# One-point distribution of the geodesic in directed last passage percolation

Zhipeng Liu

###### Abstract

We consider the geodesic of the directed last passage percolation with iid
exponential weights. We find the explicit one-point distribution of the
geodesic location joint with the last passage times, and its limit as the
parameters go to infinity under the KPZ scaling.

##  1 Introduction

In recent twenty years, there has been a huge progress towards to
understanding a universal class of random growth models, the so-called Kardar-
Parisi-Zhang (KPZ) universality class [BDJ99, Joh00, Joh03, BFPS07, TW08,
TW09, BC14, MQR17, DOV18, JR19, Liu19]. Very recently, studies about the
geodesics of these models started to appear [BSS17, Ham20, HS20, BHS18, BGH21,
BGH19, BF20, DSV20, CHHM21, DV21]. However, the explicit distributions of the
geodesic are still not well understood. As far as we know, the only known
related results are the distribution of the geodesic endpoint location
[MFQR13, Sch12, BLS12].

This is the first paper of an ongoing project to investigate the limiting
distributions of the geodesics in one representative model, the directed last
passage percolation with exponential weights, using the methods in integrable
probability. We obtain the finite time one-point distribution of the geodesic
location joint with the last passage times, see Theorem 1.1. We are also able
to find the large time limit of this distribution function, see Theorem 1.3.
We remark that our results are for the point-to-point geodesic. In the follow-
up papers, we will consider the point-to-point and point-to-line geodesics
using a different approach, and the multi-point distributions of the point-to-
point geodesic.

The limiting distributions obtained in this paper are expected to be universal
for all models in the KPZ universality class. See [DV21] for more discussions
related to the geodesics.

Below we introduce the main results of the paper. We start from a short
description of the model.

The directed last passage percolation is defined on the lattice set
$\mathbb{Z}^{2}$. We assign to each integer site $\mathbf{p}\in\mathbb{Z}^{2}$
an i.i.d. exponential random variable $w(\mathbf{p})$ with mean $1$. Assume
that $\mathbf{p}$ and $\mathbf{q}$ are two lattice points satisfying
$\mathbf{q}-\mathbf{p}\in\mathbb{Z}_{\geq 0}^{2}$, i.e., the point
$\mathbf{q}$ lies in the upper right direction of $\mathbf{p}$. The last
passage time from $\mathbf{p}$ to $\mathbf{q}$ is

$L_{\mathbf{p}}(\mathbf{q}):=\max_{\pi}\sum_{\mathbf{r}\in\pi}w(\mathbf{r}),$
(1.1)

where the maximum is over all possible up/right lattice paths from
$\mathbf{p}$ to $\mathbf{q}$.

Since the random variables $w(\mathbf{r})$’s are continuous, the last passage
time $L_{\mathbf{p}}(\mathbf{q})$ in (1.1) is almost surely obtained at a
unique up/right lattice path, which we call the geodesic from $\mathbf{p}$ to
$\mathbf{q}$ and denote ${\mathcal{G}}_{\mathbf{p}}(\mathbf{q})$.

Note that the two neighboring sites $\mathbf{r}$ and $\mathbf{r}_{+}$ with
$\mathbf{r}_{+}-\mathbf{r}\in\\{(0,1),(1,0)\\}$ are on the geodesic
${\mathcal{G}}_{\mathbf{p}}(\mathbf{q})$, if and only if the sites
$\mathbf{p},\mathbf{r},\mathbf{r}_{+},\mathbf{q}$ satisfy
$\mathbf{r}-\mathbf{p},\mathbf{q}-\mathbf{r}_{+}\in\mathbb{Z}_{\geq 0}^{2}$,
and the last passage times $L_{\mathbf{p}}(\mathbf{r})$ and
$L_{\mathbf{r}_{+}}(\mathbf{q})$ satisfy

$L_{\mathbf{p}}(\mathbf{r})+L_{\mathbf{r}_{+}}(\mathbf{q})=L_{\mathbf{p}}(\mathbf{q}).$
(1.2)

Throughout this paper, we always use $\mathbf{r}_{+}$ to denote the lattice
point following $\mathbf{r}$ in the geodesic.

###  1.1 Finite time joint probabilities of geodesic location and last passage
times

The first main result of this paper is about the joint probability that a
fixed pair of neighboring sites $\mathbf{r}$ and $\mathbf{r}^{\prime}$ are on
the geodesic ${\mathcal{G}}_{\mathbf{p}}(\mathbf{q})$, and the two last
passage times $L_{\mathbf{p}}(\mathbf{r})$,
$L_{\mathbf{r}^{\prime}}(\mathbf{q})$ lie in some intervals.

######  Theorem 1.1.

Set $\mathbf{p}=(1,1)$, $\mathbf{q}=(M,N)$. Suppose $\mathbf{r}=(m,n)$ and
$\mathbf{r^{\prime}}=(m+1,n)$, with $m,n$ satisfying $1\leq m\leq M-1$ and
$1\leq n\leq N$. Assume that $t_{1},t_{2},\epsilon_{1},\epsilon_{2}$ are all
positive real numbers. We have

$\mathbb{P}\left(\mathbf{r},\mathbf{r}^{\prime}\in{\mathcal{G}}_{\mathbf{p}}(\mathbf{q}),L_{\mathbf{p}}(\mathbf{r})\in[t_{1},t_{1}+\epsilon_{1}],L_{\mathbf{r^{\prime}}}(\mathbf{q})\in[t_{2},t_{2}+\epsilon_{2}]\right)=\int_{t_{1}}^{t_{1}+\epsilon_{1}}\int_{t_{2}}^{t_{2}+\epsilon_{2}}p(s_{1},s_{2};m,n,M,N)\mathrm{d}s_{2}\mathrm{d}s_{1},$
(1.3)

where the function $p(s_{1},s_{2};m,n,M,N)$ is defined in (1.7). Similarly, if
$\mathbf{r}=(m,n)$ and $\mathbf{r^{\prime}}=(m,n+1)$, with $m,n$ satisfying
$1\leq m\leq M$ and $1\leq n\leq N-1$, the formula (1.3) holds with
$p(s_{1},s_{2};m,n,M,N)$ replaced by $p(s_{1},s_{2};n,m,N,M)$.

######  Remark 1.2.

By setting $t_{1}=t_{2}=0$ and $\epsilon_{1}=\epsilon_{2}=\infty$, one can
derive a formula for the probability of
$\mathbf{r},\mathbf{r}^{\prime}\in{\mathcal{G}}_{\mathbf{p}}(\mathbf{q})$
without the double integral with respect to the last passage times. See
(1.11). However, we are not able to directly perform the asymptotics analysis
of this formula since the summand (1.12) diverges when the parameters go to
infinity under the KPZ scaling, the scaling of most interests to us. Moreover,
it is not very surprising that the geodesic information is intertwisted with
the last passage times. In fact, it has been proved that the geodesic
${\mathcal{G}}_{\mathbf{p}}(\mathbf{q})$ becomes more rigid (or localized)
around its expected location when the last passage time
$L_{\mathbf{p}}(\mathbf{q})$ becomes very large [BG19, Liu21]. On the other
hand, it is not concentrated around any deterministic curve when the last
passage time becomes very small [BGS19].

The proof of Theorem 1.1 is given in Section 2.

###  1.2 The probability density function $p(s_{1},s_{2};m,n,M,N)$

We first introduce three notations. Suppose
$W=(w_{1},\cdots,w_{k})\in\mathbb{C}^{k}$ is a vector, we denote

$\Delta(W):=\prod_{1\leq i<j\leq k}(w_{j}-w_{i}).$ (1.4)

If $W=(w_{1},\cdots,w_{k})\in\mathbb{C}^{k}$ and
$W^{\prime}=(w^{\prime}_{1},\cdots,w^{\prime}_{k^{\prime}})\in\mathbb{C}^{k^{\prime}}$
are two vectors, we denote

$\Delta(W;W^{\prime}):=\prod_{i=1}^{k}\prod_{i^{\prime}=1}^{k^{\prime}}(w_{i}-w^{\prime}_{i^{\prime}}).$
(1.5)

Finally, if $f:\mathbb{C}\to\mathbb{C}$ is a function and
$W=(w_{1},\cdots,w_{k})\in\mathbb{C}^{k}$ is a vector, or
$W=\\{w_{1},\cdots,w_{k}\\}$ with each element $w_{i}\in\mathbb{C}$, we write

$f(W):=\prod_{i=1}^{k}f(w_{i}).$ (1.6)

Throughout this paper, we allow the empty product and define it to be $1$.

We need to introduce six contours. Suppose
$\Sigma_{\mathrm{L},\mathrm{out}},\Sigma_{\mathrm{L}}$ and
$\Sigma_{\mathrm{L},\mathrm{in}}$ are three nested contours, from outside to
inside, enclosing $-1$ but not $0$. Similarly,
$\Sigma_{\mathrm{R},\mathrm{out}},\Sigma_{\mathrm{R}}$ and
$\Sigma_{\mathrm{R},\mathrm{in}}$ are three nested contours, from outside to
inside, enclosing $0$ but not $-1$. We further assume that the contours
enclosing $-1$ are disjoint from those enclosing $0$. In other words, the two
outermost contours $\Sigma_{\mathrm{L},\mathrm{out}}$ and
$\Sigma_{\mathrm{R},\mathrm{out}}$ do not intersect. All the closed contours
throughout this paper are counterclockwise oriented. See Figure 1 for an
illustration of these contours.

