&0&0&0&\cdots&0&0&0&-\mathbb{T}^{n,+}\\\
\Psi^{n,a}&&0&0&0&0&\cdots&0&0&0&0\end{array}\right).$

If $a=-1/2$, the last two columns do not occur. The matrix blocks in $L^{n,a}$
have the following entries:

$\left[\Psi^{n,a}\right]_{k,l}=\psi_{n-l}^{n,a}\left(x_{k}^{n,a}\left(t^{N}_{0}\right)\right),\quad{1\leq
k,l\leq n}.$
$\left[E_{m}\right]_{kl}=\left\\{\begin{array}[]{cc}\phi_{+}^{-}\left(x_{l}^{m,-}\left(t^{2m-1}_{c(2m-1)}\right),x_{m}^{m-1,+}\left(t^{2m-2}_{0}\right)\right),&k=m,\
1\leq l\leq m,\\\ 0,&1\leq k\leq n,\ k\neq m,\ 1\leq l\leq
m,\end{array}\right.$

$\left[F_{m,-}^{m,+}\right]_{k,l}=\phi^{+}_{-}\left(x_{l}^{m,+}\left(t^{2m}_{c(2m)}\right),x_{k}^{m,-}\left(t^{2m-1}_{0}\right)\right),\quad{1\leq
k,l\leq m},$
$\left[F_{m-1,+}^{m,-}\right]_{k,l}=\phi_{+}^{-}\left(x_{l}^{m,-}\left(t^{2m-1}_{c(2m-1)}\right),x_{k}^{m-1,+}\left(t^{2m-2}_{0}\right)\right),\quad{1\leq
k\leq m-1,\ 1\leq l\leq m},$

and $\mathbb{T}^{m,\pm}$ is the matrix made of blocks

$\mathbb{T}^{m,\pm}=\left(\begin{array}[]{cccc}\mathbb{T}^{m,\pm}_{1}&0&0\\\
0&\ddots&0\\\ 0&0&\mathbb{T}^{m,\pm}_{c(2m-1/2\pm 1/2)}\end{array}\right),$

where

$\left[\mathbb{T}^{m,\pm}_{b}\right]_{k,l}=\mathcal{T}_{t^{2m-1/2\pm
1/2}_{b-1},t^{2m-1/2\pm 1/2}_{b}}^{m,\pm}\left(x_{l}^{m,\pm}\left(t^{2m-1/2\pm
1/2}_{b-1}\right),x_{k}^{m,\pm}\left(t^{2m-1/2\pm 1/2}_{b}\right)\right),\\\
1\leq k,l\leq m,\ 1\leq b\leq c(2m-1/2\pm 1/2).$

The rest of the proof is along the same lines as that of Lemma 3.4 in [9]. ∎

In the following sections, we compute each term that appears in Lemma 3.1. We
first calculate the function
$\phi^{t^{2n_{2}-1/2+a_{2}}_{b_{2}},t^{2n_{1}-1/2+a_{1}}_{b_{1}}}$ in section
3.2, then we calculate matrix $M^{n,a}$ and the function
$\psi_{n-k}^{t^{2k-1/2+a_{1}}_{b},N}$ in section 3.3. In section 3.4, we
compute $\Phi_{k-j}^{t^{2k-1/2+a_{1}}_{b},N}(s)$ and
$\displaystyle{\sum_{k=1}^{n_{2}}\psi_{n_{1}-k}^{t^{2n_{1}-1/2+a_{1}}_{b_{1}},N}(s_{1})\Phi_{n_{2}-k}^{t^{2n_{2}-1/2+a_{2}}_{b_{2}},N}(s_{2})}$.
Last, in section 3.5, we combine all the functions obtained from section
3.3–3.4 to get the correlation kernel $K$ in Theorem 3.1.

###  3.2 Calculating
$\phi^{t^{2n_{2}-1/2+a_{2}}_{b_{2}},t^{2n_{1}-1/2+a_{1}}_{b_{1}}}$

Recall that $\varphi_{t}(x)=e^{t(x-1)}$, for ease of notation, let
$\varphi_{n,+}(x)=\varphi_{t^{2n}_{c(2n)}-t^{2n}_{0}}(x)$,
$\varphi_{n,-}(x)=\varphi_{t^{2n-1}_{c(2n-1)}-t^{2n-1}_{0}}(x)$ and
$\varphi_{n,\pm}(x)=\varphi_{n,+}(x)\varphi_{n,-}(x)=\varphi_{t^{2n}_{c(2n)}-t^{2n-1}_{0}}(x)$.
Let’s recall some useful identities from Lemma 2.2–2.7 in [10].

######  Lemma 3.2.

[10] Let $T(x)\in C^{1}[-1,1]$, the following identities hold:

  1. (1)

$\sum_{r=0}^{s}W^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(r)\mathcal{J}_{r}^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(x)=\mathcal{J}_{s}^{\left(\frac{1}{2},-\frac{1}{2}\right)}(x),$

  2. (2)

$\frac{1}{\pi}\int_{-1}^{1}\mathcal{J}_{r}^{\left(\frac{1}{2},-\frac{1}{2}\right)}(x)(1-x)^{-1/2}(1+x)^{-1/2}dx=1,$

  3. (3)

$\sum_{r=0}^{\infty}\frac{W^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(r)}{\pi}\int_{-1}^{1}\mathcal{J}_{r}^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(x)T(x)(1-x)^{-1/2}(1+x)^{-1/2}dx=T(1),$

  4. (4)

$\sum_{r=s+1}^{\infty}\frac{W^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(r)}{\pi}\int_{-1}^{1}\mathcal{J}_{r}^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(x)T(x)(1-x)^{-1/2}(1+x)^{-1/2}dx\\\
=\frac{1}{\pi}\int_{-1}^{1}\mathcal{J}_{s}^{\left(\frac{1}{2},-\frac{1}{2}\right)}(x)\left(T(1)-T(x)\right)(1-x)^{-1/2}(1+x)^{-1/2}dx,$

  5. (5)

$\sum_{r=s}^{\infty}\frac{W^{\left(\frac{1}{2},-\frac{1}{2}\right)}(r)}{\pi}\int_{-1}^{1}\mathcal{J}_{r}^{\left(\frac{1}{2},-\frac{1}{2}\right)}(x)T(x)(1-x)^{1/2}(1+x)^{-1/2}dx\\\
=\frac{1}{\pi}\int_{-1}^{1}\mathcal{J}_{s}^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(x)T(x)(1-x)^{-1/2}(1+x)^{-1/2}dx,$

  6. (6)

for $a=\pm\frac{1}{2}$, $-1\leq\zeta\leq 1$,

$\sum_{k=0}^{\infty}\frac{W^{\left(a,-\frac{1}{2}\right)}(k)}{\pi}\int_{-1}^{1}\mathcal{J}_{k}^{\left(a,-\frac{1}{2}\right)}(x)\mathcal{J}_{k}^{\left(a,-\frac{1}{2}\right)}(\zeta)T(x)(1-x)^{a}(1+x)^{-1/2}dx=T(\zeta).$

We start by computing some basic convolutions that will be useful later.

######  Proposition 3.2.

  1. (1)

$\mathcal{T}^{n,a}_{t^{2n-1/2+a}_{i},t_{j}^{2n-1/2+a}}=I_{a}^{\varphi_{t^{2n-1/2+a}_{j}-t_{i}^{2n-1/2+a}}}.$

  2. (2)

$\mathcal{T}^{n,+}\ast\phi_{-}^{+}(m,l)=\frac{W^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(l)}{\pi}\int_{-1}^{1}\mathcal{J}_{l}^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(x)\mathcal{J}_{m}^{\left(\frac{1}{2},-\frac{1}{2}\right)}(x)\varphi_{n,+}(x)(1-x)^{-1/2}(1+x)^{-1/2}dx.$

  3. (3)

$\mathcal{T}^{n,-}\ast\phi_{+}^{-}(m,l)=\\\
\left\\{\begin{array}[]{cc}\begin{split}\\\
\frac{W^{\left(\frac{1}{2},-\frac{1}{2}\right)}(l)}{\pi}\int_{-1}^{1}\mathcal{J}_{l}^{\left(\frac{1}{2},-\frac{1}{2}\right)}(x)\left(\mathcal{J}_{m}^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(1)\varphi_{n,-}(1)-\mathcal{J}_{m}^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(x)\varphi_{n,-}(x)\right)\times\\\
(1-x)^{-1/2}(1+x)^{-1/2}dx,\end{split}&m,l\geq 0,\\\
\mathcal{J}_{m}^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(1)\varphi_{n,-}(1),&m\geq
0,\ l=\text{virt}.\end{array}\right.$

  4. (4)

$\mathcal{T}^{n,+}\ast\phi_{-}^{+}\ast\mathcal{T}^{n,-}\ast\phi_{+}^{-}(m,l)=\\\
\left\\{\begin{array}[]{lc}\begin{split}-\frac{1}{\pi}\int_{-1}^{1}\mathcal{J}_{m}^{\left(\frac{1}{2},-\frac{1}{2}\right)}(y)\mathcal{J}_{l}^{\left(\frac{1}{2},-\frac{1}{2}\right)}(y)\varphi_{n,\pm}(y)(1-y)^{-1/2}(1+y)^{-1/2}dy\\\
+\mathcal{J}_{m}^{\left(\frac{1}{2},-\frac{1}{2}\right)}(1)\varphi_{n,\pm}(1),\end{split}&m\geq
0,\ l\geq 0,\\\
\mathcal{J}_{m}^{\left(\frac{1}{2},-\frac{1}{2}\right)}(1)\varphi_{n,\pm}(1),&m\geq
0,\ l=\text{virt}.\\\ \end{array}\right.$

###### Proof.

Results (1)–(3) follow directly from Lemma 3.2. We only show the computation
for (4). By definition, when $m,l\geq 0$,

$\mathcal{T}^{n,+}\ast\phi_{-}^{+}\ast\mathcal{T}^{n,-}\ast\phi_{+}^{-}(m,l)=\\\
\frac{1}{\pi^{2}}\sum_{k=0}^{\infty}\left(W^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(k)\int_{-1}^{1}\mathcal{J}_{k}^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(x)\mathcal{J}_{m}^{\left(\frac{1}{2},-\frac{1}{2}\right)}(x)\varphi_{n,+}(x)(1-x)^{-1/2}(1+x)^{-1/2}dx\times\right.\\\
\left.\int_{-1}^{1}\mathcal{J}_{l}^{\left(\frac{1}{2},-\frac{1}{2}\right)}(y)\left(\mathcal{J}_{k}^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(1)\varphi_{n,-}(1)-\mathcal{J}_{k}^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(y)\varphi_{n,-}(y)\right)(1-y)^{-1/2}(1+x)^{-1/2}dy\right).$

According to Lemma 3.2 (2),

$\frac{1}{\pi}\int_{-1}^{1}\mathcal{J}_{l}^{\left(\frac{1}{2},-\frac{1}{2}\right)}(y)\mathcal{J}_{k}^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(1)\varphi_{n,-}(1)(1-y)^{-1/2}(1+y)^{-1/2}dy=\mathcal{J}_{k}^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(1)\varphi_{n,-}(1),$

thus we have

$\mathcal{T}^{n,+}\ast\phi_{-}^{+}\ast\mathcal{T}^{n,-}\ast\phi_{+}^{-}(m,l)=\\\
-\frac{1}{\pi^{2}}\sum_{k=0}^{\infty}W^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(k)\left(\int_{-1}^{1}\mathcal{J}_{k}^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(x)\mathcal{J}_{m}^{\left(\frac{1}{2},-\frac{1}{2}\right)}(x)\varphi_{n,+}(x)(1-x)^{-1/2}(1+x)^{-1/2}dx\times\right.\\\
\left.\int_{-1}^{1}\mathcal{J}_{l}^{\left(\frac{1}{2},-\frac{1}{2}\right)}(y)\mathcal{J}_{k}^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(y)\varphi_{n,-}(y)(1-y)^{-1/2}(1+y)^{-1/2}dy\right)\\\
+\frac{1}{\pi}\sum_{k=0}^{\infty}W^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(k)\mathcal{J}_{k}^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(1)\varphi_{n,-}(1)\int_{-1}^{1}\mathcal{J}_{k}^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(x)\mathcal{J}_{m}^{\left(\frac{1}{2},-\frac{1}{2}\right)}(x)\varphi_{n,+}(x)(1-x)^{-1/2}(1+x)^{-1/2}dx.$
(3.3)

Apply Lemma 3.2 (3) to the second summation in (3.3),

$\frac{1}{\pi}\sum_{k=0}^{\infty}W^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(k)\mathcal{J}_{k}^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(1)\varphi_{n,-}(1)\int_{-1}^{1}\mathcal{J}_{k}^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(x)\mathcal{J}_{m}^{\left(\frac{1}{2},-\frac{1}{2}\right)}(x)\varphi_{n,+}(x)(1-x)^{-1/2}(1+x)^{-1/2}dx\\\
=\mathcal{J}_{m}^{\left(\frac{1}{2},-\frac{1}{2}\right)}(1)\varphi_{n,\pm}(1).$

For the first summation, we apply Lemma 3.2 (6),

$\frac{1}{\pi^{2}}\sum_{k=0}^{\infty}W^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(k)\left(\int_{-1}^{1}\mathcal{J}_{k}^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(x)\mathcal{J}_{m}^{\left(\frac{1}{2},-\frac{1}{2}\right)}(x)\varphi_{n,+}(x)(1-x)^{-1/2}(1+x)^{-1/2}dx\times\right.\\\
\left.\int_{-1}^{1}\mathcal{J}_{l}^{\left(\frac{1}{2},-\frac{1}{2}\right)}(y)\mathcal{J}_{k}^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(y)\varphi_{n,-}(y)(1-y)^{-1/2}(1+y)^{-1/2}dy\right)\\\
=\frac{1}{\pi}\int_{-1}^{1}\mathcal{J}_{m}^{\left(\frac{1}{2},-\frac{1}{2}\right)}(y)\mathcal{J}_{l}^{\left(\frac{1}{2},-\frac{1}{2}\right)}(y)\varphi_{n,\pm}(y)(1-y)^{-1/2}(1+y)^{-1/2}dy.$

When $m\geq 0$ and $\ l=\text{virt}$, the proof is the same. ∎

In the following, we omit computation of convolutions if it’s similar to that
of Proposition 3.2.