$\mathbb{R}$$\mathrm{i}\mathbb{R}$$0$$-1$ Figure 1: Illustration of the
contours: The three contours around $-1$ from outside to inside are
$\Sigma_{\mathrm{L},\mathrm{out}},\Sigma_{\mathrm{L}}$ and
$\Sigma_{\mathrm{L},\mathrm{in}}$ respectively, and the three contours around
$0$ from outside to inside are
$\Sigma_{\mathrm{R},\mathrm{out}},\Sigma_{\mathrm{R}}$ and
$\Sigma_{\mathrm{R},\mathrm{in}}$ respectively.

We also introduce the notation of an integral along a small loop around a
point $z_{0}$ in the complex plane

$\oint_{z_{0}}f(z)\mathrm{d}z:=\int_{|z-z_{0}|=\epsilon}f(z)\mathrm{d}z,$

where $f(z)$ is an arbitrary meromorphic function defined in a neighborhood of
$z_{0}$ and $\epsilon$ is a sufficiently small constant.

The probability density function $p(s_{1},s_{2};m,n,M,N)$ is defined to be

$p(s_{1},s_{2};m,n,M,N):=\oint_{0}\frac{{\mathrm{d}}\mathrm{z}}{2\pi{\mathrm{i}}\mathrm{(1-z)^{2}}}\sum_{k_{1},k_{2}\geq
1}\frac{1}{(k_{1}!k_{2}!)^{2}}T_{k_{1},k_{2}}(\mathrm{z};s_{1},s_{2};m,n,M,N)$
(1.7)

with

$\begin{split}&T_{k_{1},k_{2}}(\mathrm{z};s_{1},s_{2};m,n,M,N)\\\
&:=\prod_{i_{1}=1}^{k_{1}}\left(\frac{1}{1-\mathrm{z}}\int_{\Sigma_{\mathrm{L},\mathrm{in}}}\frac{{\mathrm{d}}u^{(1)}_{i_{1}}}{2\pi{\mathrm{i}}}-\frac{\mathrm{z}}{1-\mathrm{z}}\int_{\Sigma_{\mathrm{L},\mathrm{out}}}\frac{{\mathrm{d}}u^{(1)}_{i_{1}}}{2\pi{\mathrm{i}}}\right)\left(\frac{1}{1-\mathrm{z}}\int_{\Sigma_{\mathrm{R},\mathrm{in}}}\frac{{\mathrm{d}}v^{(1)}_{i_{1}}}{2\pi{\mathrm{i}}}-\frac{\mathrm{z}}{1-\mathrm{z}}\int_{\Sigma_{\mathrm{R},\mathrm{out}}}\frac{{\mathrm{d}}v^{(1)}_{i_{1}}}{2\pi{\mathrm{i}}}\right)\\\
&\quad\cdot\prod_{i_{2}=1}^{k_{2}}\int_{\Sigma_{\mathrm{L}}}\frac{{\mathrm{d}}u^{(2)}_{i_{2}}}{2\pi{\mathrm{i}}}\int_{\Sigma_{\mathrm{R}}}\frac{{\mathrm{d}}v^{(2)}_{i_{2}}}{2\pi{\mathrm{i}}}\cdot\left(1-\mathrm{z}\right)^{k_{2}}\left(1-\frac{1}{\mathrm{z}}\right)^{k_{1}}\cdot\frac{f_{1}(U^{(1)};s_{1})f_{2}(U^{(2)};s_{2})}{f_{1}(V^{(1)};s_{1})f_{2}(V^{(2)};s_{2})}\cdot
H(U^{(1)},U^{(2)};V^{(1)},V^{(2)})\\\
&\quad\cdot\prod_{\ell=1}^{2}\frac{\left(\Delta(U^{(\ell)})\right)^{2}\left(\Delta(V^{(\ell)})\right)^{2}}{\left(\Delta(U^{(\ell)};V^{(\ell)})\right)^{2}}\cdot\frac{\Delta(U^{(1)};V^{(2)})\Delta(V^{(1)};U^{(2)})}{\Delta(U^{(1)};U^{(2)})\Delta(V^{(1)};V^{(2)})},\end{split}$
(1.8)

where the vectors $U^{(\ell)}=(u_{1}^{(\ell)},\cdots,u_{k_{\ell}}^{(\ell)})$
and $V^{(\ell)}=(v_{1}^{(\ell)},\cdots,v^{(\ell)}_{k_{\ell}})$ for
$\ell\in\\{1,2\\}$, the functions $f_{1},f_{2}$ are defined by

$\begin{split}f_{1}(w;s)&:=(w+1)^{-m}w^{n}e^{sw},\\\
f_{2}(w;s)&:=(w+1)^{-M+m}w^{N-n}e^{sw},\end{split}$ (1.9)

and the function $H$ is defined by

$\begin{split}&H(U^{(1)},U^{(2)};V^{(1)},V^{(2)})\\\
&:=\frac{1}{2}\left(\sum_{i_{1}=1}^{k_{1}}(u^{(1)}_{i_{1}}-v^{(1)}_{i_{1}})-\sum_{i_{2}=1}^{k_{2}}(u^{(2)}_{i_{2}}-v^{(2)}_{i_{2}})\right)^{2}\left(1+\prod_{i_{1}=1}^{k_{1}}\frac{v_{i_{1}}^{(1)}}{u_{i_{1}}^{(1)}}\prod_{i_{2}=1}^{k_{2}}\frac{u_{i_{2}}^{(2)}}{v_{i_{2}}^{(2)}}\right)\\\
&\quad+\frac{1}{2}\left(-\sum_{i_{1}=1}^{k_{1}}\left((u^{(1)}_{i_{1}})^{2}-(v^{(1)}_{i_{1}})^{2}\right)+\sum_{i_{2}=1}^{k_{2}}\left((u^{(2)}_{i_{2}})^{2}-(v^{(2)}_{i_{2}})^{2}\right)\right)\left(1-\prod_{i_{1}=1}^{k_{1}}\frac{v_{i_{1}}^{(1)}}{u_{i_{1}}^{(1)}}\prod_{i_{2}=1}^{k_{2}}\frac{u_{i_{2}}^{(2)}}{v_{i_{2}}^{(2)}}\right).\end{split}$
(1.10)

We remark that the formula (1.7) has a very similar structure with the two-
point distribution formula of TASEP in [Liu19] (with step initial condition),
except that we have different $\mathrm{z}$ factors in the integral, and that
we have an extra factor $H(U^{(1)},U^{(2)};V^{(1)},V^{(2)})$. See equations
(2.1) and (2.14) in [Liu19]. It is not hard to prove that $T_{k_{1},k_{2}}$
becomes zero when $k_{1}$ or $k_{2}$ becomes large, hence the formula (1.7)
only involves finite many nonzero terms in the summation and is well defined.
111In fact, we can view the integrand of (1.8) as a function of $V^{(1)}$ and
$V^{(2)}$, which equals to the product of the following three terms:
$\Delta(V^{(1)})\Delta(V^{(2)})$, a Cauchy-type factor
$\mathcal{C}(V^{(1)};V^{(2)})=\Delta(V^{(1)})\Delta(V^{(2)})/\Delta(V^{(1)};V^{(2)})$
(see the definition in (2.47)), and some function which is meromorphic for
each $v_{i_{\ell}}^{(\ell)}$ with a possible pole at $0$ but the degree of
this pole is at most $\max\\{n,N-n+1\\}$. Note that expanding the first term
$\Delta(V^{(1)})\Delta(V^{(2)})$ gives a sum of terms
$\prod_{1\leq\ell_{1}\leq
k_{1}}(v_{\sigma(\ell_{1})}^{(1)})^{k_{1}-\ell_{1}}\prod_{1\leq\ell_{2}\leq
k_{2}}(v_{\pi(\ell_{2})}^{(2)})^{k_{2}-\ell_{2}}$ over permutations $\sigma\in
S_{k_{1}}$ and $\pi\in S_{k_{2}}$, here $S_{k}$ denotes the permutation group
of $\\{1,2,\cdots,k\\}$. If $k_{1}$ is large enough (the case when $k_{2}$ is
large is similar), for example if $k_{1}>N$, the integrand is analytic for
$v_{\sigma(1)}^{(1)}$ at $0$ by checking the degrees. So when we integrate
$v_{\sigma(1)}^{(1)}$, the only possible nontrivial contribution is from the
residues $v_{\sigma(1)}^{(1)}=v_{j}^{(2)}$ if $v_{j}^{(2)}$ lies inside the
contour of $v_{\sigma(1)}^{(1)}$ due to the Cauchy-type factor. However, if we
further integrate $v_{j}^{(2)}$ we find each residue contribution is also zero
by checking the degree of $v_{j}^{(2)}$ which is
$k_{1}-1-n-(N-n+1)=k_{1}-N>0$. We remark that the proof does not rely on the
explicit formula of $H$ or the variable $\mathrm{z}$, and it is similar to the
argument for the two-point distribution formula of TASEP (see Remark 2.8 of
[Liu19]) where they do not have the factor $H$.

Finally, by exchanging the integral and summations, and using the identity
$\int_{0}^{\infty}\frac{f_{\ell}(U^{(\ell)};s_{\ell})}{f_{\ell}(V^{(\ell)};s_{\ell})}\mathrm{d}s_{\ell}=\frac{f_{\ell}(U^{(\ell)};0)}{f_{\ell}(V^{(\ell)};0)}\cdot\frac{1}{\sum_{i_{\ell}=1}^{k_{\ell}}(v_{i_{\ell}}^{(\ell)}-u_{i_{\ell}}^{(\ell)})}$
since $\mathrm{Re}(v_{i_{\ell}}^{(\ell)}-u_{i_{\ell}}^{(\ell)})<0$ due to the
locations of the contours, we obtain

$\begin{split}\mathbb{P}\left(\mathbf{r},\mathbf{r}^{\prime}\in{\mathcal{G}}_{\mathbf{p}}(\mathbf{q})\right)&=\int_{0}^{\infty}\int_{0}^{\infty}p(s_{1},s_{2};m,n,M,N)\mathrm{d}s_{1}\mathrm{d}s_{2}\\\
&=\oint_{0}\frac{{\mathrm{d}}\mathrm{z}}{2\pi{\mathrm{i}}\mathrm{(1-z)^{2}}}\sum_{k_{1},k_{2}\geq
1}\frac{1}{(k_{1}!k_{2}!)^{2}}\mathcal{T}_{k_{1},k_{2}}(\mathrm{z};m,n,M,N),\end{split}$
(1.11)

where

$\begin{split}&\mathcal{T}_{k_{1},k_{2}}(\mathrm{z};m,n,M,N)\\\
&:=\prod_{i_{1}=1}^{k_{1}}\left(\frac{1}{1-\mathrm{z}}\int_{\Sigma_{\mathrm{L},\mathrm{in}}}\frac{{\mathrm{d}}u^{(1)}_{i_{1}}}{2\pi{\mathrm{i}}}-\frac{\mathrm{z}}{1-\mathrm{z}}\int_{\Sigma_{\mathrm{L},\mathrm{out}}}\frac{{\mathrm{d}}u^{(1)}_{i_{1}}}{2\pi{\mathrm{i}}}\right)\left(\frac{1}{1-\mathrm{z}}\int_{\Sigma_{\mathrm{R},\mathrm{in}}}\frac{{\mathrm{d}}v^{(1)}_{i_{1}}}{2\pi{\mathrm{i}}}-\frac{\mathrm{z}}{1-\mathrm{z}}\int_{\Sigma_{\mathrm{R},\mathrm{out}}}\frac{{\mathrm{d}}v^{(1)}_{i_{1}}}{2\pi{\mathrm{i}}}\right)\\\
&\quad\cdot\prod_{i_{2}=1}^{k_{2}}\int_{\Sigma_{\mathrm{L}}}\frac{{\mathrm{d}}u^{(2)}_{i_{2}}}{2\pi{\mathrm{i}}}\int_{\Sigma_{\mathrm{R}}}\frac{{\mathrm{d}}v^{(2)}_{i_{2}}}{2\pi{\mathrm{i}}}\cdot\left(1-\mathrm{z}\right)^{k_{2}}\left(1-\frac{1}{\mathrm{z}}\right)^{k_{1}}\cdot\frac{f_{1}(U^{(1)};0)f_{2}(U^{(2)};0)}{f_{1}(V^{(1)};0)f_{2}(V^{(2)};0)}\cdot\frac{1}{\prod_{\ell=1}^{2}\sum_{i_{\ell}=1}^{k_{\ell}}(u_{i_{\ell}}^{(\ell)}-v_{i_{\ell}}^{(\ell)})}\\\
&\quad\cdot
H(U^{(1)},U^{(2)};V^{(1)},V^{(2)})\cdot\prod_{\ell=1}^{2}\frac{\left(\Delta(U^{(\ell)})\right)^{2}\left(\Delta(V^{(\ell)})\right)^{2}}{\left(\Delta(U^{(\ell)};V^{(\ell)})\right)^{2}}\cdot\frac{\Delta(U^{(1)};V^{(2)})\Delta(V^{(1)};U^{(2)})}{\Delta(U^{(1)};U^{(2)})\Delta(V^{(1)};V^{(2)})}.\end{split}$
(1.12)

###  1.3 Limiting joint distribution of geodesic location and last passage
times

For any two lattice points $\mathbf{p}=(\mathrm{p}_{1},\mathrm{p}_{2})$ and
$\mathbf{q}=(\mathrm{q}_{1},\mathrm{q}_{2})$ satisfying
$\mathrm{p}_{1}\leq\mathrm{q}_{1}$ and $\mathrm{p}_{2}\leq\mathrm{q}_{2}$, we
define

$d(\mathbf{p},\mathbf{q}):=\left(\sqrt{\mathrm{q}_{1}-\mathrm{p}_{1}}+\sqrt{\mathrm{q}_{2}-\mathrm{p}_{2}}\right)^{2}.$
(1.13)

We say a geodesic ${\mathcal{G}}_{\mathbf{p}}(\mathbf{q})$ exits a set $A$ at
a point $\mathbf{r}$, if and only if the geodesic intersects $A$ and
$\mathbf{r}$ is the last point of the intersection, i.e.,
$\mathbf{r}\in{\mathcal{G}}_{\mathbf{p}}(\mathbf{q})\cap A$ and
$\mathbf{r}_{+}\in{\mathcal{G}}_{\mathbf{p}}(\mathbf{q})\setminus A$.

######  Theorem 1.3.

Suppose $\alpha>0$, $\gamma\in(0,1)$ are fixed constants. Assume
$x_{1},x_{2},x^{\prime}_{1},x^{\prime}_{2}$ are four real numbers satisfying
$x_{1}>x^{\prime}_{1}$ and $x_{2}<x^{\prime}_{2}$. Let

$\begin{split}M&=[\alpha N],\\\ m&=[\gamma\alpha
N+x_{1}\alpha^{2/3}(1+\sqrt{\alpha})^{2/3}N^{2/3}],\\\ n&=[\gamma
N+x_{2}\alpha^{-1/3}(1+\sqrt{\alpha})^{2/3}N^{2/3}],\\\
m^{\prime}&=[\gamma\alpha
N+x^{\prime}_{1}\alpha^{2/3}(1+\sqrt{\alpha})^{2/3}N^{2/3}],\\\
n^{\prime}&=[\gamma
N+x^{\prime}_{2}\alpha^{-1/3}(1+\sqrt{\alpha})^{2/3}N^{2/3}],\end{split}$
(1.14)

where $[x]$ denotes the largest integer which is smaller than or equal to $x$.
Suppose $\pi$ is an up/left lattice path from $(m,n)$ to
$(m^{\prime},n^{\prime})$. Then

$\lim_{N\to\infty}\mathbb{P}\left(\begin{array}[]{l}{\mathcal{G}}_{(1,1)}{(M,N)}\text{
intersects $\pi$,}\\\ \mbox{and }L_{(1,1)}(\mathbf{p})\geq
d((1,1),\mathbf{p})+\mathrm{t}_{1}\cdot\alpha^{-1/6}(1+\sqrt{\alpha})^{4/3}N^{1/3},\\\
\mbox{and }L_{\mathbf{p}_{+}}(M,N)\geq
d(\mathbf{p}_{+},(M,N))+\mathrm{t}_{2}\cdot\alpha^{-1/6}(1+\sqrt{\alpha})^{4/3}N^{1/3},\\\
\mbox{where $\mathbf{p}$ denotes the exit point of
${\mathcal{G}}_{(1,1)}{(M,N)}$ on $\pi$}\\\ \end{array}\right)$ (1.15)

exists and is independent of the choice of $\pi$. The limit equals to

$\int_{x_{2}-x_{1}}^{x^{\prime}_{2}-x^{\prime}_{1}}\int_{\mathrm{t}_{1}}^{\infty}\int_{\mathrm{t}_{2}}^{\infty}\mathrm{p}(\mathrm{s}_{1},\mathrm{s}_{2},\mathrm{x};\gamma)\mathrm{d}\mathrm{s}_{2}\mathrm{d}\mathrm{s}_{1}\mathrm{d}\mathrm{x},$
(1.16)

where the joint probability density function
$\mathrm{p}(\mathrm{s}_{1},\mathrm{s}_{2},\mathrm{x};\gamma)$ is defined in
(1.22).

$(m,n)$$(m^{\prime},n^{\prime})$$(1,1)$$(M,N)$

Figure 2: The thick path denotes the geodesic ${\mathcal{G}}_{(1,1)}(M,N)$.
The spring-shaped lattice path denotes $\pi$. The star-shaped point is the
exit point of ${\mathcal{G}}_{(1,1)}(M,N)$ on $\pi$, and the square-shaped
point is the next point on ${\mathcal{G}}_{(1,1)}(M,N)$ after the exit point.