It is beneficial to write the results in Proposition 3.2 as contour integrals.
Note that

$\mathcal{J}_{m}^{\left(\pm\frac{1}{2},-\frac{1}{2}\right)}(x)\varphi_{n,\pm}(x)=\frac{1}{2\pi
i}\oint\frac{\mathcal{J}_{m}^{\left(\pm\frac{1}{2},-\frac{1}{2}\right)}(u)\varphi_{n,\pm}(u)}{u-x}du,$

where the $u$-contour is a positively oriented simple loop that encircles
point $x$. Then, if we enlarge the $u$-contour such that it encircles the
interval $[-1,1]$,

$\mathcal{T}^{n,-}\ast\phi_{n-1,+}^{n,-}(m,l)=\\\
\left\\{\begin{array}[]{lc}\frac{W^{\left(\frac{1}{2},-\frac{1}{2}\right)}(l)}{\pi}\int_{-1}^{1}\mathcal{J}_{l}^{\left(\frac{1}{2},-\frac{1}{2}\right)}(x)\frac{1}{2\pi
i}\oint\frac{\mathcal{J}_{m}^{\left(-\frac{1}{2},-\frac{1}{2}\right)}(u)\varphi_{n,+}(u)}{(u-1)(u-x)}(1-x)^{1/2}(1+x)^{-1/2}dudx,&m,l\geq
0,\\\ \frac{1}{2\pi
i}\oint\frac{\mathcal{J}_{m}^{\left(\frac{1}{2},-\frac{1}{2}\right)}(u)\varphi_{n,+}(u)}{u-1}du,&m\geq
0,\ l=\text{virt}.\end{array}\right.$

$\mathcal{T}^{n,+}\ast\phi_{n,-}^{n,+}\ast\mathcal{T}^{n,-}\ast\phi_{n-1,+}^{n,-}(m,l)=\\\
\left\\{\begin{array}[]{lc}\frac{1}{\pi}\int_{-1}^{1}\mathcal{J}_{l}^{\left(\frac{1}{2},-\frac{1}{2}\right)}(x)\frac{1}{2\pi
i}\oint\frac{\mathcal{J}_{m}^{\left(\frac{1}{2},-\frac{1}{2}\right)}(u)\varphi_{n,\pm}(u)}{(u-1)(u-x)}(1-x)^{1/2}(1+x)^{-1/2}dudx,&m,l\geq
0,\\\ \frac{1}{2\pi
i}\oint\frac{\mathcal{J}_{m}^{\left(\frac{1}{2},-\frac{1}{2}\right)}(u)\varphi_{n,\pm}(u)}{u-1}du,&m\geq
0,\ l=\text{virt}.\\\ \end{array}\right.$

Following the same computation, we compute iterative convolutions of the
functions.

######  Proposition 3.3.

For $k\geq 0$,

$\mathcal{T}^{n,+}\ast\phi_{-}^{+}\ast\cdots\ast\mathcal{T}^{n-k,-}\ast\phi_{+}^{-}(m,l)=\\\
\left\\{\begin{array}[]{lc}\begin{split}\frac{1}{\pi}\int_{-1}^{1}\mathcal{J}_{l}^{\left(\frac{1}{2},-\frac{1}{2}\right)}(x)\frac{1}{(2\pi
i)^{k+1}}\oint\cdots\oint\frac{H^{n,k}_{m}\left(u_{n},u_{n-1},\ldots,u_{n-k}\right)}{(u_{n-k}-x)}du_{n}\cdots
du_{n-k}\\\ \times(1-x)^{1/2}(1+x)^{-1/2}dx,\end{split}&m\geq 0,\ l\geq 0,\\\
\frac{1}{(2\pi i)^{k+1}}\oint\cdots\oint
H^{n,k}_{m}(u_{n},u_{n-1},\ldots,u_{n-k})du_{n}\cdots du_{n-k},&m\geq 0,\
l=\text{virt},\\\ \end{array}\right.$

where

$H^{n,k}_{m}\left(u_{n},u_{n-1},\ldots,u_{n-k+1}\right)=\frac{J_{m}^{\left(\frac{1}{2},-\frac{1}{2}\right)}(u_{n})\varphi_{n,\pm}(u_{n})\varphi_{n-1,\pm}(u_{n-1})\cdots\varphi_{n-k,\pm}(u_{n-k})}{(u_{n}-1)(u_{n}-u_{n-1})(u_{n-1}-1)(u_{n-1}-u_{n-2})\cdots(u_{n-k}-1)}$

and the $u_{i}$-contour is a positively oriented simple loop that encircles
point $1$ together with the $u_{i-1}$-contour when $n-k+1\leq i\leq n$, the
$u_{n-k}$-contour is a positively oriented simple loop that encircles the
interval $[-1,1]$.

Now we are ready to compute
$\phi^{t^{2n_{2}-1/2+a_{2}}_{b_{2}},t^{2n_{1}-1/2+a_{1}}_{b_{1}}}$.

######  Proposition 3.4.

Suppose $t^{2n_{1}-1/2+a_{1}}_{b_{1}}>t^{2n_{2}-1/2+a_{2}}_{b_{2}}$, if
$n_{1}=n_{2}$,

$\phi^{t^{2n_{2}-1/2+a_{2}}_{b_{2}},t^{2n_{1}-1/2+a_{1}}_{b_{1}}}(s_{1},s_{2})=\frac{W^{\left(a_{1},-\frac{1}{2}\right)}(s_{1})}{\pi}\int_{-1}^{1}\mathcal{J}_{s_{1}}^{\left(a_{1},-\frac{1}{2}\right)}(x)J_{s_{2}}^{\left(a_{2},-\frac{1}{2}\right)}(x)\times\\\
\varphi_{t^{2n_{1}-1/2+a_{1}}_{b_{1}}-t^{2n_{2}-1/2+a_{2}}_{b_{2}}}(x)(1-x)^{a_{1}}(1+x)^{-1/2}dx;$

if $n_{1}<n_{2}$,

$\phi^{t^{2n_{2}-1/2+a_{2}}_{b_{2}},t^{2n_{1}-1/2+a_{1}}_{b_{1}}}(s_{1},s_{2})\\\
=\frac{W^{\left(a_{1},-\frac{1}{2}\right)}(s_{1})}{\pi}\int_{-1}^{1}\mathcal{J}_{s_{1}}^{\left(a_{1},-\frac{1}{2}\right)}(x)g^{a_{2}}_{n_{1}+1,n_{2}}(x)\varphi_{t^{2n_{1}-1/2+a_{1}}_{b_{1}}-t^{2n_{1}}_{0}}(x)(1-x)^{a_{1}}(1+x)^{-1/2}dx,$

where

$g^{a_{2}}_{n_{1}+1,n_{2}}(x)=\frac{1}{(2\pi i)^{n_{2}-n_{1}}}\times\\\
\oint\cdots\oint\frac{J_{s_{2}}^{\left(a_{2},-\frac{1}{2}\right)}(u_{n_{2}})\varphi_{t^{2n_{2}-1}_{c(2n_{2}-1)}-t^{2n_{2}-1/2+a_{2}}_{b_{2}}}(u_{n_{2}})\varphi_{n_{2}-1,\pm}(u_{n_{2}-1})\cdots\varphi_{n_{1}+1,\pm}(u_{n_{1}+1})}{(u_{n_{2}}-1)(u_{n_{2}}-u_{n_{2}-1})\cdots(u_{n_{1}+1}-1)(u_{n_{1}+1}-x)}du_{n_{2}}\cdots
du_{n_{1}+1}.$

###### Proof.

Convoluting the leftover terms with the expression for
$\mathcal{T}^{n,+}\ast\phi_{n,-}^{n,+}\ast\cdots\ast\mathcal{T}^{n-k,-}\ast\phi_{n-k-1,+}^{n-k,-}$
in Proposition 3.3 finishes the proof. ∎

###  3.3 The matrix $M^{n,a}$ and $\psi_{k-l}^{t^{2k-1/2+a_{1}}_{b},N}$

######  Proposition 3.5.

When $k\geq l$,

$\psi_{k-l}^{t^{2k-1/2+a_{1}}_{b},N}(s)=\frac{W^{\left(a_{1},-\frac{1}{2}\right)}(s)}{\pi}\int_{-1}^{1}E^{\omega}(y)\mathcal{J}_{s}^{(a_{1},-\frac{1}{2})}(y)\varphi_{t^{2k-1/2+a_{1}}_{b}-t^{N}_{0}}(y)\times\\\
(y-1)^{k-l}(1-y)^{a_{1}}(1+y)^{-1/2}dy,$

When $k<l$,

$\psi_{k-l}^{t^{2k-1/2+a_{1}}_{b},N}(s)=\frac{W^{\left(a_{1},-\frac{1}{2}\right)}(s)}{\pi}\frac{1}{(2\pi
i)^{l-k}}\int_{-1}^{1}\mathcal{J}_{s}^{(a_{1},-\frac{1}{2})}(x)\times\\\
\oint\cdots\oint\frac{E^{\omega}(u_{l})\varphi_{t^{2l-1}_{c(2l-1)}-t^{N}_{0}}(u_{l})\varphi_{l-1,\pm}(u_{l-1})\cdots\varphi_{k+1,\pm}(u_{k+1})}{(u_{l}-1)(u_{l}-u_{l-1})\cdots(u_{k+1}-1)(u_{k+1}-x)}du_{l}\cdots
du_{k+1}\times\\\
\varphi_{t^{2k-1/2+a_{1}}_{b}-t^{2k}_{0}}(x)(1-x)^{a_{1}}(1+x)^{-1/2}dx,$

where the $u_{i}$-contour ($k+2\leq i\leq l$) is a positively oriented simple
loop that encircles point $1$ and $u_{i-1}$-contour, the $u_{k+1}$-contour is
a positively oriented simple loop that encircles interval $[-1,1]$.

###### Proof.

The proof can be done by direct calculation of convolutions, or by the
induction method as in the proof of Theorem 4.4 of [10]. ∎

######  Lemma 3.3.

The matrix $M^{n,a}$ is upper triangular and invertible.

###### Proof.

We first show that when $k>l$, $M_{k,l}^{n,a}=0$. Recall that

$M_{k,l}^{n,a}=\psi^{n,a}_{n-l}\ast\mathcal{T}^{n,a}\ast\phi_{-a}^{a}\ast\mathcal{T}^{n-1/2+a,-a}\ast\cdots\ast\mathcal{T}^{k,-}\ast\phi_{+}^{-}(\text{virt}).$

So,

$M_{k,l}^{n,a}=\frac{1}{(2\pi
i)^{n-k+1}}\oint\cdots\oint\frac{E^{\omega}(u_{n})(u_{n}-1)^{n-l}\varphi_{n,a}(u_{N})\cdots\varphi_{k,\pm}(u_{k})}{(u_{n}-1)(u_{n}-u_{n-1})\cdots(u_{k}-1)}du_{n}\cdots
du_{k}=0.$

When $k\leq l$,

$M_{k,l}^{n,a}=\frac{1}{(2\pi
i)^{l-k+1}}\oint\cdots\oint\frac{E^{\omega}(u_{l})\varphi_{t^{2l-1}_{c(2l-1)}-t^{2N-1/2+a}_{0}}(u_{l})\varphi_{l-1,\pm}(u_{l-1})\cdots\varphi_{k,\pm}(u_{k})}{(u_{l}-1)(u_{l}-u_{l-1})\cdots(u_{k}-1)}du_{l}\cdots
du_{k}.$

Thus the diagonal elements of $M^{n,a}$ are nonzero and $M^{n,a}$ is
invertible. ∎

###  3.4 $\Phi_{k-j}^{t^{2k-1/2+a_{1}}_{b},N}(s)$ and
$\displaystyle{\sum_{k=1}^{n_{2}}\psi_{n_{1}-k}^{t^{2n_{1}-1/2+a_{1}}_{b_{1}},N}(s_{1})\Phi_{n_{2}-k}^{t^{2n_{2}-1/2+a_{2}}_{b_{2}},N}(s_{2})}$