See Figure 2 for an illustration. The proof of Theorem 1.3 is provided in
Section 3.

We expect that the geodesic is around a straight line from $(1,1)$ to $(M,N)$.
The line is of slope $\alpha^{-1}\approx N/M$. Then $x_{2}-x_{1}$ and
$x^{\prime}_{2}-x^{\prime}_{1}$ can be viewed as (after appropriate scaling)
the shifts of moving $(m,n)$ and $(m^{\prime},n^{\prime})$ to the line.
Similarly, in the density function
$\mathrm{p}(\mathrm{s}_{1},\mathrm{s}_{2},\mathrm{x};\gamma)$, $\mathrm{x}$
can be viewed as the shift of moving the exit point $\mathbf{p}$ to the line.
See Figure 4 at the beginning of Section 3 for an illustration.

It might look surprising at a first glance that the limiting distribution is
independent of $\pi$, but only depends on the locations of the endpoints. Here
we provide an intuitive explanation. Suppose we have a different up/left
lattice path $\pi^{\prime}$ from $(m,n)$ to $(m^{\prime},n^{\prime})$. For any
point $\mathbf{q}\in\pi$, we can find a unique point
$\mathbf{q^{\prime}}\in\pi^{\prime}$ such that
$\mathbf{q}-\mathbf{q}^{\prime}\in\\{(\alpha y,y):y\in\mathbb{R}\\}$. Note
that the distance between $\mathbf{q}$ and $\mathbf{q}^{\prime}$ is at most of
order $O(N^{2/3})\ll o(N)$. By the uniform slow decorrelation of the directed
last passage percolation [CFP12, CLW16],
$N^{-1/3}({L_{(1,1)}(\mathbf{q})-d((1,1),\mathbf{q})})-N^{-1/3}({L_{(1,1)}(\mathbf{q}^{\prime})-d((1,1),\mathbf{q}^{\prime})})$
converges to $0$ in probability as $N\to\infty$. Moreover, with appropriate
scaling, the limiting process of the last passage times from $(1,1)$ (and from
$(M,N)$ similarly) to the points of $\pi$ has the same law as that to the
points of $\pi^{\prime}$. Therefore we expect the limit of (1.15) is
independent of $\pi$. This probabilistic argument is heuristic but it might be
possible to make it rigorous. In this paper, we will use an analytical way to
show this independence instead. See the argument after Proposition 3.1 in
Section 3.

Note that the geodesic ${\mathcal{G}}_{(1,1)}{(M,N)}$ intersects a rectangle
with vertices $(m,n),(m,n^{\prime}),(m^{\prime},n^{\prime})$ and
$(m^{\prime},n)$ if and only if ${\mathcal{G}}_{(1,1)}{(M,N)}$ intersects a
lattice path from $(m,n)$ to $(m^{\prime},n^{\prime})$. Thus by setting
$\mathrm{t}_{1},\mathrm{t}_{2}\to-\infty$ we immediately have

$\begin{split}&\lim_{N\to\infty}\mathbb{P}\left({\mathcal{G}}_{(1,1)}{(M,N)}\text{
intersects the rectangle with vertices
$(m,n),(m,n^{\prime}),(m^{\prime},n^{\prime})$ and $(m^{\prime},n)$}\right)\\\
&=\int_{x_{2}-x_{1}}^{x^{\prime}_{2}-x^{\prime}_{1}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\mathrm{p}(\mathrm{s}_{1},\mathrm{s}_{2},\mathrm{x};\gamma)\mathrm{d}\mathrm{s}_{2}\mathrm{d}\mathrm{s}_{1}\mathrm{d}\mathrm{x}.\end{split}$
(1.17)

Now we discuss an application of Theorem 1.3.

######  Corollary 1.4.

Let $\mathcal{A}^{(1)}$ and $\mathcal{A}^{(2)}$ be two independent Airy2
processes. Denote the parabolic Airy2 processes
$\mathcal{\hat{A}}^{(\ell)}(\mathrm{x})=\mathcal{A}^{(\ell)}(\mathrm{x})-\mathrm{x}^{2}$,
$\ell=1,2$. Suppose $\gamma\in(0,1)$ is a fixed constant. Denote

$\mathcal{T}=\mathrm{argmax}_{\mathrm{x}}\left(\gamma^{1/3}\mathcal{\hat{A}}^{(1)}\left(\frac{\mathrm{x}}{2\gamma^{2/3}}\right)+(1-\gamma)^{1/3}\mathcal{\hat{A}}^{(2)}\left(\frac{\mathrm{x}}{2(1-\gamma)^{2/3}}\right)\right).$

Then $\mathrm{p}(\mathrm{s}_{1},\mathrm{s}_{2},\mathrm{x};\gamma)$ is the
joint probability density function of
$\gamma^{1/3}\mathcal{A}^{(1)}\left(\frac{\mathcal{T}}{2\gamma^{2/3}}\right),(1-\gamma)^{1/3}\mathcal{A}^{(2)}\left(\frac{\mathcal{T}}{2(1-\gamma)^{2/3}}\right)$
and $\mathcal{T}$.

###### Proof.

Denote $\pi$ the line $\\{(x,y):x+y=2\gamma N\\}$. It is known [Joh03] that
the processes of the last passage times from $(1,1)$ (or $(N,N)$) to the
points on $\pi$ after appropriate scaling converge to two independent
parabolic Airy2 processes as $N\to\infty$. More explicitly, for any constant
$K$,

$\frac{L_{(1,1)}(\gamma N-2^{-1/3}\mathrm{x}N^{2/3},\gamma
N+2^{-1/3}\mathrm{x}N^{2/3})-4\gamma
N}{2^{4/3}N^{1/3}}\to\gamma^{1/3}\mathcal{\hat{A}}^{(1)}\left(\frac{\mathrm{x}}{2\gamma^{2/3}}\right),\quad|\mathrm{x}|\leq
K$ (1.18)

and

$\frac{L_{(\gamma N-2^{-1/3}\mathrm{x}N^{2/3},\gamma
N+2^{-1/3}\mathrm{x}N^{2/3})}(N,N)-4(1-\gamma)N}{2^{4/3}N^{1/3}}\to(1-\gamma)^{1/3}\mathcal{\hat{A}}^{(2)}\left(\frac{\mathrm{x}}{2(1-\gamma^{2/3})}\right),\quad|\mathrm{x}|\leq
K$ (1.19)

as $N\to\infty$. Both processes are tight in the space of continuous functions
on $[-K,K]$ (see [FO18, Theorem 2.3] for example). Note that the geodesic
passes through a point $\mathbf{q}$ on the line $\pi$ if and only if
$L_{(1,1)}(\mathbf{q})+L_{\mathbf{q}}(N,N)$ reaches the maximum. And the
probability that this intersection point $\mathbf{q}$ lies outside of
$\\{(\gamma N-2^{-1/3}\mathrm{x}N^{2/3},\gamma
N+2^{-1/3}\mathrm{x}N^{2/3}):|\mathrm{x}|\leq K\\}$ decays exponentially as
$N\to\infty$ and $K$ becomes large (see [BL16, Proposition 2.1] for example).
Also note that the argmax $\mathcal{T}$ is unique since it represents the
geodesic location in the limiting directed landscape and the geodesic is
unique (see [DV21]). Using the above facts, we conclude that the location of
the intersection of ${\mathcal{G}}_{(1,1)}(N,N)$ and $\pi$, the argmax of the
left hand side of (1.18)$+$(1.19), converges to $\mathcal{T}$. Now we apply
Theorem 1.3 with $\alpha=1$ and use the facts that

$d_{(1,1)}(\gamma N-2^{-1/3}\mathrm{x}N^{2/3},\gamma
N+2^{-1/3}\mathrm{x}N^{2/3})=4\gamma
N+\frac{\mathrm{x}^{2}}{2^{2/3}\gamma}N^{1/3}+o(1)$

and

$d_{(\gamma N-2^{-1/3}\mathrm{x}N^{2/3},\gamma
N+2^{-1/3}\mathrm{x}N^{2/3})}(N,N)=4(1-\gamma)N+\frac{\mathrm{x}^{2}}{2^{2/3}(1-\gamma)}N^{1/3}+o(1).$

Corollary 1.4 follows immediately. ∎

The explicit distribution of $\mathcal{T}$ was an interesting open problem in
the community before, see [DOV18, Problem 14.4(a)] for example. Our result
above resolves this problem. It is also possible to apply this result and the
formula of $\mathrm{p}(\mathrm{s}_{1},\mathrm{s}_{2},\mathrm{x};\gamma)$ to
obtain some properties of the directed landscape, the limiting four-parameter
random field of the directed last passage percolation. For example, in a
follow-up paper [Liu21] we proved that when the height of the directed
landscape at a point is sufficiently large, the geodesic to this point is
rigid and the location has a Gaussian distribution under appropriate scaling.

We remark that the density function
$\mathrm{p}(\mathrm{s}_{1},\mathrm{s}_{2},\mathrm{x};\gamma)$ can be related
to the well-known GUE Tracy-Widom distribution. Note that the max of
$\gamma^{1/3}\mathcal{\hat{A}}^{(1)}\left(\frac{\mathrm{x}}{2\gamma^{2/3}}\right)+(1-\gamma)^{1/3}\mathcal{\hat{A}}^{(2)}\left(\frac{\mathrm{x}}{2(1-\gamma)^{2/3}}\right)$
satisfies

$\mathbb{P}\left(\max_{\mathrm{x}\in\mathbb{R}}\left\\{\gamma^{1/3}\mathcal{\hat{A}}^{(1)}\left(\frac{\mathrm{x}}{2\gamma^{2/3}}\right)+(1-\gamma)^{1/3}\mathcal{\hat{A}}^{(2)}\left(\frac{\mathrm{x}}{2(1-\gamma)^{2/3}}\right)\right\\}\leq\mathrm{s}\right)=F_{GUE}(\mathrm{s}),$

where $F_{GUE}(\mathrm{s})$ is the GUE Tracy-Widom distribution. See [BL14,
BL13] for more details. By applying the Corollary 1.4 and noting
$\mathcal{\hat{A}}^{(\ell)}(\mathrm{x})=\mathcal{A}^{(\ell)}(\mathrm{x})-\mathrm{x}^{2}$,
we have

$\int_{\mathbb{R}}\mathrm{d}\mathrm{x}\iint_{\mathrm{s}_{1}+\mathrm{s_{2}}\leq\mathrm{s}}\mathrm{ds_{1}ds_{2}}\mathrm{p}\left(\mathrm{s}_{1}+\frac{\mathrm{x}^{2}}{4\gamma},\mathrm{s}_{2}+\frac{\mathrm{x}^{2}}{4(1-\gamma)},\mathrm{x};\gamma\right)=F_{GUE}(\mathrm{s}).$
(1.20)

One might be able to obtain the tail estimates for the geodesic using the
formula (1.3). After a preliminary calculation, we have the following
conjecture.

######  Conjecture 1.5.

Let $M,N$ and $m,n$ be numbers satisfying the scaling (3.1) in Theorem 1.3,
then

$\lim_{N\to\infty}\mathbb{P}\left({\mathcal{G}}_{(1,1)}(M,N)\text{ lies above
}(m,n)\right)=e^{-c\mathrm{x}^{3}+o(\mathrm{x})}\quad\text{with}\quad
c=\frac{1}{6(\gamma(1-\gamma))^{3/2}},$ (1.21)

when $\mathrm{x}=x_{2}-x_{1}$ becomes large.

It also might be possible to obtain a more accurate estimate from this
formula. We leave it as a future project.

###  1.4 The limiting density function
$\mathrm{p}(\mathrm{s}_{1},\mathrm{s}_{2},\mathrm{x};\gamma)$

The limiting density function
$\mathrm{p}(\mathrm{s}_{1},\mathrm{s}_{2},\mathrm{x};\gamma)$ has a similar
structure as the finite time probability density function
$p(s_{1},s_{2};m,n,M,N)$. Before we write down the formula, we introduce some
contours. Suppose $\Gamma_{\mathrm{L},\mathrm{in}},\Gamma_{\mathrm{L}}$ and
$\Gamma_{\mathrm{L},\mathrm{out}}$ are three disjoint contours on the left
half plane each of which starts from $e^{-2\pi\mathrm{i}/3}\infty$ and ends to
$e^{2\pi\mathrm{i}/3}\infty$. Here $\Gamma_{\mathrm{L},\mathrm{in}}$ is the
leftmost contour and $\Gamma_{\mathrm{L},\mathrm{out}}$ is the rightmost
contour. The index “$\mathrm{in}$” and “$\mathrm{out}$” refer to the relative
location compared with $-\infty$. Similarly, suppose
$\Gamma_{\mathrm{R},\mathrm{in}},\Gamma_{\mathrm{R}}$ and
$\Gamma_{\mathrm{R},\mathrm{out}}$ are three disjoint contours on the right
half plane each of which starts from $e^{-\pi\mathrm{i}/3}\infty$ and ends to
$e^{\pi\mathrm{i}/3}\infty$. Here the index “$\mathrm{in}$” and
“$\mathrm{out}$” refer to the relative location compared with $+\infty$, hence
$\Gamma_{\mathrm{R},\mathrm{in}}$ is the rightmost contour and
$\Gamma_{\mathrm{R},\mathrm{out}}$ is the leftmost contour. See Figure 3 for
an illustration of these contours.

$\mathbb{R}$$\mathrm{i}\mathbb{R}$$0$ Figure 3: Illustration of the contours:
The three contours in the left half plane from left to right are
$\Gamma_{\mathrm{L},\mathrm{in}},\Gamma_{\mathrm{L}}$ and
$\Gamma_{\mathrm{L},\mathrm{out}}$ respectively, and the three contours in the
right half plane from left to right are
$\Gamma_{\mathrm{R},\mathrm{out}},\Gamma_{\mathrm{R}}$ and
$\Gamma_{\mathrm{R},\mathrm{in}}$ respectively.

The probability density function
$\mathrm{p}(\mathrm{s}_{1},\mathrm{s}_{2},\mathrm{x};\gamma)$ is defined to be

$\mathrm{p}(\mathrm{s}_{1},\mathrm{s}_{2},\mathrm{x};\gamma):=\oint_{0}\frac{{\mathrm{d}}\mathrm{z}}{2\pi{\mathrm{i}}\mathrm{(1-z)^{2}}}\sum_{k_{1},k_{2}\geq
1}\frac{1}{(k_{1}!k_{2}!)^{2}}\mathrm{T}_{k_{1},k_{2}}(\mathrm{z};\mathrm{s}_{1},\mathrm{s}_{2},\mathrm{x};\gamma)$
(1.22)

with

$\begin{split}&\mathrm{T}_{k_{1},k_{2}}(\mathrm{z};\mathrm{s}_{1},\mathrm{s}_{2},\mathrm{x};\gamma)\\\
&:=\prod_{i_{1}=1}^{k_{1}}\left(\frac{1}{1-\mathrm{z}}\int_{\Gamma_{\mathrm{L},\mathrm{in}}}\frac{{\mathrm{d}}\xi^{(1)}_{i_{1}}}{2\pi{\mathrm{i}}}-\frac{\mathrm{z}}{1-\mathrm{z}}\int_{\Gamma_{\mathrm{L},\mathrm{out}}}\frac{{\mathrm{d}}\xi^{(1)}_{i_{1}}}{2\pi{\mathrm{i}}}\right)\left(\frac{1}{1-\mathrm{z}}\int_{\Gamma_{\mathrm{R},\mathrm{in}}}\frac{{\mathrm{d}}\eta^{(1)}_{i_{1}}}{2\pi{\mathrm{i}}}-\frac{\mathrm{z}}{1-\mathrm{z}}\int_{\Gamma_{\mathrm{R},\mathrm{out}}}\frac{{\mathrm{d}}\eta^{(1)}_{i_{1}}}{2\pi{\mathrm{i}}}\right)\\\
&\quad\cdot\prod_{i_{2}=1}^{k_{2}}\int_{\Gamma_{\mathrm{L}}}\frac{{\mathrm{d}}\xi^{(2)}_{i_{2}}}{2\pi{\mathrm{i}}}\int_{\Gamma_{\mathrm{R}}}\frac{{\mathrm{d}}\eta^{(2)}_{i_{2}}}{2\pi{\mathrm{i}}}\cdot\left(1-\mathrm{z}\right)^{k_{2}}\left(1-\frac{1}{\mathrm{z}}\right)^{k_{1}}\cdot\frac{\mathrm{f}_{1}(\boldsymbol{\xi}^{(1)};\mathrm{s}_{1})\mathrm{f}_{2}(\boldsymbol{\xi}^{(2)};\mathrm{s}_{2})}{\mathrm{f}_{1}(\boldsymbol{\eta}^{(1)};\mathrm{s}_{1})\mathrm{f}_{2}(\boldsymbol{\eta}^{(2)};\mathrm{s}_{2})}\cdot\mathrm{H}(\boldsymbol{\xi}^{(1)},\boldsymbol{\eta}^{(1)};\boldsymbol{\xi}^{(2)},\boldsymbol{\eta}^{(2)})\\\
&\quad\cdot\prod_{\ell=1}^{2}\frac{\left(\Delta(\boldsymbol{\xi}^{(\ell)})\right)^{2}\left(\Delta(\boldsymbol{\eta}^{(\ell)})\right)^{2}}{\left(\Delta(\boldsymbol{\xi}^{(\ell)};\boldsymbol{\eta}^{(\ell)})\right)^{2}}\cdot\frac{\Delta(\boldsymbol{\xi}^{(1)};\boldsymbol{\eta}^{(2)})\Delta(\boldsymbol{\eta}^{(1)};\boldsymbol{\xi}^{(2)})}{\Delta(\boldsymbol{\xi}^{(1)};\boldsymbol{\xi}^{(2)})\Delta(\boldsymbol{\eta}^{(1)};\boldsymbol{\eta}^{(2)})},\end{split}$
(1.23)

where the vectors
$\boldsymbol{\xi}^{(\ell)}=(\xi_{1}^{(\ell)},\cdots,\xi_{i_{\ell}}^{(\ell)})$
and
$\boldsymbol{\eta}^{(\ell)}=(\eta_{1}^{(\ell)},\cdots,\eta^{(\ell)}_{i_{\ell}})$
for $\ell\in\\{1,2\\}$, the functions $\mathrm{f}_{1},\mathrm{f}_{2}$ are
defined by