######  Lemma 3.4.

For any $1\leq j\leq k\leq N$, define

$\Phi_{k-j}^{t^{2k-1/2+a_{1}}_{b},N}(s)=\frac{1}{2\pi
i}\oint\frac{J_{s}^{\left(a_{1},-\frac{1}{2}\right)}(u)}{E^{\omega}(u)\varphi_{t^{2k-1/2+a_{1}}_{b}-t^{N}_{0}}(u)}\frac{1}{(u-1)^{k-j+1}}du.$

Then,

  1. (1)

$\left\\{\Phi_{k-j}^{t^{2k-1/2+a_{1}}_{b},N}(s)\right\\}_{j=1}^{k}$ is a basis
of the linear span of

$\left\\{\mathcal{T}_{t^{2k-1/2+a_{1}}_{b},t^{2k-1/2+a_{1}}_{c\left(2k-1/2+a_{1}\right)}}^{k,a_{1}}\ast\cdots\ast\mathcal{T}_{t^{2j-1}_{0},t^{2j-1}_{c\left(2j-1\right)}}^{j,-}\ast\phi_{+}^{-}(s,\text{virt})\right\\}_{j=1}^{k}.$

  2. (2)

For $0\leq j_{1},j_{2}\leq k-1$,

$\sum_{s\geq
0}\Phi_{j_{1}}^{t^{2k-1/2+a_{1}}_{b},N}(s)\psi_{j_{2}}^{t^{2k-1/2+a_{1}}_{b},N}(s)=\delta_{j_{1},j_{2}}.$

###### Proof.

To prove (1), we use the fact that $\left\\{\frac{\partial^{j}}{\partial
u^{j}}J_{s}^{\left(a_{1},-\frac{1}{2}\right)}(u)\Big{|}_{u=1}\right\\}_{j=0}^{k-1}$
is a set of polynomials in variable $s$ of degree $2j+1/2+a_{1}$, which is a
linear basis of both the linear span of

$\left\\{\mathcal{T}_{t^{2k-1/2+a_{1}}_{b},t^{2k-1/2+a_{1}}_{c\left(2k-1/2+a_{1}\right)}}^{k,a_{1}}\ast\cdots\ast\mathcal{T}_{t^{2j-1}_{0},t^{2j-1}_{c\left(2j-1\right)}}^{j,-}\ast\phi_{+}^{-}(s,\text{virt})\right\\}_{j=1}^{k}$

and $\left\\{\Phi_{k-j}^{t^{2k-1/2+a_{1}}_{b},N}(s)\right\\}_{j=1}^{k}$.
(2) follows from Lemma 3.2 (6). ∎

Next, we calculate
$\sum_{k=1}^{n_{2}}\psi_{n_{1}-k}^{t^{2n_{1}-1/2+a_{1}}_{b_{1}},N}(s_{1})\Phi_{n_{2}-k}^{t^{2n_{2}-1/2+a_{2}}_{b_{2}},N}(s_{2})$.
When $n_{1}<n_{2}$, let

$f^{N}_{t^{2n_{2}-1/2+a_{2}}_{b_{2}}}(u)=\frac{J_{s_{2}}^{\left(a_{2},-\frac{1}{2}\right)}(u)}{E^{\omega}(u)\varphi_{t^{2n_{2}-1/2+a_{2}}_{b_{2}}-t^{N}_{0}}(u)}$

and

$h^{N}_{n_{1}+1,k}(x)=\frac{1}{(2\pi
i)^{k-n_{1}}}\oint\cdots\oint\frac{E^{\omega}(u_{k})\varphi_{t^{2k-1}_{c(2k-1)}-t^{N}_{0}}(u_{k})\cdots\varphi_{n_{1}+1,\pm}(u_{n_{1}+1})}{(u_{k}-1)(u_{k}-u_{k-1})\cdots(u_{n_{1}+1}-1)(u_{n_{1}+1}-x)}du_{k}\cdots
du_{n_{1}+1}.$

Then

$g^{a_{2}}_{n_{1}+1,n_{2}}(x)=\frac{1}{(2\pi i)^{n_{2}-n_{1}}}\times\\\
\oint\cdots\oint\frac{f^{N}_{t^{2n_{2}-1/2+a_{2}}_{b_{2}}}(u_{n_{2}})E^{\omega}(u_{n_{2}})\varphi_{t^{2n_{2}-1}_{c(2n_{2}-1)}-t^{N}_{0}}(u_{n_{2}})\cdots\varphi_{n_{1}+1,\pm}(u_{n_{1}+1})}{(u_{n_{2}}-1)(u_{n_{2}}-u_{n_{2}-1})\cdots(u_{n_{1}+1}-1)(u_{n_{1}+1}-x)}du_{n_{2}}\cdots
du_{n_{1}+1},$ $\Phi_{n_{2}-k}^{t^{2n_{2}-1/2+a_{2}}_{b},N}=\frac{1}{2\pi
i}\oint\frac{f^{N}_{t^{2n_{2}-1/2+a_{2}}_{b_{2}}}(u)}{(u-1)^{n_{2}-k+1}}du.$

######  Lemma 3.5.

When $n_{1}<n_{2}$,

$\sum_{k=n_{1}+1}^{n_{2}}h^{N}_{n_{1}+1,k}(x)*\Phi_{n_{2}-k}^{t^{2n_{2}-1/2+a_{2}}_{b},N}=\frac{1}{2\pi
i}\oint\frac{f^{N}_{t^{2n_{2}-1/2+a_{2}}_{b_{2}}}(u)}{(u-1)^{n_{2}-n_{1}}(x-u)}duE^{\omega}(x)\varphi_{t^{2n_{1}+1}_{c(2n_{1}+1)}-t^{N}_{0}}(x)+g^{a_{2}}_{n_{1}+1,n_{2}}(x).\\\
$

###### Proof.

First, when $n_{2}=n_{1}+1$, the equality holds. Now suppose this holds for
fixed $n_{1}$ and $n_{2}$, we show this holds for $n_{2}$ and $n_{1}-1$ as
well. Then, it suffices to show that

$\frac{1}{2\pi
i}\oint\frac{E^{\omega}(u_{n_{1}})\varphi_{t^{2n_{1}-1}_{c(2n_{1}-1)}-t^{N}_{0}}(u_{n_{1}})}{(u_{n_{1}}-1)(u_{n_{1}}-x)}du_{n_{1}}\cdot\Phi_{n_{2}-n_{1}}^{t^{2n_{2}-1/2+a_{2}}_{b},N}\\\
+\sum_{k=n_{1}+1}^{n_{2}}\oint\frac{h^{N}_{n_{1},k}(u_{n_{1}})\varphi_{n_{1},\pm}(u_{n_{1}})}{(u_{n_{1}}-1)(u_{n_{1}}-x)}du_{n_{1}}\cdot\Phi_{n_{2}-k}^{t^{2n_{2}-1/2+a_{2}}_{b},N}\\\
=\frac{1}{2\pi
i}\oint\frac{f^{N}_{t^{2n_{2}-1/2+a_{2}}_{b_{2}}}(u)}{(u-1)^{n_{2}-n_{1}+1}(x-u)}duE^{\omega}(x)\varphi_{t^{2n_{1}-1}_{c(2n_{1}-1)}-t^{N}_{0}}(x)+\frac{1}{2\pi
i}\oint\frac{g^{a_{2}}_{n_{1}+1,n_{2}}(u_{n_{1}})\varphi_{n_{1}}(u_{n_{1}})}{(u_{n_{1}}-1)(u_{n_{1}}-x)}du_{n_{1}}.$

By induction assumption,

$\sum_{k=n_{1}+1}^{n_{2}}\left(\oint\frac{h^{N}_{n_{1},k}(u_{n_{1}})\varphi_{n_{1},\pm}(u_{n_{1}})}{(u_{n_{1}}-1)(u_{n_{1}}-x)}du_{n_{1}}\cdot\Phi_{n_{2}-k}^{t^{2n_{2}-1/2+a_{2}}_{b},N}\right)-\frac{1}{2\pi
i}\oint\frac{g^{a_{2}}_{n_{1}+1,n_{2}}(u_{n_{1}})\varphi_{n_{1},\pm}(u_{n_{1}})}{(u_{n_{1}}-1)(u_{n_{1}}-x)}du_{n_{1}}\\\
=\frac{\varphi_{n_{1},\pm}(1)(-g^{a_{2}}_{n_{1}+1,n_{2}}(1)+\sum_{k=n_{1}+1}^{n_{2}}h^{N}_{n_{1},k}(1)\Phi_{n_{2}-k}^{t^{2n_{2}-1/2+a_{2}}_{b},N})}{1-x}\\\
-\frac{\varphi_{n_{1},\pm}(x)(-g^{a_{2}}_{n_{1}+1,n_{2}}(x)+\sum_{k=n_{1}+1}^{n_{2}}h^{N}_{n_{1},k}(x)\Phi_{n_{2}-k}^{t^{2n_{2}-1/2+a_{2}}_{b},N})}{1-x}\\\
=-\frac{1}{1-x}\cdot\frac{1}{2\pi
i}\oint\frac{f^{N}_{t^{2n_{2}-1/2+a_{2}}_{b_{2}}}(u)}{(u-1)^{n_{2}-n_{1}+1}}duE^{\omega}(1)\varphi_{t^{2n_{1}-1}_{c(2n_{1}-1)}-t^{N}_{0}}(1)\\\
-\frac{1}{1-x}\cdot\frac{1}{2\pi
i}\oint\frac{f^{N}_{t^{2n_{2}-1/2+a_{2}}_{b_{2}}}(u)}{(u-1)^{n_{2}-n_{1}}(x-u)}duE^{\omega}(x)\varphi_{t^{2n_{1}-1}_{c(2n_{1}-1)}-t^{N}_{0}}(x),$

while

$\frac{1}{2\pi
i}\oint\frac{E^{\omega}(u_{n_{1}})\varphi_{t^{2n_{1}-1}_{c(2n_{1}-1)}-t^{N}_{0}}(u_{n_{1}})}{(u_{n_{1}}-1)(u_{n_{1}}-x)}du_{n_{1}}\cdot\Phi_{n_{2}-n_{1}}^{t^{2n_{2}-1/2+a_{2}}_{b},N}\\\
-\frac{1}{2\pi
i}\oint\frac{f^{N}_{t^{2n_{2}-1/2+a_{2}}_{b_{2}}}(u)}{(u-1)^{n_{2}-n_{1}+1}(x-u)}duE^{\omega}(x)\varphi_{t^{2n_{1}-1}_{c(2n_{1}-1)}-t^{N}_{0}}(x)\\\
=\frac{E^{\omega}(1)\varphi_{t^{2n_{1}-1}_{c(2n_{1}-1)}-t^{N}_{0}}(1)-E^{\omega}(x)\varphi_{t^{2n_{1}-1}_{c(2n_{1}-1)}-t^{N}_{0}}(x)}{1-x}\cdot\frac{1}{2\pi
i}\oint\frac{f^{N}_{t^{2n_{2}-1/2+a_{2}}_{b_{2}}}(u)}{(u-1)^{n_{2}-n_{1}+1}}du\\\
-\frac{1}{2\pi
i}\oint\frac{f^{N}_{t^{2n_{2}-1/2+a_{2}}_{b_{2}}}(u)}{(u-1)^{n_{2}-n_{1}+1}(x-u)}duE^{\omega}(x)\varphi_{t^{2n_{1}-1}_{c(2n_{1}-1)}-t^{N}_{0}}(x)\\\
=\frac{1}{1-x}\cdot\frac{1}{2\pi
i}\oint\frac{f^{N}_{t^{2n_{2}-1/2+a_{2}}_{b_{2}}}(u)}{(u-1)^{n_{2}-n_{1}+1}}duE^{\omega}(1)\varphi_{t^{2n_{1}-1}_{c(2n_{1}-1)}-t^{N}_{0}}(1)\\\
+\frac{1}{1-x}\cdot\frac{1}{2\pi
i}\oint\frac{f^{N}_{t^{2n_{2}-1/2+a_{2}}_{b_{2}}}(u)}{(u-1)^{n_{2}-n_{1}}(x-u)}duE^{\omega}(x)\varphi_{t^{2n_{1}-1}_{c(2n_{1}-1)}-t^{N}_{0}}(x).$

∎

######  Proposition 3.6.

If $n_{1}\geq n_{2}\geq 1$ and and $s_{1},s_{2}\in\mathbb{Z}_{\geq 0}$, we
have

$\sum_{k=1}^{n_{2}}\psi_{n_{1}-k}^{t^{2n_{1}-1/2+a_{1}}_{b_{1}},N}(s_{1})\Phi_{n_{2}-k}^{t^{2n_{2}-1/2+a_{2}}_{b_{2}},N}(s_{2})=\\\
\frac{W^{\left(a_{1},-\frac{1}{2}\right)}(s_{1})}{\pi}\frac{1}{2\pi
i}\int_{-1}^{1}\oint\mathcal{J}_{s_{1}}^{\left(a_{1},-\frac{1}{2}\right)}(y){J}_{s_{2}}^{\left(a_{2},-\frac{1}{2}\right)}(u)\frac{E^{\omega}(y)}{E^{\omega}(u)}\frac{\varphi_{t^{2n_{1}-1/2+a_{1}}_{b_{1}}-t^{N}_{0}}(y)}{\varphi_{t^{2n_{2}-1/2+a_{2}}_{b_{2}}-t^{N}_{0}}(u)}\times\\\
\frac{(y-1)^{n_{1}}}{(u-1)^{n_{2}}}\frac{(1-y)^{a_{1}}(1+y)^{-\frac{1}{2}}}{y-u}dudy\\\
+\frac{W^{\left(a_{1},-\frac{1}{2}\right)}(s_{1})}{\pi}\int_{-1}^{1}\mathcal{J}_{s_{1}}^{\left(a_{1},-\frac{1}{2}\right)}(y){J}_{s_{2}}^{\left(a_{2},-\frac{1}{2}\right)}(y)\frac{\varphi_{t^{2n_{1}-1/2+a_{1}}_{b_{1}}-t^{N}_{0}}(y)}{\varphi_{t^{2n_{2}-1/2+a_{2}}_{b_{2}}-t^{N}_{0}}(y)}(y-1)^{n_{1}-n_{2}}(1-y)^{a_{1}}(1+y)^{-\frac{1}{2}}dy.$