$\begin{split}\mathrm{f}_{1}(\zeta;\mathrm{s})&:=\exp\left(-\frac{\gamma}{3}\zeta^{3}-\frac{1}{2}\mathrm{x}\zeta^{2}+\left(\mathrm{s}-\frac{\mathrm{x}^{2}}{4\gamma}\right)\zeta\right),\\\
\mathrm{f}_{2}(\zeta;\mathrm{s})&:=\exp\left(-\frac{(1-\gamma)}{3}\zeta^{3}+\frac{1}{2}\mathrm{x}\zeta^{2}+\left(\mathrm{s}-\frac{\mathrm{x}^{2}}{4(1-\gamma)}\right)\zeta\right),\end{split}$
(1.24)

and the function $\mathrm{H}$ is defined by

$\mathrm{H}(\boldsymbol{\xi}^{(1)},\boldsymbol{\eta}^{(1)};\boldsymbol{\xi}^{(2)},\boldsymbol{\eta}^{(2)})=\frac{1}{12}\mathrm{S}_{1}^{4}+\frac{1}{4}\mathrm{S}_{2}^{2}-\frac{1}{3}\mathrm{S}_{1}\mathrm{S}_{3}$
(1.25)

with

$\mathrm{S}_{\ell}=\mathrm{S}_{\ell}(\boldsymbol{\xi}^{(1)},\boldsymbol{\eta}^{(1)};\boldsymbol{\xi}^{(2)},\boldsymbol{\eta}^{(2)})=\sum_{i_{1}=1}^{k_{1}}\left(\left(\xi_{i_{1}}^{(1)}\right)^{\ell}-\left(\eta_{i_{1}}^{(1)}\right)^{\ell}\right)-\sum_{i_{2}=1}^{k_{2}}\left(\left(\xi_{i_{2}}^{(2)}\right)^{\ell}-\left(\eta_{i_{2}}^{(2)}\right)^{\ell}\right).$
(1.26)

######  Remark 1.6.

It can be directly verified that $\mathrm{T}$ is symmetric on $\mathrm{x}$,
i.e., it satisfies
$\mathrm{T}_{k_{1},k_{2}}(\mathrm{z};\mathrm{s}_{1},\mathrm{s}_{2},\mathrm{x};\gamma)=\mathrm{T}_{k_{1},k_{2}}(\mathrm{z};\mathrm{s}_{1},\mathrm{s}_{2},-\mathrm{x};\gamma)$.
In fact, one can see it clearly by changing variables
$\xi_{i_{\ell}}^{(\ell)}=-\tilde{\eta}_{i_{\ell}}^{(\ell)}$ and
$\eta_{i_{\ell}}^{(\ell)}=-\tilde{\xi}_{i_{\ell}}^{(\ell)}$ for $1\leq
i_{\ell}\leq k_{\ell}$ and $\ell=1,2$.

One can prove that the summation is absolutely convergent in (1.22) due to the
super-exponential decay of $\mathrm{f}_{\ell}$ along the integral contours.
The proof is similar to that of Lemma 3.3 so we omit it.

## Acknowledgments

We would like to thank Jinho Baik, Ivan Corwin, Duncan Dauvergne, Patrik
Ferrari, Kurt Johansson, Daniel Remenik and Bálint Virág for the comments and
suggestions. The work was supported by the University of Kansas Start Up
Grant, the University of Kansas New Faculty General Research Fund, Simons
Collaboration Grant No. 637861, and NSF grant DMS-1953687.

##  2 Finite time formulas and proof of Theorem 1.1

###  2.1 Outline of the proof

Theorem 1.1 states two formulas for different locations of
$\mathbf{r}^{\prime}$. The equation (1.3) holds when
$\mathbf{r}^{\prime}=(m+1,n)$, i.e., when $\mathbf{r}^{\prime}$ is at the same
row as $\mathbf{r}$. The case when $\mathbf{r}^{\prime}$ is at the same column
as $\mathbf{r}$ follows by switching the rows and columns of the model. Thus
it is sufficient to show the equation (1.3) with
$\mathbf{r}^{\prime}=(m+1,n)$.

The proof involves a few computations and identities. We would like to split
the proof into three steps, each of which ends with an identity about the
probability density function $p(s_{1},s_{2};m,n,M,N)$. We will outline the
steps and state these main identities in this subsection and leave their
proofs in subsequent subsections.

In the first step, we obtain a formula for $p(s_{1},s_{2};m,n,M,N)$. The main
idea is to convert the desired probability to a sum of the product of two
transition probabilities, and evaluate the sum explicitly. There are two types
of transition probabilities for the exponential directed last passage
percolation. One is the transition probability by viewing its equivalent
model, the so-called TASEP, as a Markov process with respect to time [Sch97].
The second one is the transition probability by viewing the model as a Markov
chain along one dimension on the space [Joh10]. It turns out that only the
later one can be used to find an exact formula for $p(s_{1},s_{2};m,n,M,N)$.
If one uses the transition probabilities of TASEP instead, there will be an
$\mathcal{O}(1)$ error on the finite time formulas but the resulting limit
probability densities
$\mathrm{p}(\mathrm{s}_{1},\mathrm{s}_{2},\mathrm{x};\gamma)$ is the same. We
will consider this approach in a follow-up paper.

Using the transition probability formula of [Joh10] and an summation identity
for the product of two eigenfunctions, we obtain the following proposition.

######  Proposition 2.1.

We have the following formula for $p(s_{1},s_{2};m,n,M,N)$

$\begin{split}&p(s_{1},s_{2};m,n,M,N)\\\
&=\frac{(-1)^{N(N-1)/2}}{(N!)^{2}}\oint_{0}\frac{{\mathrm{d}}\mathrm{z}}{2\pi{\mathrm{i}}\mathrm{z}^{n}}\prod_{i_{1}=1}^{N}\int_{|w_{i_{1}}^{(1)}|=R_{1}}\frac{{\mathrm{d}}w_{i_{1}}^{(1)}}{2\pi{\mathrm{i}}}\prod_{i_{2}=1}^{N}\int_{|w_{i_{2}}^{(2)}|=R_{2}}\frac{{\mathrm{d}}w_{i_{2}}^{(2)}}{2\pi{\mathrm{i}}}\Delta\left(W^{(1)}\right)\Delta\left(W^{(2)}\right)\\\
&\quad\cdot\tilde{f}_{1}\left(W^{(1)}\right)\tilde{f}_{2}\left(W^{(2)}\right)\cdot\sum_{\ell_{1},\ell_{2}=1}^{N}(-1)^{\ell_{1}+\ell_{2}}\frac{\left(w_{\ell_{1}}^{(1)}\right)^{n}e^{s_{1}w_{\ell_{1}}^{(1)}}}{\left(w_{\ell_{2}}^{(2)}\right)^{n-1}e^{s_{1}w_{\ell_{2}}^{(2)}}}\det\left[C_{\mathrm{z}}\left(w_{i_{1}}^{(1)},w_{i_{2}}^{(2)}\right)+D_{\mathrm{z}}\left(w_{i_{1}}^{(1)},w_{i_{2}}^{(2)}\right)\right]_{\begin{subarray}{c}i_{1}\neq\ell_{1},\\\
i_{2}\neq\ell_{2}\end{subarray}}.\end{split}$ (2.1)

Here the radii of the contours satisfy $R_{1}>R_{2}>1$. The vectors $W^{(1)}$
and $W^{(2)}$ are defined by

$W^{(1)}=(w_{1}^{(1)},\cdots,w_{N}^{(1)}),\quad
W^{(2)}=(w_{1}^{(2)},\cdots,w_{N}^{(2)}).$

Recall our conventions $\Delta(W)$ and $f(W)$ as in (1.4) and (1.6). The
functions $\tilde{f}_{1}$ and $\tilde{f}_{2}$ are defined by

$\tilde{f}_{1}(w):=w^{-N}(w+1)^{-m},\quad\tilde{f}_{2}(w):=(w+1)^{-M+m}e^{(s_{1}+s_{2})w}.$
(2.2)

The functions $C$ and $D$ appearing in the determinant are defined by

$C_{\mathrm{z}}(w_{1},w_{2}):=\frac{\mathrm{z}}{w_{1}-w_{2}}\frac{w_{1}^{n-1}e^{s_{1}w_{1}}}{w_{2}^{n-1}e^{s_{1}w_{2}}}+\frac{1}{-w_{1}+w_{2}}\frac{w_{1}^{n+1}e^{s_{1}w_{1}}}{w_{2}^{n+1}e^{s_{1}w_{2}}},$
(2.3)

and

$D_{\mathrm{z}}(w_{1},w_{2}):=\frac{\mathrm{z}}{-w_{1}+w_{2}}\frac{w_{1}}{w_{2}}+\frac{1}{w_{1}-w_{2}}\frac{w_{1}^{N}e^{(s_{1}+s_{2})w_{1}}}{w_{2}^{N}e^{(s_{1}+s_{2})w_{2}}}.$
(2.4)

The proof of Proposition 2.1 is provided in the next subsection 2.2.