If $1\leq n_{1}<n_{2}$,

$\sum_{k=1}^{n_{2}}\psi_{n_{1}-k}^{t^{2n_{1}-1/2+a_{1}}_{b_{1}},N}(s_{1})\Phi_{n_{2}-k}^{t^{2n_{2}-1/2+a_{2}}_{b_{2}},N}(s_{2})=\\\
\frac{W^{\left(a_{1},-\frac{1}{2}\right)}(s_{1})}{\pi}\frac{1}{2\pi
i}\int_{-1}^{1}\oint\mathcal{J}_{s_{1}}^{\left(a_{1},-\frac{1}{2}\right)}(y)\mathcal{J}_{s_{2}}^{\left(a_{2},-\frac{1}{2}\right)}(u)\frac{E^{\omega}(y)}{E^{\omega}(u)}\frac{\varphi_{t^{2n_{1}-1/2+a_{1}}_{b_{1}}-t^{N}_{0}}(y)}{\varphi_{t^{2n_{2}-1/2+a_{2}}_{b_{2}}-t^{N}_{0}}(u)}\frac{(y-1)^{n_{1}}}{(u-1)^{n_{2}}}\frac{(1-y)^{a_{1}}(1+y)^{-\frac{1}{2}}}{y-u}dudy\\\
+\phi^{t^{2n_{2}-1/2+a_{2}}_{b_{2}},t^{2n_{1}-1/2+a_{1}}_{b_{1}}}(s_{1},s_{2}).$

###### Proof.

The calculation for

$\sum_{k=1}^{\text{min}\\{n_{1},n_{2}\\}}\psi_{n_{1}-k}^{t^{2n_{1}-1/2+a_{1}}_{b_{1}},N}(s_{1})\Phi_{n_{2}-k}^{t^{2n_{2}-1/2+a_{2}}_{b_{2}},N}(s_{2})$

follows the same arguments as the proof for Proposition 4.6 in [10]. We only
show the proof when $1\leq n_{1}<n_{2}$.

First,

$\sum_{k=1}^{n_{1}}\psi_{n_{1}-k}^{t^{2n_{1}-1/2+a_{1}}_{b_{1}},N}(s_{1})\Phi_{n_{2}-k}^{t^{2n_{2}-1/2+a_{2}}_{b_{2}},N}(s_{2})=\\\
\frac{W^{\left(a_{1},-\frac{1}{2}\right)}(s_{1})}{\pi}\frac{1}{2\pi
i}\int_{-1}^{1}\oint\mathcal{J}_{s_{1}}^{\left(a_{1},-\frac{1}{2}\right)}(y)\mathcal{J}_{s_{2}}^{\left(a_{2},-\frac{1}{2}\right)}(u)\frac{E^{\omega}(y)}{E^{\omega}(u)}\frac{\varphi_{t^{2n_{1}-1/2+a_{1}}_{b_{1}}-t^{N}_{0}}(y)}{\varphi_{t^{2n_{2}-1/2+a_{2}}_{b_{2}}-t^{N}_{0}}(u)}\times\\\
\frac{(y-1)^{n_{1}}}{(u-1)^{n_{2}}}\left(1-\left(\frac{u-1}{y-1}\right)^{n_{1}}\right)\frac{(1-y)^{a_{1}}(1+y)^{-\frac{1}{2}}}{y-u}dudy.$

Now we only need to find
$\sum_{k=n_{1}+1}^{n_{2}}\psi_{n_{1}-k}^{t^{2n_{1}-1/2+a_{1}}_{b_{1}},N}(s_{1})\Phi_{n_{2}-k}^{t^{2n_{2}-1/2+a_{2}}_{b_{2}},N}(s_{2})$.
By Lemma 3.5,

$\sum_{k=n_{1}+1}^{n_{2}}\psi_{n_{1}-k}^{t^{2n_{1}-1/2+a_{1}}_{b_{1}},N}(s_{1})\Phi_{n_{2}-k}^{t^{2n_{2}-1/2+a_{2}}_{b_{2}},N}(s_{2})\\\
=\frac{W^{\left(a_{1},-\frac{1}{2}\right)}(s_{1})}{\pi}\int_{-1}^{1}\mathcal{J}_{s_{1}}^{\left(a_{1},-\frac{1}{2}\right)}(x)\left(\sum_{k=n_{1}+1}^{n_{2}}h^{N}_{n_{1}+1,k}(x)*\Phi_{n_{2}-k}^{t^{2n_{2}-1/2+a_{2}}_{b},N}\right)\varphi_{t^{2n_{1}-1/2+a_{1}}_{b}-t^{2n_{1}}_{0}}(x)(1-x)^{a_{1}}(1+x)^{-1/2}dx\\\
=\frac{W^{\left(a_{1},-\frac{1}{2}\right)}(s_{1})}{\pi(2\pi
i)}\int_{-1}^{1}\oint\frac{\mathcal{J}_{s_{1}}^{\left(a_{1},-\frac{1}{2}\right)}(x)f^{N}_{t^{2n_{2}-1/2+a_{2}}_{b_{2}}}(u)}{(u-1)^{n_{2}-n_{1}}(x-u)}E^{\omega}(x)\varphi_{t^{2n_{1}-1/2+a_{1}}_{b_{1}}-t^{N}_{0}}(x)(1-x)^{a_{1}}(1+x)^{-1/2}dudx\\\
+\frac{W^{\left(a_{1},-\frac{1}{2}\right)}(s_{1})}{\pi}\int_{-1}^{1}\mathcal{J}_{s}^{\left(a_{1},-\frac{1}{2}\right)}(x)g^{a_{2}}_{n_{1}+1,n_{2}}(x)\varphi_{t^{2n_{1}-1/2+a_{1}}_{b_{1}}-t^{2n_{1}}_{0}}(x)(1-x)^{a_{1}}(1+x)^{-1/2}dx.$

Note that

$\frac{W^{\left(a_{1},-\frac{1}{2}\right)}(s_{1})}{\pi}\int_{-1}^{1}\mathcal{J}_{s_{1}}^{\left(a_{1},-\frac{1}{2}\right)}(x)g^{a_{2}}_{n_{1}+1,n_{2}}(x)\varphi_{t^{2n_{1}-1/2+a_{1}}_{b_{1}}-t^{2n_{1}}_{0}}(x)(1-x)^{a_{1}}(1+x)^{-1/2}dx\\\
=\phi^{t^{2n_{2}-1/2+a_{2}}_{b_{2}},t^{2n_{1}-1/2+a_{1}}_{b_{1}}}(s_{1},s_{2}),$

Thus,

$\sum_{k=n_{1}+1}^{n_{2}}\psi_{n_{1}-k}^{t^{2n_{1}-1/2+a_{1}}_{b_{1}},N}(s_{1})\Phi_{n_{2}-k}^{t^{2n_{2}-1/2+a_{2}}_{b_{2}},N}(s_{2})\\\
=\frac{W^{\left(a_{1},-\frac{1}{2}\right)}(s_{1})}{\pi(2\pi
i)}\int_{-1}^{1}\oint\frac{\mathcal{J}_{s_{1}}^{\left(a_{1},-\frac{1}{2}\right)}(x)f^{N}_{t^{2n_{2}-1/2+a_{2}}_{b_{2}}}(u)}{(u-1)^{n_{2}-n_{1}}(x-u)}E^{\omega}(x)\varphi_{t^{2n_{1}-1/2+a_{1}}_{b_{1}}-t^{N}_{0}}(x)(1-x)^{a_{1}}(1+x)^{-1/2}dudx\\\
+\phi^{t^{2n_{2}-1/2+a_{2}}_{b_{2}},t^{2n_{1}-1/2+a_{1}}_{b_{1}}}(s_{1},s_{2}).$

Adding the above two summations together, we get

$\sum_{k=1}^{n_{2}}\psi_{n_{1}-k}^{t^{2n_{1}-1/2+a_{1}}_{b_{1}},N}(s_{1})\Phi_{n_{2}-k}^{t^{2n_{2}-1/2+a_{2}}_{b_{2}},N}(s_{2})=\\\
\frac{W^{\left(a_{1},-\frac{1}{2}\right)}(s_{1})}{\pi}\frac{1}{2\pi
i}\int_{-1}^{1}\oint\mathcal{J}_{s_{1}}^{\left(a_{1},-\frac{1}{2}\right)}(y)\mathcal{J}_{s_{2}}^{\left(a_{2},-\frac{1}{2}\right)}(u)\frac{E^{\omega}(y)}{E^{\omega}(u)}\frac{\varphi_{t^{2n_{1}-1/2+a_{1}}_{b_{1}}-t^{N}_{0}}(y)}{\varphi_{t^{2n_{2}-1/2+a_{2}}_{b_{2}}-t^{N}_{0}}(u)}\frac{(y-1)^{n_{1}}}{(u-1)^{n_{2}}}\frac{(1-y)^{a_{1}}(1+y)^{-\frac{1}{2}}}{y-u}dudy\\\
+\phi^{t^{2n_{2}-1/2+a_{2}}_{b_{2}},t^{2n_{1}-1/2+a_{1}}_{b_{1}}}(s_{1},s_{2}).$

∎

###  3.5 Computing the kernel

In this section, we apply Lemma 3.1 to derive the correlation kernel $K$.
Adding Proposition 3.4 and Proposition 3.6, we get the following:

When $(n_{1},a_{1},t_{1})\succ(n_{2},a_{2},t_{2})$, which means $t_{1}\leq
t_{2}$, $2n_{1}-1/2+a_{1}\geq 2n_{2}-1/2+a_{2}$ and
$(n_{1},a_{1},t_{1})\neq(n_{2},a_{2},t_{2})$. Let
$t^{2n_{i}-1/2+a_{i}}_{b_{i}}=t_{i}$, $i=1,2$. By the fact that
$\phi^{t_{2},t_{1}}=0$ when $t_{1}\leq t_{2}$, we have

$K(n_{1},a_{1},t_{1},s_{1};n_{2},a_{2},t_{2},s_{2})=\sum_{k=1}^{n_{2}}\psi_{n_{1}-k}^{t^{2n_{1}-1/2+a_{1}}_{b_{1}},N}(s_{1})\Phi_{n_{2}-k}^{t^{2n_{2}-1/2+a_{2}}_{b_{2}},N}(s_{2})\\\
=\frac{W^{\left(a_{1},-\frac{1}{2}\right)}(s_{1})}{\pi}\frac{1}{2\pi
i}\int_{-1}^{1}\oint\mathcal{J}_{s_{1}}^{\left(a_{1},-\frac{1}{2}\right)}(y){J}_{s_{2}}^{\left(a_{2},-\frac{1}{2}\right)}(u)\frac{E^{\omega}(y)}{E^{\omega}(u)}\frac{e^{t_{1}(y-1)}}{e^{t_{2}(u-1)}}\frac{(y-1)^{n_{1}}}{(u-1)^{n_{2}}}\frac{(1-y)^{a_{1}}(1+y)^{-\frac{1}{2}}}{y-u}dudy\\\
+\frac{W^{\left(a_{1},-\frac{1}{2}\right)}(s_{1})}{\pi}\int_{-1}^{1}\mathcal{J}_{s_{1}}^{\left(a_{1},-\frac{1}{2}\right)}(y){J}_{s_{2}}^{\left(a_{2},-\frac{1}{2}\right)}(y)\frac{e^{t_{1}(y-1)}}{e^{t_{2}(y-1)}}(y-1)^{n_{1}-n_{2}}(1-y)^{a_{1}}(1+y)^{-\frac{1}{2}}dy.$

When $(n_{1},a_{1},t_{1})\nsucc(n_{2},a_{2},t_{2})$, which means
$2n_{1}-1/2+a_{1}\leq 2n_{2}-1/2+a_{2}$ and $t_{1}>t_{2}$,

$K(n_{1},a_{1},t_{1},s_{1};n_{2},a_{2},t_{2},s_{2})=-\phi^{t^{2n_{2}-1/2+a_{2}}_{b_{2}},t^{2n_{1}-1/2+a_{1}}_{b_{1}}}(s_{1},s_{2})+\sum_{k=1}^{n_{2}}\psi_{n_{1}-k}^{t^{2n_{1}-1/2+a_{1}}_{b_{1}},N}(s_{1})\Phi_{n_{2}-k}^{t^{2n_{2}-1/2+a_{2}}_{b_{2}},N}(s_{2})\\\
=\frac{W^{\left(a_{1},-\frac{1}{2}\right)}(s_{1})}{\pi}\frac{1}{2\pi
i}\int_{-1}^{1}\oint\mathcal{J}_{s_{1}}^{\left(a_{1},-\frac{1}{2}\right)}(y)\mathcal{J}_{s_{2}}^{\left(a_{2},-\frac{1}{2}\right)}(u)\frac{E^{\omega}(y)}{E^{\omega}(u)}\frac{e^{t_{1}(y-1)}}{e^{t_{2}(u-1)}}\frac{(y-1)^{n_{1}}}{(u-1)^{n_{2}}}\frac{(1-y)^{a_{1}}(1+y)^{-\frac{1}{2}}}{y-u}dudy.$

##  4 Non-commutative random walk on $U(\mathfrak{so}_{N+1})$

In this section, we construct a non-commutative random walk on
$U(\mathfrak{so}_{N+1})$, which is analogues to the non-commutative random
walk on $U(\mathfrak{gl}_{N})$ constructed in [15].