It seems that the formula (2.1) is not suitable for asymptotic analysis by the
following two reasons. The first reason is that this formula involves some
unneeded information. Note that the two terms in $D_{\mathrm{z}}(w_{1},w_{2})$
have factors $(w_{1}/w_{2})^{1}$ and $(w_{1}/w_{2})^{N}$ whose exponents $1$
and $N$ indeed represent the bounds of the possible locations of the geodesic.
However, we expect that the geodesic only fluctuates of order $N^{2/3}$ around
its expected location. In other words, changing the far endpoints $1$ and $N$
will not affect the asymptotics. Therefore, $D_{\mathrm{z}}(w_{1},w_{2})$
should not appear in the limit and we need to reformulate (2.1) and remove the
term $D_{\mathrm{z}}(w_{1},w_{2})$. The second reason is that the formula
(2.1) contains some determinants of size $\mathcal{O}(N)$, such as the
Vandermonde determinants $\Delta\left(W^{(1)}\right)$ and
$\Delta\left(W^{(2)}\right)$, and the determinant
$\det(C_{\mathrm{z}}+D_{\mathrm{z}})$. It is typically hard to find the
asymptotics of these determinants when the size $N\to\infty$. We will need to
rewrite it to a formula which is more suitable for asymptotic analysis.

In the second step, we take the term $D_{\mathrm{z}}(w_{1},w_{2})$ away at the
cost of changing the integral contours, and then evaluate the summation over
$\ell_{1},\ell_{2}$. We obtain

######  Proposition 2.2.

The equation (2.1) is equivalent to

$\begin{split}&p(s_{1},s_{2};m,n,M,N)=\frac{1}{(N!)^{2}}\oint_{0}\frac{(1-\mathrm{z})^{N-2}\mathrm{d}\mathrm{z}}{2\pi\mathrm{i}\mathrm{z}^{n}}\prod_{i_{1}=1}^{N}\left(\frac{-\mathrm{z}}{1-\mathrm{z}}\int_{\Sigma_{\mathrm{out}}}\frac{{\mathrm{d}}w_{i_{1}}^{(1)}}{2\pi{\mathrm{i}}}+\frac{1}{1-\mathrm{z}}\int_{\Sigma_{\mathrm{in}}}\frac{{\mathrm{d}}w_{i_{1}}^{(1)}}{2\pi{\mathrm{i}}}\right)\prod_{i_{2}=1}^{N}\int_{\Sigma}\frac{{\mathrm{d}}w_{i_{2}}^{(2)}}{2\pi{\mathrm{i}}}\\\
&\quad\hat{f}_{1}\left(W^{(1)}\right)\hat{f}_{2}\left(W^{(2)}\right)\frac{\left(\Delta\left(W^{(1)}\right)\right)^{2}\left(\Delta\left(W^{(2)}\right)\right)^{2}}{\Delta\left(W^{(2)};W^{(1)}\right)}\cdot\left(\hat{H}\left(W^{(1)};W^{(2)}\right)+\mathrm{z}\frac{\prod_{i_{2}=1}^{N}w_{i_{2}}^{(2)}}{\prod_{i_{1}=1}^{N}w_{i_{1}}^{(1)}}\hat{H}\left(W^{(2)};W^{(1)}\right)\right),\end{split}$
(2.5)

where the contours $\Sigma_{\mathrm{out}}$, $\Sigma$, and
$\Sigma_{\mathrm{in}}$ are three nested closed contours, from outside to
inside, all of which enclose both $0$ and $-1$. The vectors
$W^{(1)}:=(w_{1}^{(1)},\cdots,w_{N}^{(1)})$ and
$W^{(2)}:=(w_{1}^{(2)},\cdots,w_{N}^{(2)})$. The functions

$\hat{f}_{1}(w):=(w+1)^{-m}w^{-N+n}e^{s_{1}w},\quad\hat{f}_{2}(w):=(w+1)^{-M+m}w^{-n}e^{s_{2}w},$
(2.6)

and

$\hat{H}\left(W;W^{\prime}\right):=\frac{1}{2}\left(\sum_{i}w_{i}-\sum_{i^{\prime}}w^{\prime}_{i^{\prime}}\right)^{2}-\frac{1}{2}\left(\sum_{i}w_{i}^{2}-\sum_{i^{\prime}}\left(w^{\prime}_{i^{\prime}}\right)^{2}\right)$
(2.7)

for any vectors $W=(\cdots,w_{i},\cdots)$ and
$W^{\prime}=(\cdots,w^{\prime}_{i^{\prime}},\cdots)$ of finite sizes.

We remark that the idea of changing the integral contours is constructive. It
results in a compact formula which effectively removes the terms including the
information of the geodesic bounds. Formulas from similar summations (for
product of two eigenfunctions in TASEP as we did in the proof of Proposition
2.1) without including the information of the summation bounds were also
obtained in the periodic version of the directed last passage percolation
[BL18, BL19, BL21] and its large period limit [Liu19]. Heuristically, in the
periodic model it turned out that the upper bound (in the previous period)
cancels out the lower bound (in the current period) in the summation. While in
this paper, we construct contours $\Sigma_{\mathrm{in}}$ and
$\Sigma_{\mathrm{out}}$ which play similar roles as different periods:
integral of the terms involving the upper bound along one contour cancels that
involving the lower bound along the other contour.

The proof of Proposition 2.2 is provided in subsection 2.3.

In the last step, we rewrite the formula (2.5) in the form with a structure
similar to a Fredholm determinant expansion, which is the formula (1.7).

######  Proposition 2.3.

The formula (2.5) is equivalent to (1.7).

The proof of Proposition 2.3 is provided in subsection 2.4. It involves an
extension of a Cauchy-type summation formula in [Liu19]. We first convert the
integral into discrete summations over a so-called Bethe roots, then
reformulate the summation as a Fredholm-determinant-like expansion, and
finally convert the discrete summation back into integrals. It would be nice
to see a more direct proof for Proposition 2.3 but it seems quite complicated
considering the differences between the two formulas.

###  2.2 Proof of Proposition 2.1

As we mentioned in the previous subsection, we need a transition probability
formula by viewing the directed last passage percolation as a Markov chain.
Such a formula was obtained in [Joh10] for the geometric directed last passage
percolation, which is a discrete version of the model we are considering in
this paper. We will introduce the model below. Then we will show how to
compute an analogous probability for the geodesic in the geometric model, and
take the limit to get the results for exponential directed last passage
percolation.

The geometric last passage percolation model is defined as follows. We assign
to each site $\mathbf{p}\in\mathbb{Z}^{2}$ an i.i.d. geometric random
variables $\tilde{w}(\mathbf{p})$ with parameter $q\in(0,1)$

$\mathbb{P}\left(\tilde{w}(\mathbf{p})=i\right)=(1-q)q^{i},\qquad
i=0,1,2,\cdots$ (2.8)

for each integer site $\mathbf{p}$. Note that if we take $q=1-\epsilon$ and
let $\epsilon\to 0$, $\epsilon\tilde{w}(\mathbf{p})$ converges to an
exponential random variable.

Similar to (1.1), if a lattice point $\mathbf{q}$ lies in the upper right
direction of another lattice point $\mathbf{p}$, we define the last passage
time from $\mathbf{p}$ to $\mathbf{q}$ as

$G_{\mathbf{p}}(\mathbf{q}):=\max_{\pi}\sum_{\mathbf{r}\in\pi}\tilde{w}(\mathbf{r}),$
(2.9)

where the maximum is over all possible up/right lattice paths from
$\mathbf{p}$ to $\mathbf{q}$. We remark that the maximal path is not necessary
unique in this model. We call these maximal paths the geodesics from
$\mathbf{p}$ to $\mathbf{q}$.