We take the universal enveloping algebra of the Lie group
$\mathfrak{so}_{N+1}$ as the state space and define a semi-group of the non-
commutative Markov operator $\\{P_{t}\\}_{t\geq 0}$ on
$U(\mathfrak{so}_{N+1})$.

For each class function $\kappa\in L^{2}(SO(N+1))$, we can define a state
$\langle\cdot\rangle_{\kappa}$ on $U(\mathfrak{so}_{N+1})$ by $\langle
X\rangle_{\kappa}=D(X)\kappa(U)|_{U=I}$, where $D$ is the canonical
isomorphism from $U(\mathfrak{so}_{N+1})$ to the algebra of left–invariant
differential operators on $SO(N+1)$ with complex coefficients (see e.g. [23]).

It is not hard to see (see e.g. [15]) that if the class function $\kappa$
decomposes as

$\kappa=\sum_{\lambda}\hat{\kappa}(\lambda)\frac{\chi^{\lambda}}{\text{dim}\lambda},$

where $\lambda$ ranges over all irreducible representations of $SO(N+1)$ and
$\chi^{\lambda}$ are the corresponding characters, then

$\langle
X\rangle_{\kappa}=\sum_{\lambda}\hat{\kappa}(\lambda)\sum_{i=1}^{\text{dim
}\lambda}\text{Tr}\left(X|_{V_{\lambda}^{i}}\right).$ (4.1)

In what follows, we let $\kappa_{t}(O)=e^{t\text{Tr}(O-Id)}$ and write
$\langle\cdot\rangle_{t}$ for $\langle\cdot\rangle_{\kappa_{t}}$.

If $X=F_{i_{1}j_{1}}\cdots F_{i_{k}j_{k}}$, then the state can be computed
with the following formula (see e.g. page 101 of [22]):

$D(X)\kappa(O)=\partial_{t_{1}}\cdots\partial_{t_{k}}\kappa(Oe^{t_{1}F_{i_{1}j_{1}}}\cdots
e^{t_{k}F_{i_{k}j_{k}}})|_{t_{1}=\cdots=t_{k}=0},$ (4.2)

where $e^{tF}$ is the usual exponential of matrices. In particular, we have

$e^{tF_{ij}}=\left\\{\begin{array}[]{lr}Id+tF_{ij},&i\neq\pm j,\ i,j\neq 0,\\\
Id+tF_{ij}-\frac{t^{2}}{2}E_{-j,j},&i=0,\\\
Id+tF_{ij}-\frac{t^{2}}{2}E_{i,-i},&j=0,\\\
Id+(e^{t}-1)E_{ii}+(e^{-t}-1)E_{-i,-i},&i=j.\end{array}\right.$

Since (4.2) only involves linear terms in $t_{j}$, we can replace
$e^{tF_{ij}}$ with $Id+tF_{ij}$. Applying Faa di Bruno formula (see e.g.
[15]), we have

$\langle F_{i_{1},j_{1}}\cdots
F_{i_{m}j_{m}}\rangle_{t}=\sum_{\pi\in\Pi}t^{|\pi|}\prod_{B\in\pi,B=\\{b_{1},\cdots,b_{k}\\}}\text{Tr}\left(\prod_{b\in
B,B=\\{b_{1},\cdots,b_{k}\\}}F_{i_{b}j_{b}}\right),$

where $\Pi$ is the set of partitions of the set $\\{1,2,\ldots,m\\}$ and
$B\in\pi$ means that $B$ is a block in partition $\pi$.

It is not hard to see that the non-commutative Markov operator $P_{t}$ defined
in [15, 14] also defines a Markov operator on $U(\mathfrak{so}_{N+1})$.

######  Theorem 4.1.

(Theorem 3.1 in [15]) Define
$P_{t}=(id\otimes\langle\cdot\rangle_{\kappa_{t}})\circ\Delta$, then

  1. (1)

$P_{t}$ satisfies the semi-group property $P_{t+s}=P_{t}\circ P_{s}$.

  2. (2)

$P_{t}$ preserves $Z(U(\mathfrak{so}_{N+1}))$, i.e.
$P_{t}Z(U(\mathfrak{so}_{N+1}))\subset Z(U(\mathfrak{so}_{N+1}))$.

  3. (3)

For all $Y\in Z(U(\mathfrak{so}_{N+1}))$, $\langle P_{t}Y\rangle_{s}=\langle
Y\rangle_{s+t}$.

Since $P_{t}$ preserves $Z(U(\mathfrak{so}_{N+1}))$, we can expand
$P_{t}\Phi_{2k}^{N+1}$ in terms of generators of $Z(U(\mathfrak{so}_{N+1}))$.

######  Example 2.

  1. (1)

$P_{t}\Phi_{2}^{N+1}=\Phi_{2}^{N+1}+\text{constant}.$

  2. (2)

$P_{t}\Phi_{4}^{N+1}=\Phi_{4}^{N+1}+16tn\Phi_{2}^{N+1}+\text{constant}.$

If we define $Q^{n,a}_{t}$ to be the Markov operator for the point process
$\left\\{x_{k}^{n,a}|\ 1\leq k\leq n,\ a\in\left\\{\pm 1/2\right\\},\ n\geq
1\right\\}$ projected onto $\mathbb{Z}_{\geq 0}\times\\{n\\}\times\\{a\\}$,
then $Q^{n,a}_{t}$ also defines a Markov operator on
$Z(U(\mathfrak{so}_{N+1}))$ through the Harish-Chandra isomorphism.

######  Proposition 4.1.

For any $Y\in Z(U(\mathfrak{so}_{N+1}))$, there exists a polynomial
$\overline{p}^{N+1}$ such that

$\langle
Y\rangle_{\frac{t}{2}}=\mathbb{E}\left[\overline{p}^{N+1}_{Y}\left(x^{n,a}_{1}(t),\cdots,x^{n,a}_{n}(t)\right)\right].$

In addition, $\langle Q^{n,a}_{t}Y\rangle_{{}_{\frac{s}{2}}}=\langle
Y\rangle_{{}_{\frac{s+t}{2}}}$.

###### Proof.

Suppose $Y\in Z(U(\mathfrak{so}_{N+1}))$ is sent to the polynomial
$p_{Y}^{N+1}$ by the Harish-Chandra isomorphism. When $N+1=2n+1$, by (2.3),
for any $O\in SO(N+1)$,

$e^{\frac{1}{2}t\text{Tr}(O-I)}=\sum_{\lambda}\text{Prob}\left(x_{k}^{n,\frac{1}{2}}(t)=\lambda_{k}+n-k,1\leq
k\leq
n\right)\frac{\chi^{\lambda}_{SO(2n+1)}(O)}{\text{dim}_{SO(2n+1)}\lambda}.$

Recall that $l_{k}=\lambda_{k}-k+\frac{1}{2}$ when $N+1=2n+1$, so we define a
polynomial $\overline{p}^{N+1}_{Y}$ with a change of variables, i.e.
$\overline{p}^{N+1}_{Y}\left(l_{1}+n-\frac{1}{2},\cdots,l_{n}+n-\frac{1}{2}\right)=p^{N+1}_{Y}\left(\lambda_{1},\cdots,\lambda_{n}\right)$.
Thus, by linearity and (4.1),

$\langle
Y\rangle_{\frac{t}{2}}=\sum_{\lambda}\text{Prob}\left(x_{k}^{n,\frac{1}{2}}(t)=\lambda_{k}+n-k\right)\frac{\langle
Y\rangle_{\chi^{\lambda}}}{\text{dim}\lambda}\\\
=\sum_{\lambda}\text{Prob}\left(x_{k}^{n,\frac{1}{2}}(t)=\lambda_{k}+n-k,1\leq
k\leq n\right)p^{N+1}_{Y}(\lambda_{1},\cdots,\lambda_{n})\\\
=\sum_{l}\text{Prob}\left(x_{k}^{n,\frac{1}{2}}(t)=\lambda_{k}+n-k,1\leq k\leq
n\right)\overline{p}^{N+1}_{Y}\left(l_{1}+n-\frac{1}{2},\cdots,l_{n}+n-\frac{1}{2}\right)\\\
=\mathbb{E}\left[\overline{p}^{N+1}_{Y}\left(x_{1}^{n,\frac{1}{2}}(t),\cdots,x^{n,\frac{1}{2}}_{n}(t)\right)\right].$

Thus,

$\langle
Q^{n,\frac{1}{2}}_{t}Y\rangle_{\frac{s}{2}}=\mathbb{E}\left[Q^{n,\frac{1}{2}}_{t}\overline{p}^{N+1}_{Y}\left(x_{1}^{n,\frac{1}{2}}(s),\cdots,x^{n,\frac{1}{2}}_{n}(s)\right)\right]=\mathbb{E}\left[\overline{p}^{N+1}_{Y}\left(x_{1}^{n,\frac{1}{2}}(t+s),\cdots,x^{n,\frac{1}{2}}_{n}(t+s)\right)\right]=\langle
Y\rangle_{\frac{s+t}{2}}.$

Similarly, when $N+1=2n$, for any $O\in SO(N+1)$,

$e^{\frac{1}{2}t\text{Tr}(O-I)}=\sum_{\lambda}\text{Prob}\left(x_{k}^{n,-\frac{1}{2}}(t)=\lambda_{k}+n-k,1\leq
k\leq
n\right)\frac{\chi^{\lambda}_{SO(2n)}(O)+\chi^{\lambda^{*}}_{SO(2n)}(O)}{2\text{dim}_{SO(2n)}\lambda}$

by (2.4).

$\langle
Y\rangle_{\frac{t}{2}}=\sum_{\lambda}\text{Prob}\left(x_{k}^{n,-\frac{1}{2}}(t)=\lambda_{k}+n-k,1\leq
k\leq n\right)\frac{\langle Y\rangle_{\chi^{\lambda}}+\langle
Y\rangle_{\chi^{\lambda^{*}}}}{2\text{dim}\lambda}\\\
=\sum_{\lambda}\text{Prob}\left(x_{k}^{n,-\frac{1}{2}}(t)=\lambda_{k}+n-k,1\leq
k\leq
n\right)\frac{p^{N+1}_{Y}(\lambda_{1},\cdots,\lambda_{n})+p^{N+1}_{Y}(\lambda_{1},\cdots,-\lambda_{n})}{2}.$

Since $l_{k}=\lambda_{k}-k+1$ when $N+1=2n$, we define
$\overline{p}^{N+1}_{Y}$ for even $N+1$ such that

$\overline{p}^{N+1}_{Y}(l_{1}+n-1,\cdots,l_{n}+n-1)=\frac{p^{N+1}_{Y}(\lambda_{1},\cdots,\lambda_{n})+p^{N+1}_{Y}(\lambda_{1},\cdots,-\lambda_{n})}{2}.$

Then,

$\langle
Y\rangle_{\frac{t}{2}}=\sum_{l}\text{Prob}\left(x_{k}^{n,-\frac{1}{2}}(t)=\lambda_{k}+n-k,1\leq
k\leq n\right)\overline{p}^{N+1}_{Y}(l_{1}+n-1,\cdots,l_{n}+n-1)\\\
=\mathbb{E}\left[\overline{p}^{N+1}_{Y}\left(x_{1}^{n,-\frac{1}{2}}(t),\cdots,x^{n,-\frac{1}{2}}_{n}(t)\right)\right].$

Again, we have $\langle
Q^{n,-\frac{1}{2}}_{t}Y\rangle_{{}_{\frac{s}{2}}}=\langle
Y\rangle_{{}_{\frac{s+t}{2}}}$. ∎

######  Theorem 4.2.

$P_{\frac{t}{2}}X=Q^{n,a}_{t}X$ for all $X\in Z(U(\mathfrak{so}_{N+1}))$. In
particular, $P_{\frac{t}{2}}$ is the Markov operator of the process
$\left(x_{1}^{n,a}(t)>\cdots>x_{n}^{n,a}(t)\right)$.

###### Proof.

From Theorem 4.1 and Proposition 4.1 we have $\langle
P_{\frac{t}{2}}Y\rangle_{{}_{\frac{s}{2}}}=\langle
Q^{n,a}_{t}Y\rangle_{{}_{\frac{s}{2}}}=\langle Y\rangle_{{}_{\frac{s+t}{2}}}$.
The rest of proof follows the standard arguments as that of Proposition 4.4 in
[15]. We only sketch the proof here. To show $P_{\frac{t}{2}}X=Q^{n,a}_{t}X$
for all $X\in Z(U(\mathfrak{so}_{N+1}))$, we show that if there exists a $Y\in
Z(U(\mathfrak{so}_{N+1}))$ such that $\langle Y\rangle_{t}=0$ for all $t\geq
0$, then $Y=0$ by a contradiction. ∎

When restricting our non–commutative random walk to the Gelfand–Tsetlin
subalgebra, which is the subalgebra of $U(\mathfrak{so}_{N+1})$ generated by
the centres $Z(U(\mathfrak{so}_{k}))$, $1\leq k\leq N+1$, it also matches the
two–dimensional particle system along space–like paths:

######  Theorem 4.3.

Suppose $Y_{1}\in Z(U(\mathfrak{so}_{N_{1}+1})),\ldots,Y_{r}\in
Z(U(\mathfrak{so}_{N_{r}+1}))$ are mapped to the shifted polynomials
$\overline{p}^{N_{1}+1}_{Y_{1}},\ldots,\overline{p}^{N_{r}+1}_{Y_{r}}$ as in
Proposition 4.1 under the Harish-Chandra isomorphism. Assume
$N_{1}\geq\cdots\geq N_{r}$ and $t_{1}\leq\ldots\leq t_{r}$, then

$\left\langle
Y_{1}\left(P_{\frac{t_{2}-t_{1}}{2}}Y_{2}\right)\cdots\left(P_{\frac{tr-t1}{2}}Y_{r}\right)\right\rangle_{\frac{t_{1}}{2}}=\mathbb{E}\left[\overline{p}^{N_{1}+1}_{Y_{1}}\left(\mathbf{x}^{n_{1},a_{1}}(t_{1})\right)\cdots\overline{p}^{N_{r}+1}_{Y_{r}}\left(\mathbf{x}^{n_{r},a_{r}}(t_{r})\right)\right],$

where $\mathbf{x}^{n_{r},a_{r}}$ is the vector of $x^{n_{r},a_{r}}_{j}$ with
$1\leq j\leq n_{r}$.

###### Proof.

The proof only needs the Gibbs property of the Markov process
$\left\\{x_{k}^{n,a}|\ 1\leq k\leq n,\ a\in\left\\{\pm 1/2\right\\},\ n\geq
1\right\\}$ (see e.g. [10]) and follows the same lines as that of Theorem 4.5
in [15]. ∎

##  5 Three dimensional Gaussian fluctuations

In this section, we show that certain elements of the Gelfand–Tsetlin
subalgebra are asymptotically Gaussian with an explicit covariance along
space-like paths and time-like paths.

For a Laurent polynomial $p(v)$, let $p(v)[v^{r}]$ denotes the coefficient of
$v^{r}$ in $p(v)$. Then, the main theorem is

######  Theorem 5.1.

Suppose $N_{j}=\lfloor\eta_{j}L\rfloor$, $t_{j}=\tau_{j}L$ for $1\leq j\leq
r$. Assume $\min\\{\tau_{1},\ldots,\tau_{r}\\}=\tau_{1}$. Then as
$L\xrightarrow{}\infty$,

$\left(\frac{\Phi_{2k_{1}}^{N_{1}+1}-\left\langle\Phi_{2k_{1}}^{N_{1}+1}\right\rangle_{\frac{t_{1}}{2}}}{2L^{2k_{1}}},\ldots,\frac{P_{\frac{t_{r}-t_{1}}{2}}\Phi_{2k_{r}}^{N_{r}+1}-\left\langle
P_{\frac{t_{r}-t_{1}}{2}}\Phi_{2k_{r}}^{N_{r}+1}\right\rangle_{\frac{t_{1}}{2}}}{2L^{2k_{r}}}\right)\xrightarrow{}(\xi_{1},\ldots,\xi_{r}),$

where the convergence is with respect to the state
$\langle\cdot\rangle_{\frac{t_{1}}{2}}$ and $(\xi_{1},\ldots,\xi_{r})$ is a
Gaussian vector with covariance

$\mathbb{E}[\xi_{i}\xi_{j}]=\left\\{\begin{array}[]{lc}\frac{1}{(2\pi
i)^{2}}\iint\displaylimits_{|v|>|u|}\left(\frac{(v+2)(\eta_{i}/2+\tau_{i}v)^{2}}{v}\right)^{k_{i}}\left(\frac{(u+2)(\eta_{j}/2+\tau_{j}u)^{2}}{u}\right)^{k_{j}}\frac{1}{(v-u)^{2}}dvdu,&\eta_{i}\geq\eta_{j},\tau_{i}\leq\tau_{j},\\\
\\\ \frac{\sum_{l=1}^{k_{j}}c_{k_{j},l}(\tau_{j},\tau_{i},\eta_{j})}{(2\pi
i)^{2}}\iint\displaylimits_{|v|<|u|}\left(\frac{(v+2)(\eta_{i}/2+\tau_{i}v)^{2}}{v}\right)^{k_{i}}\left(\frac{(u+2)(\eta_{j}/2+\tau_{i}u)^{2}}{u}\right)^{l}\frac{1}{(v-u)^{2}}dvdu,&\eta_{i}<\eta_{j},\tau_{i}\leq\tau_{j},\end{array}\right.$

where $c_{k,l}$ is defined as the coefficient in the following expansion: for
$r\leq-1$,

$\sum_{l=1}^{k}c_{k,l}(\tau_{2},\tau_{1},\eta_{2})\left(\frac{(v+2)\left(\frac{\eta_{2}}{2}+\tau_{1}v\right)^{2}}{v}\right)^{l}[v^{r}]=\left(\frac{(v+2)\left(\frac{\eta_{2}}{2}+\tau_{2}v\right)^{2}}{v}\right)^{k}[v^{r}].$

######  Remark 5.1.

It is straightforward from inspection that the covariance for the random
surface growth when $\eta_{i}\geq\eta_{j},\tau_{i}\leq\tau_{j}$ is different
form the covariance for the spectra of overlapping stochastic Wishart matrices
in [16] due to the lack of a second integral.

It is not obvious that the covariances along time-like paths are different.
Heuristically, if they were the same, the covariance formula in Theorem 5.1
when $\eta_{i}<\eta_{j},\tau_{i}\leq\tau_{j}$ could be written as

$\frac{C_{1}}{(2\pi
i)^{2}}\iint\displaylimits_{|v|<|u|}\left(\frac{(v+2)(\eta_{i}/2+\tau_{i}v)^{2}}{v}\right)^{k_{i}}\left(\frac{(u+2)(\eta_{j}/2+\tau_{j}u)^{2}}{u}\right)^{k_{j}}\frac{1}{(v-u)^{2}}dvdu\\\
+C_{2}\cdot\text{Residue}\left(\left(\frac{(v+2)(\eta_{i}/2+\tau_{i}v)^{2}}{v}\right)^{k_{i}}\right)\text{Residue}\left(\left(\frac{(u+2)(\eta_{j}/2+\tau_{j}u)^{2}}{u}\right)^{k_{j}}\right),$

where $C_{1}$ and $C_{2}$ are constants which are independent of $k_{i}$ and
$k_{j}$. However, after checking a few examples, the constants $C_{1}$ and
$C_{2}$ do not exist. For example, letting $k_{i}=1,2$ and $k_{j}=1$ and
solving for $C_{1,2}$ yields $C_{1}=\tau_{i}^{2}/\tau_{j}^{2}$ and
$C_{2}=2\tau_{1}(\tau_{2}-\tau_{1})/(\eta_{2}\tau_{2})$. However, if
$C_{1}=\tau_{i}^{2}/\tau_{j}^{2}$ and
$C_{2}=2\tau_{1}(\tau_{2}-\tau_{1})/(\eta_{2}\tau_{2})$, the covariances are
not equal when $k_{i}=1$ and $k_{j}=2$.

###  5.1 Gaussian fluctuations along space-like paths

In this section, we focus on the space-like paths. It was proved that the
random surface growth converges to a deterministic limit shape and the
fluctuations around the limit shape are described by the Gaussian free field
fluctuations (see e.g. [10, 13]).

Let $G(z)=G(\nu,\eta,\tau;z)$ be the function

$G(\nu,\eta,\tau;z)=\tau\frac{z+z^{-1}}{2}+\eta\log\left(\frac{z+z^{-1}}{2}-1\right)-\nu\log
z,$

and $\mathcal{D}$ be the connected domain consisting of all triples
$(\nu,\eta,\tau)$ such that $G(\nu,\eta,\tau;z)$ has a unique critical point
in the region $\mathbb{H-D}=\left\\{z|\Im z>0,\ |z|>1\right\\}$. Let
$\Upsilon$ be the map sending $(\nu,\eta,\tau)\in\mathcal{D}$ to the critical
point of $G(\nu,\eta,\tau;z)$ in $\mathbb{H-D}$. Specifically,

$\mathcal{D}=\left\\{(\nu,\eta,\tau)|l(\eta,\tau)<\nu<r(\eta,\tau),\tau,\eta>0\right\\},$

where

$\begin{array}[]{l}r(\eta,\tau)=\sqrt{-\frac{\tau^{2}}{2}+5\tau\eta+1+\frac{\tau^{2}}{2}\left(1+\frac{4\eta}{\tau}\right)^{3/2}},\\\
\\\ l(\eta,\tau)=\left\\{\begin{array}[]{lc}0,&\frac{\eta}{\tau}\leq 2,\\\
\sqrt{-\frac{\tau^{2}}{2}+5\tau\eta+1-\frac{\tau^{2}}{2}\left(1+\frac{4\eta}{\tau}\right)^{3/2}},&\frac{\eta}{\tau}>2.\end{array}\right.\end{array}$

Let $H(x,N,t)$ be the height function of $\mathbf{x}$, which is defined as the
number of particles to the right of $(x,n,a)$ at time $t$. We recall the limit
shape of the height function $H$ from [10].

######  Theorem 5.2.

[10] For any $\left(\nu,\frac{\eta}{2},\tau\right)\in\mathcal{D}$, suppose
$\Upsilon\left(\nu,\frac{\eta}{2},\tau\right)=z_{0}$, then the limit shape
exists,

$h(z_{0})=\lim_{L\xrightarrow[]{}\infty}\frac{1}{L}\mathbb{E}H(\lfloor\nu
L\rfloor,\lfloor\eta L\rfloor,\tau
L)=\Im\left(\frac{G\left(z_{0}\right)}{\pi}\right).$ (5.1)

By Theorem 1.1 in [13], the height fluctuations converge to a Gaussian free
field on the domain $\mathcal{D}$. The proof is based on the fact that the
interacting particle system is a determinantal point process. Analogous to [9,
13], the determinantal structure derived in Theorem 3.1 also leads to the
convergence of the moments of height fluctuations to that of a Gaussian free
field along space-like paths. As a result, we could generalize Theorem 1.1 in
[13] as the following.

######  Theorem 5.3.

For any $r\in\mathbb{N}^{+}$, let
$\varkappa_{j}=\left(\nu_{j},\frac{\eta_{j}}{2},\tau_{j}\right)\in\mathcal{D}$
for $1\leq j\leq r$. Define

$H_{L}(\nu,\eta,\tau)=\sqrt{\pi}\left(H(\nu L,\lfloor\eta L\rfloor,\tau
L)-\mathbb{E}H(\nu L,\lfloor\eta L\rfloor,\tau L)\right)$

and $\Upsilon_{j}=\Upsilon(\varkappa_{j})$. Assume
$\\{\varkappa_{j}\\}_{j=1}^{r}$ lie on a space-like path, that is
$\eta_{1}\geq\ldots\geq\eta_{r}$ and $\tau_{1}\leq\ldots\leq\tau_{r}$, then

$\lim_{L\xrightarrow[]{}\infty}\mathbb{E}(H_{L}(\varkappa_{1})\cdots
H_{L}(\varkappa_{r}))=\left\\{\begin{array}[]{lr}\sum_{\sigma}\prod_{i=1}^{r/2}\mathcal{G}(\Upsilon_{\sigma(2i-1)},\Upsilon_{\sigma(2i)}),&r\
\text{even},\\\ 0,&r\ \text{odd},\end{array}\right.$

where the sum is over all fixed point free involutions $\sigma$ on
$\\{1,\ldots,r\\}$ and $\mathcal{G}$ is the funciton

$\mathcal{G}(z,w)=\frac{1}{2\pi}\log\left|\frac{z+z^{-1}-\overline{w}-\overline{w}^{-1}}{z+z^{-1}-w-w^{-1}}\right|.$

Let $p_{2k}^{N+1}=\sum_{i=1}^{n}l_{i}^{2k}$, which is the image of
$\frac{\Phi^{N+1}_{2k}}{2}$ under the Harish-Chandra isomorphism. We first
relate the height function and the polynomials $p_{2k}^{N+1}$ as in [3]. The
definition of the height function implies that

$\frac{d}{du}H(u,N)=-\sqrt{\pi}\sum_{s=1}^{n}\delta(u-(\lambda_{s}^{N}-s+n)).$

Recall that for $N+1=2n+1/2+a$, $l_{i}=\lambda_{i}-i+\frac{3}{4}-\frac{a}{2}$,

$\frac{d}{du}H(u,N)=-\sqrt{\pi}\sum_{s=1}^{n}\delta\left(u-\left(l_{s}+n-\frac{3}{4}+\frac{a}{2}\right)\right).$
(5.2)

Then, let $u=Lx$, $N=\lfloor L\eta\rfloor$, $t=L\tau$,

$p_{2k}^{N+1}=\int_{0}^{\infty}\left(u-n+\frac{3}{4}-\frac{a}{2}\right)^{2k}\left(\sum_{s=1}^{n}\delta\left(u-n-
l_{s}+\frac{3}{4}-\frac{a}{2}\right)\right)du\\\
=-\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}\left(u-n+\frac{3}{4}-\frac{a}{2}\right)^{2k}\frac{d}{du}H(u,N)du\\\
\approx\frac{1}{\sqrt{\pi}}\left(n^{2k+1}+2k\int_{0}^{\infty}(u-n)^{2k-1}H(u,N)du\right).$

The relation between height function and the polynomial $p_{2k}^{N+1}$
indicates the convergence to a Gaussian free field.

######  Proposition 5.1.

Suppose $N_{j}=\lfloor\eta_{j}L\rfloor$, $t_{j}=\tau_{j}L$ for $1\leq j\leq
r$. Assume $\eta_{1}\geq\ldots\geq\eta_{r}$ and
$\tau_{1}\leq\ldots\leq\tau_{r}$, then as $L\xrightarrow{}\infty$,

$\left(\frac{\Phi_{2k_{1}}^{N_{1}+1}-\left\langle\Phi_{2k_{1}}^{N_{1}+1}\right\rangle_{\frac{t_{1}}{2}}}{2L^{2k_{1}}},\ldots,\frac{P_{\frac{t_{r}-t_{1}}{2}}\Phi_{2k_{r}}^{N_{r}+1}-\left\langle
P_{\frac{t_{r}-t_{1}}{2}}\Phi_{2k_{r}}^{N_{r}+1}\right\rangle_{\frac{t_{1}}{2}}}{2L^{2k_{r}}}\right)\xrightarrow{}(\xi_{1},\ldots,\xi_{r}),$

where the convergence is with respect to the state
$\langle\cdot\rangle_{\frac{t_{1}}{2}}$ and $(\xi_{1},\ldots,\xi_{r})$ is a
Gaussian vector.

###### Proof.

The convergence to a Gaussian vector (along both space-like and time-like
paths) for elements in $Z(U(\mathfrak{so}_{N+1}))$ can be proved using
combinatorial arguments analogous to Proposition 5.1 of [3]. Another way to
see that is to apply Theorem 4.3 and Theorem 5.3 to show that the limit as
$L\xrightarrow[]{}\infty$ of joint moments of

$\frac{\Phi_{2k_{j}}^{N_{j}+1}-\left\langle\Phi_{2k_{j}}^{N_{j}+1}\right\rangle_{\frac{t_{j}}{2}}}{2L^{2k_{j}}}$

satisfy the Wick’s formula, which implies the convergence to a Gaussian
vector. ∎

Next, we calculate the covariance structure for elements in
$Z(U(\mathfrak{so}_{N+1}))$.

######  Proposition 5.2.

Suppose $\eta_{1}\geq\eta_{2}$ and $\tau_{1}\leq\tau_{2}$,

$\lim_{L\xrightarrow{}\infty}\left\langle\frac{\Phi_{2k}^{\lfloor\eta_{1}L\rfloor+1}-\left\langle\Phi_{2k}^{\lfloor\eta_{1}L\rfloor+1}\right\rangle_{\tau_{1}L/2}}{2L^{2k}}\cdot\frac{P_{(\tau_{2}-\tau_{1})/2L}\Phi_{2l}^{\lfloor\eta_{2}L\rfloor+1}-\left\langle
P_{(\tau_{2}-\tau_{1})/2L}\Phi_{2l}^{\lfloor\eta_{2}L\rfloor+1}\right\rangle_{\tau_{1}L/2}}{2L^{2l}}\right\rangle_{\tau_{1}L/2}\\\
=\frac{1}{(2\pi
i)^{2}}\iint\displaylimits_{|v|>|u|}\left(\frac{(v+2)(\eta_{1}/2+\tau_{1}v)^{2}}{v}\right)^{k}\left(\frac{(u+2)(\eta_{2}/2+\tau_{2}u)^{2}}{u}\right)^{l}\frac{1}{(v-u)^{2}}dvdu.$
(5.3)

###### Proof.

By Theorem 4.3, it suffices to compute the limit of

$\frac{\mathbb{E}\left[\left(p_{2k}^{\lfloor
L\eta_{1}\rfloor+1}(\tau_{1}L)-\mathbb{E}p_{2k}^{\lfloor
L\eta_{1}\rfloor+1}(\tau_{1}L)\right)\left(p_{2l}^{\lfloor
L\eta_{2}\rfloor+1}(\tau_{2}L)-\mathbb{E}p_{2l}^{\lfloor
L\eta_{2}\rfloor+1}(\tau_{2}L)\right)\right]}{L^{2k+2l}}$

as $L\xrightarrow{}\infty$. Apply Theorem 5.3 and dominated convergence
theorem,

$\lim_{L\xrightarrow{}\infty}\frac{\mathbb{E}\left[\left(p_{2k}^{\lfloor
L\eta_{1}\rfloor+1}(\tau_{1}L)-\mathbb{E}p_{2k}^{\lfloor
L\eta_{1}\rfloor+1}(\tau_{1}L)\right)\left(p_{2l}^{\lfloor
L\eta_{2}\rfloor+1}(\tau_{2}L)-\mathbb{E}p_{2l}^{\lfloor
L\eta_{2}\rfloor+1}(\tau_{2}L)\right)\right]}{L^{2k+2l}}\\\
=\displaystyle{\frac{4kl}{\pi}\int\displaylimits_{\left(x,\frac{\eta_{1}}{2},\tau_{1}\right)\in\mathcal{D}}\int\displaylimits_{\left(y,\frac{\eta_{2}}{2},\tau_{2}\right)\in\mathcal{D}}\left(x-\frac{\eta_{1}}{2}\right)^{2k-1}\left(y-\frac{\eta_{2}}{2}\right)^{2l-1}\mathcal{G}\left(\Upsilon\left(y,\frac{\eta_{2}}{2},\tau_{2}\right),\Upsilon\left(x,\frac{\eta_{1}}{2},\tau_{1}\right)\right)dxdy.}$
(5.4)

Recall that

$\mathcal{G}(z,w)=\frac{1}{2\pi}\log\left|\frac{z+z^{-1}-\overline{w}-\overline{w}^{-1}}{z+z^{-1}-w-w^{-1}}\right|$

and $\Upsilon:\mathcal{D}\xrightarrow{}\mathbb{H-D}$ is defined by sending
$(d,l,t)\in\mathcal{D}$ to the solution of

$\left\\{\begin{array}[]{cc}z+\overline{z}-\frac{1}{z\overline{z}}=-\frac{2l-2d-t}{t},\\\
\frac{1}{z}+\frac{1}{\overline{z}}-z\overline{z}=-\frac{2l+2d-t}{t}.\end{array}\right.$

Now, let $z(x,\eta_{1},\tau_{1})=\Upsilon(x,\eta_{1}/2,\tau_{1})$,
$w(y,\eta_{2},\tau_{2})=\Upsilon(y,\eta_{2}/2,\tau_{2})$, then
$x(z)=\eta_{1}+\frac{\tau_{1}}{2}\frac{z^{3}-z^{2}+(\frac{2\eta_{1}}{\tau_{1}}-1)z+1}{z(z-1)}$
and
$y(w)=\eta_{2}+\frac{\tau_{2}}{2}\frac{w^{3}-w^{2}+(\frac{2\eta_{2}}{\tau_{2}}-1)w+1}{w(w-1)}$.

Using the symmetry of the integrand, the right-hand-side of (5.4) can be
written as

$\frac{4kl}{(2\pi
i)^{2}}\int\displaylimits_{\gamma_{1}}\int\displaylimits_{\gamma_{2}}\left(x(z)-\frac{\eta_{1}}{2}\right)^{2k-1}\left(y(w)-\frac{\eta_{2}}{2}\right)^{2l-1}\log(z+z^{-1}-w-w^{-1})\frac{d(x(z))}{dz}\frac{d(y(w))}{dw}dzdw,$

where for each fixed $\eta_{i}$ and $\tau_{i}$, the integration contour
$\gamma_{i}$ is a positively orientated path given by all points in the set
$\left\\{z(x,\eta_{i},\tau_{i}),\overline{z(x,\eta_{i},\tau_{i})}|\left(x,\frac{\eta_{i}}{2},\tau_{i}\right)\in\mathcal{D}\right\\}$
(see figure 2).

Figure 2: Integration contour $\gamma_{i}$ when $\eta_{i}/\tau_{i}=4$

Integrate by part in $z$ and $w$ and let $v=\frac{z+z^{-1}}{2}-1$,
$u=\frac{w+w^{-1}}{2}-1$, we have

$\frac{1}{(2\pi
i)^{2}}\oint\displaylimits_{\gamma_{1}^{\prime}}\oint\displaylimits_{\gamma_{2}^{\prime}}\left(\frac{(v+2)(\eta_{1}/2+\tau_{1}v)^{2}}{v}\right)^{k}\left(\frac{(u+2)(\eta_{2}/2+\tau_{2}u)^{2}}{u}\right)^{l}\frac{1}{(v-u)^{2}}dvdu.$
(5.5)

Here $\gamma_{1}^{\prime}$ and $\gamma_{2}^{\prime}$ are images of
$\gamma_{1}$ and $\gamma_{2}$ under the change of variable. Last, modify
$\gamma_{1}^{\prime}$ and $\gamma_{2}^{\prime}$ to two concentric circles
centered at $0$, then the double integral could be written as

$\frac{1}{(2\pi
i)^{2}}\iint\displaylimits_{|v|>|u|}\left(\frac{(v+2)(\eta_{1}/2+\tau_{1}v)^{2}}{v}\right)^{k}\left(\frac{(u+2)(\eta_{2}/2+\tau_{2}u)^{2}}{u}\right)^{l}\frac{1}{(v-u)^{2}}dvdu.$

∎

Note that

$\frac{(v+2)(\eta_{1}/2+\tau_{1}v)^{2}}{v}=\frac{\eta_{1}^{2}}{2v}+\tau_{1}^{2}v^{2}+(2\tau_{1}^{2}+\eta_{1}\tau_{1})v+\frac{\eta_{1}^{2}}{4}+2\eta_{1}\tau_{1},$

then the double integral can be computed using residue theorem and the Taylor
series

$(v-u)^{-2}=v^{-2}\left(1+2\frac{u}{v}+3\frac{u^{2}}{v^{2}}+\cdots\right).$

######  Example 3.

Recall
$\Phi_{2}^{N+1}=2\sum_{m=1}^{n}\left((F_{mm}+\rho_{m})^{2}+2\sum_{-m<i<m}F_{mi}F_{im}\right)$.
When $k=l=1$, (5.3) equals
$\tau_{1}^{2}\eta_{2}^{2}+1/2\tau_{1}\eta_{1}\eta_{2}^{2}$.

By direct computation,

$P_{(\tau_{2}-\tau_{1})L}\Phi_{2}^{N+1}=\Phi_{2}^{N+1}+2\sum_{m=1}^{n}\left(2(\tau_{2}-\tau_{1})L+\sum_{-m<i<m}2(\tau_{2}-\tau_{1})L\right)$

and

$\left\langle\Phi_{2}^{N+1}\right\rangle_{\tau_{1}L}=2\sum_{m=1}^{n}\left(\left(2\tau_{1}L+\rho_{m}^{2}\right)+2\sum_{-m<i<m}2\tau_{1}L\right),$

thus

$P_{(\tau_{2}-\tau_{1})L}\Phi_{2}^{N+1}-\left\langle
P_{(\tau_{2}-\tau_{1})L}\Phi_{2}^{N+1}\right\rangle_{\tau_{1}L}=\Phi_{2}^{N+1}-\left\langle\Phi_{2}^{N+1}\right\rangle_{\tau_{1}L},$

moreover,

$\lim_{L\xrightarrow{}\infty}\left\langle\frac{\Phi_{2}^{\lfloor\eta_{1}L\rfloor+1}-\left\langle\Phi_{2}^{\lfloor\eta_{1}L\rfloor+1}\right\rangle_{\tau_{1}L/2}}{2L^{2k}}\cdot\frac{P_{(\tau_{2}-\tau_{1})/2L}\Phi_{2}^{\lfloor\eta_{2}L\rfloor+1}-\left\langle
P_{(\tau_{2}-\tau_{1})/2L}\Phi_{2}^{\lfloor\eta_{2}L\rfloor+1}\right\rangle_{\tau_{1}L/2}}{2L^{2l}}\right\rangle_{\tau_{1}L/2}\\\
=\lim_{L\xrightarrow{}\infty}\frac{1}{L^{4}}\left\langle\left(\frac{\Phi_{2}^{\lfloor\eta_{1}L\rfloor+1}}{2}-\left\langle\frac{\Phi_{2}^{\lfloor\eta_{1}L\rfloor+1}}{2}\right\rangle_{\frac{\tau_{1}L}{2}}\right)\cdot\left(\frac{\Phi_{2}^{\lfloor\eta_{2}L\rfloor+1}}{2}-\left\langle\frac{\Phi_{2}^{\lfloor\eta_{2}L\rfloor+1}}{2}\right\rangle_{\frac{\tau_{1}L}{2}}\right)\right\rangle_{\frac{\tau_{1}L}{2}}=\tau_{1}^{2}\eta_{2}^{2}+1/2\tau_{1}\eta_{1}\eta_{2}^{2},$

which matches (5.3).

###  5.2 Gaussian fluctuations along time-like paths

Next, we prove an analogy of Proposition 5.3 in [15], which provides an
expression for the covariance along the time-like paths.

Given a partition $\rho=(\rho_{1},\ldots,\rho_{l})$, let
$\Phi_{\rho}=\prod_{i=1}^{l}\Phi_{2\rho_{i}}$, denote
$|\rho|=2(\rho_{1}+\cdots+\rho_{l})$ and $\text{wt}(\rho)=|\rho|+l$.

######  Proposition 5.3.

Let $\eta,\tau>0$, set $N=\lfloor\eta L\rfloor$, and $t=\tau L$, then

  1. (1)

$<\Phi_{\rho}^{N+1}>_{t}=\Theta(L^{\text{wt}(\rho)})$ .

  2. (2)

There exist constants $c^{\prime}_{k,\rho}(\tau,\eta)$ such that

$P_{\tau
L}\Phi_{2k}^{N+1}=\sum_{\rho}\left(c^{\prime}_{k,\rho}(\tau,\eta)+o(1)\right)L^{2k+1-\text{wt}(\rho)}\Phi_{\rho}^{N+1},$

where the sum is over $\rho$ such that $\text{wt}(\rho)\leq 2k+1$.

  3. (3)

For any $\tau_{2}\geq\tau_{1}$, there exist constants
$c_{k,j}(\tau_{2},\tau_{1},\eta_{2})$ such that

$\lim_{L\xrightarrow{}\infty}\left<\frac{\Phi_{2m}^{\lfloor\eta_{1}L\rfloor+1}-\left<\Phi_{2m}^{\lfloor\eta_{1}L\rfloor+1}\right>_{\tau_{1}L}}{L^{2m}}\cdot\frac{P_{(\tau_{2}-\tau_{1})/2L}\Phi_{2k}^{\lfloor\eta_{2}L\rfloor+1}-\left<P_{(\tau_{2}-\tau_{1})L}\Phi_{2k}^{\lfloor\eta_{2}L\rfloor+1}\right>_{\tau_{1}L}}{L^{2k}}\right>_{\tau_{1}L}\\\
=\lim_{L\xrightarrow{}\infty}\sum_{j=1}^{k}c_{k,j}(\tau_{2},\tau_{1},\eta_{2})\left\langle\frac{\Phi_{2m}^{\lfloor\eta_{1}L\rfloor+1}-\left\langle\Phi_{2m}^{\lfloor\eta_{1}L\rfloor+1}\right\rangle_{\tau_{1}L}}{L^{2m}}\cdot\frac{\Phi_{2j}^{\lfloor\eta_{2}L\rfloor+1}-\left\langle\Phi_{2j}^{\lfloor\eta_{2}L\rfloor+1}\right\rangle_{\tau_{1}L}}{L^{2j}}\right\rangle_{\tau_{1}L}.$

###### Proof.

We can prove all the results following the same arguments in [15]. Another
intuitive way to see that (1) is true is to use Proposition 4.1, Theorem 5.2
and Theorem 5.3. Recall the limit of height function in frozen region is given
by $(\frac{\eta}{2}-x)$, and in liquid region $\mathcal{D}$ is expected to be
the function $h\left(\Upsilon\left(x,\frac{\eta}{2},\tau\right)\right)$
defined in (5.1). Note that
$l\left(\frac{\eta}{2},\tau\right)<\frac{\eta}{2}<r\left(\frac{\eta}{2},\tau\right)$
when $\eta>0$. Let $N=\lfloor\eta L\rfloor$, $t=\tau L$, then interchange the
expectation and integration, we have

$\lim_{L\xrightarrow{}\infty}\left\langle\frac{\Phi^{N+1}_{2k}}{2L^{2k+1}}\right\rangle_{\frac{t}{2}}=\lim_{L\xrightarrow{}\infty}\frac{\mathbb{E}[p_{2k}^{N+1}]}{L^{2k+1}}=\frac{1}{\sqrt{\pi}}\bigg{(}2k\int_{l\left(\frac{\eta}{2},\tau\right)}^{r\left(\frac{\eta}{2},\tau\right)}\left(x-\frac{\eta}{2}\right)^{2k-1}h\left(\Upsilon\left(x,\frac{\eta}{2},\tau\right)\right)dx\\\
-2k\int_{0}^{l\left(\frac{\eta}{2},\tau\right)}\left(x-\frac{\eta}{2}\right)^{2k}dx+\left(\frac{\eta}{2}\right)^{2k+1}\bigg{)}.$
(5.6)

The right-hand-side of (5.6) is strictly bounded below by

$\frac{1}{\sqrt{\pi}}\left(2k\int_{l\left(\frac{\eta}{2},\tau\right)}^{\frac{\eta}{2}}\left(x-\frac{\eta}{2}\right)^{2k-1}\left(\frac{\eta}{2}-l\left(\frac{\eta}{2},\tau\right)\right)dx-2k\int_{0}^{l\left(\frac{\eta}{2},\tau\right)}\left(x-\frac{\eta}{2}\right)^{2k}dx+\left(\frac{\eta}{2}\right)^{2k+1}\right)\\\
=\frac{\left(\frac{\eta}{2}\right)^{2k+1}-\left(\frac{\eta}{2}-l\left(\frac{\eta}{2},\tau\right)\right)^{2k+1}}{\sqrt{\pi}(2k+1)}.$

Thus, when $\eta>0$, the right-hand side of (5.6) is strictly larger than $0$
and is finite, which means
$\left\langle\Phi^{N+1}_{2k}\right\rangle_{t}=\Theta(L^{2k+1})$. Also by
Theorem 5.1, we know that

$\left\langle\left(\Phi^{N+1}_{2k}\right)^{2}\right\rangle_{t}=\Theta(L^{2(2k+1)}).$

Thus by Cauchy-Schwartz inequality,
$\langle\Phi_{\rho}\rangle_{t}=\mathcal{O}(L^{\text{wt}(\rho)})$.

To get a lower bound, one notice that
$\lim_{L\xrightarrow{}\infty}\frac{p_{2k}^{N+1}(\lambda(t))}{L^{2k+1}}>0$ a.e.
with respect to $\mathbb{E}$,

$\lim_{L\xrightarrow{}\infty}\left\langle\prod_{i=1}^{l}\frac{\Phi^{N+1}_{2\rho_{i}}}{2L^{2\rho_{i}+1}}\right\rangle_{t/2}=\lim_{L\xrightarrow{}\infty}\mathbb{E}\left(\prod_{i=1}^{l}\frac{p^{N+1}_{2k_{i}}}{L^{2\rho_{i}+1}}\right)>0.$

(2) follows from the fact that $\langle
P_{\tau_{1}L}\Phi_{2k}^{N+1}\rangle_{\tau_{2}L}=\Theta(L^{\text{2k+1}})$, thus
only $\text{wt}(\rho)\leq 2k+1$ terms have nonzero coefficients.

To prove (3), we first apply (2) to the left-hand-side. Note that
$\frac{\Phi_{2k}^{N+1}-\langle\Phi_{2k}^{N+1}\rangle_{t}}{L^{2k}}$ corresponds
to the moment of the fluctuation of height function, thus converges to a
Gaussian random variable. Heuristically,
$\Phi_{2k}^{N+1}\approx\langle\Phi_{2k}^{N+1}\rangle_{t}+\xi L^{2k}$, where
$\xi$ is a Gaussian random variable.

$\prod_{i=1}^{l}\Phi_{2\rho_{i}}-\left\langle\prod_{i=1}^{l}\Phi_{2\rho_{i}}\right\rangle_{\tau
L}=\sum_{j=1}^{l}\left\langle\Phi_{2\rho_{1}}\right\rangle_{\tau
L}\cdots\widehat{\left\langle\Phi_{2\rho_{j}}\right\rangle_{\tau
L}}\cdots\left\langle\Phi_{2\rho_{l}}\right\rangle_{\tau
L}\left(\Phi_{2\rho_{j}}-\left\langle\Phi_{2\rho_{j}}\right\rangle_{\tau
L}\right)+\text{ smaller order terms}$

Thus in the asymptotic limit, we could replace $P_{t}\Phi_{\rho}$ with a
linear combination of $\Phi_{2k}$ with $2k+1\leq\text{wt}(\rho)$. ∎

Expand $(v-u)^{-2}$ as
$v^{-2}\left(1+2\frac{u}{v}+3\frac{u^{2}}{v^{2}}+\cdots\right)$ in Proposition
5.2 and take residues, one obtains

$\sum_{j=1}^{k}c_{kj}(\tau_{2},\tau_{1},\eta_{2})\left(\frac{(v+2)\left(\frac{\eta_{2}}{2}+\tau_{1}v\right)^{2}}{v}\right)^{j}[v^{r}]=\left(\frac{(v+2)\left(\frac{\eta_{2}}{2}+\tau_{2}v\right)^{2}}{v}\right)^{k}[v^{r}]$
(5.7)

for $r\leq-1$. We make use of the expansion (5.7) to compute the coefficients
$c_{k,j}$, which provides a formula for the covariance along the time-like
paths.

######  Example 4.

When $k=2$, solving (5.7) for $r=-1,-2$, we have $c_{2,2}=1$ and
$c_{2,1}=4\eta_{2}(\tau_{2}-\tau_{1})$, which agrees with the expansion of
$P_{t}\Phi_{4}^{N+1}$ derived in Example 2:

$P_{t}\Phi_{4}^{N+1}=\Phi_{4}^{N+1}+16tn\Phi_{2}^{N+1}+\text{constant}.$

######  Example 5.

  1. (1)

When $k=3$,
$c_{3,3}=1$,
$c_{3,2}=-6\eta_{2}(\tau_{1}-\tau_{2})$,
$c_{3,1}=-\frac{3}{2}\left(\eta_{2}^{3}\tau_{1}-6\eta_{2}^{2}\tau_{1}^{2}-\eta_{2}^{3}\tau_{2}+16\eta_{2}^{2}\tau_{1}\tau_{2}-10\eta_{2}^{2}\tau_{2}^{2}\right)$.

  2. (2)

When $k=4$,
$c_{4,4}=1$,
$c_{4,3}=-8\eta_{2}(\tau_{1}-\tau_{2})$,
$c_{4,2}=-2\left(\eta_{2}^{3}\tau_{1}-10\eta_{2}^{2}\tau_{1}^{2}-\eta_{2}^{3}\tau_{2}+24\eta_{2}^{2}\tau_{1}\tau_{2}-14\eta_{2}^{2}\tau_{2}^{2}\right)$,
$c_{4,1}=-\frac{1}{2}\left(-\eta_{2}^{5}\tau_{1}+12\eta_{2}^{4}\tau_{1}^{2}-32\eta_{2}^{3}\tau_{1}^{3}+\eta_{2}^{5}\tau_{2}-40\eta_{2}^{4}\tau_{1}\tau_{2}+144\eta_{2}^{3}\tau_{1}^{2}\tau_{2}+28\eta_{2}^{4}\tau_{2}^{2}-224\eta_{2}^{3}\tau_{1}\tau_{2}^{2}+122\eta_{2}^{3}\tau_{2}^{3}\right)$.

Last, we briefly show the convergence to a Gaussian vector along time-like
paths.

######  Proposition 5.4.

Suppose $N_{j}=\lfloor\eta_{j}L\rfloor$, $t_{j}=\tau_{j}L$ for $1\leq j\leq
r$. Assume $\eta_{1}\leq\ldots\leq\eta_{r}$ and
$\tau_{1}\leq\ldots\leq\tau_{r}$, then as $L\xrightarrow{}\infty$,

$\left(\frac{\Phi_{2k_{1}}^{N_{1}+1}-\left\langle\Phi_{2k_{1}}^{N_{1}+1}\right\rangle_{\frac{t_{1}}{2}}}{2L^{2k_{1}}},\ldots,\frac{P_{\frac{t_{r}-t_{1}}{2}}\Phi_{2k_{r}}^{N_{r}+1}-\left\langle
P_{\frac{t_{r}-t_{1}}{2}}\Phi_{2k_{r}}^{N_{r}+1}\right\rangle_{\frac{t_{1}}{2}}}{2L^{2k_{r}}}\right)\xrightarrow{}(\xi_{1},\ldots,\xi_{r}),$

where the convergence is with respect to the state
$\langle\cdot\rangle_{\frac{t_{1}}{2}}$ and $(\xi_{1},\ldots,\xi_{r})$ is a
Gaussian vector.

###### Proof.

Same as the proof for space-like paths in Proposition 5.1, we could apply
Theorem 4.3, 5.3 and Proposition 5.3 to derive the explicit formula for the
joint moments, which satisfies the Wick’s formula and implies the convergence
to a Gaussian vector.

∎

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