We consider the following event

$A=\left\\{\begin{array}[]{l}G_{(1,1)}(m,n)+G_{(m+1,n)}(M,N)=G_{(1,1)}(M,N),\\\
G_{(1,1)}(m,n)=x,\\\ G_{(m+1,n)}(M,N)=y.\end{array}\right\\}.$ (2.10)

Here $x$ and $y$ are nonnegative integers. As we mentioned before, there may
be more than one geodesic. The event $A$ means that there is one geodesic that
passes through the two points $(m,n)$ and $(m+1,n)$, and these two points
split the last passage time $G_{(1,1)}(M,N)$ into two parts $G_{(1,1)}(m,n)=x$
and $G_{(m+1,n)}(M,N)=y$. Later we will show

######  Lemma 2.4.

We have

$\begin{split}&\mathbb{P}\left(A\right)\\\
&=(-1)^{N(N-1)/2}\frac{(1-q)^{MN}}{(N!)^{2}}\oint_{0}\frac{{\mathrm{d}}\mathrm{z}}{2\pi{\mathrm{i}}\mathrm{z}^{n}}\prod_{i_{1}=1}^{N}\int_{|w_{i_{1}}^{(1)}|=R_{1}}\frac{{\mathrm{d}}w_{i_{1}}^{(1)}}{2\pi{\mathrm{i}}}\prod_{i_{2}=1}^{N}\int_{|w_{i_{2}}^{(2)}|=R_{2}}\frac{{\mathrm{d}}w_{i_{2}}^{(2)}}{2\pi{\mathrm{i}}}\Delta\left(W^{(1)}\right)\Delta\left(W^{(2)}\right)\\\
&\quad\tilde{F}_{1}\left(W^{(1)}\right)\tilde{F}_{2}\left(W^{(2)}\right)\sum_{\ell_{1},\ell_{2}=1}^{N}(-1)^{\ell_{1}+\ell_{2}}\frac{\left(w_{\ell_{1}}^{(1)}+1\right)^{x}\left(w_{\ell_{1}}^{(1)}\right)^{n}}{\left(w_{\ell_{2}}^{(2)}+1\right)^{x+1}\left(w_{\ell_{2}}^{(2)}\right)^{n-1}}\det\left[\mathcal{C}_{\mathrm{z}}(w_{i_{1}}^{(1)},w_{i_{2}}^{(2)})+\mathcal{D}_{\mathrm{z}}(w_{i_{1}}^{(1)},w_{i_{2}}^{(2)})\right]_{\begin{subarray}{c}i_{1}\neq\ell_{1}\\\
i_{2}\neq\ell_{2}\end{subarray}},\end{split}$ (2.11)

where the radii $R_{1}$ and $R_{2}$ are distinct and both larger than $1$. The
functions $\tilde{F}_{1}$ and $\tilde{F}_{2}$ are defined by

$\tilde{F}_{1}(w):=(w+1)^{m-1}w^{-N}(w+1-q)^{-m},\quad\tilde{F}_{2}(w):=(w+1)^{x+y+M-m}(w+1-q)^{-M+m}.$
(2.12)

Recall the conventions $\tilde{F}_{\ell}\left(W^{(\ell)}\right)$ and
$\Delta\left(W^{(\ell)}\right)=\prod_{i>j}\left(w_{i}^{(\ell)}-w_{j}^{(\ell)}\right)=\det\left[\left(w_{i}^{(\ell)}\right)^{j-1}\right]_{i,j=1}^{N}$
as introduced in (1.4) and (1.6). Finally, the functions
$\mathcal{C}_{\mathrm{z}}$ and $\mathcal{D}_{\mathrm{z}}$ are given by

$\mathcal{C}_{\mathrm{z}}(w_{1},w_{2}):=\frac{\mathrm{z}}{w_{1}-w_{2}}\cdot\frac{w_{1}^{n-1}(w_{1}+1)^{x+1}}{w_{2}^{n-1}(w_{2}+1)^{x}}+\frac{1}{-w_{1}+w_{2}}\cdot\frac{w_{1}^{n+1}(w_{1}+1)^{x}}{w_{2}^{n+1}(w_{2}+1)^{x-1}}$
(2.13)

and

$\mathcal{D}_{\mathrm{z}}(w_{1},w_{2}):=\frac{\mathrm{z}}{-w_{1}+w_{2}}\cdot\frac{w_{1}(w_{2}+1)}{w_{2}}+\frac{1}{w_{1}-w_{2}}\cdot\frac{w_{1}^{N}(w_{1}+1)^{x+y+1}}{w_{2}^{N}(w_{2}+1)^{x+y}}.$
(2.14)

We postpone the proof of this lemma later in this subsection. Assuming Lemma
2.4, we are ready to prove Proposition 2.1. Below we write $A$ as $A(x,y)$ in
(2.10) to emphasize the parameters $x$ and $y$. As we mentioned before, if we
take $q=1-\epsilon$ and let $\epsilon\to 0$, the geometric directed last
passage percolation becomes an exponential one. More explicitly,
$\epsilon\tilde{w}(\mathbf{p})$ converges to an exponential random variable in
distribution as $\epsilon\to 0$. Moreover, for any fixed interval
$I_{1}=[t_{1},t_{1}+\epsilon_{1}]$ and $I_{2}=[t_{2},t_{2}+\epsilon_{2}]$, we
have

$\mathbb{P}\left(\bigcup_{s_{1}\in I_{1}}\bigcup_{s_{2}\in
I_{2}}A\left(\frac{s_{1}}{\epsilon},\frac{s_{2}}{\epsilon}\right)\right)=\mathbb{P}\left\\{\begin{array}[]{l}G_{(1,1)}(m,n)+G_{(m+1,n)}(M,N)=G_{(1,1)}(M,N),\\\
\epsilon G_{(1,1)}(m,n)\in I_{1},\\\ \epsilon G_{(m+1,n)}(M,N)\in
I_{2}.\end{array}\right\\}$ (2.15)

converges as $\epsilon\to 0$ to the analogous probability that in the
exponential directed last passage percolation, the geodesic
${\mathcal{G}}_{(1,1)}(M,N)$ passes through two points $(m,n)$ and $(m+1,n)$,
and the analogous last passage times satisfy $L_{(1,1)}(m,n)\in I_{1}$ and
$L_{(m+1,n)}(M,N)\in I_{2}$. In other words, the limit of (2.15) is the left
hand side of (1.3). We remark that although it is possible that there are more
than one geodesics in the geometric last passage percolation, after taking the
small $\epsilon$ limit the chance of getting more geodesics becomes zero.

Now we evaluate the limit of (2.15). The left hand side of (2.15) is

$\sum_{i\epsilon\in I_{1},j\epsilon\in
I_{2}}\mathbb{P}(A(i,j))=\int_{I_{1}}\int_{I_{2}}\frac{1}{\epsilon^{2}}\mathbb{P}\left(A\left(\frac{s_{1}}{\epsilon},\frac{s_{2}}{\epsilon}\right)\right)\mathrm{d}\mu_{\epsilon}(s_{2})\mathrm{d}\mu_{\epsilon}(s_{1}),$
(2.16)

where
$\mathrm{d}\mu_{\epsilon}(s)=\epsilon\delta_{\frac{s}{\epsilon}\in\mathbb{Z}}$.
We will prove

$\lim_{\epsilon\to
0}\frac{1}{\epsilon^{2}}\mathbb{P}(A(s_{1}/\epsilon,s_{2}/\epsilon))=p(s_{1},s_{2};m,n,M,N)$
(2.17)

uniformly on $I_{1}\times I_{2}$, with $p(s_{1},s_{2};m,n,M,N)$ defined in
(2.1). Then by using the continuity of the function $p(s_{1},s_{2};m,n,M,N)$
we immediately obtain that the limit of (2.15) equals to
$\int_{I_{1}}\int_{I_{2}}p(s_{1},s_{2};m,n,M,N)\mathrm{d}s_{2}\mathrm{d}s_{1}$.
Hence we prove Proposition 2.1.

Now we prove (2.17). We insert $q=1-\epsilon$, $x=s_{1}/\epsilon$, and
$y=s_{2}/\epsilon$ in (2.11). Note that all other parameters are fixed, and
$s_{1}\in I_{1},s_{2}\in I_{2}$ are nonnegative. We observe that the exponents
of $(w_{i_{1}}^{(1)}+1)$ for each $1\leq i_{1}\leq N$ in the integrand are at
least $m-1+\min\\{x,1\\}\geq m-1\geq 0$, and the exponents of
$(w_{i_{2}}^{(2)}+1)$ for each $1\leq i_{2}\leq N$ are at least
$x+y+M-m-\max\\{x+1,x+y\\}\geq M-m-1\geq 0$. Therefore the integrand is
analytic at $-1$ for each $w_{i_{1}}^{(1)}$ and $w_{i_{2}}^{(2)}$. There are
possible poles at $0$ and $q-1=-\epsilon$ both of which are close to $0$ as
$\epsilon\to 0$. We hence can deform the contours sufficiently close to the
origin. More precisely, we replace $R_{1}$ and $R_{2}$ by
$\epsilon\hat{R}_{1}$ and $\epsilon\hat{R}_{2}$ where
$\hat{R}_{1},\hat{R}_{2}$ are distinct constants and both larger than $1$, and
change variables $w_{i_{1}}^{(1)}=\epsilon\hat{w}_{i_{1}}^{(1)}$ and
$w_{i_{2}}^{(2)}=\epsilon\hat{w}_{i_{2}}^{(2)}$. Then

$\begin{split}&\Delta\left(W^{(1)}\right)\Delta\left(W^{(2)}\right)=\epsilon^{N(N-1)}\Delta\left(\hat{W}^{(1)}\right)\Delta\left(\hat{W}^{(2)}\right),\\\
&\tilde{F}_{1}(w)=\epsilon^{-N-m}(\hat{w}^{-N}(\hat{w}+1)^{-m}+\mathcal{O}(\epsilon))=\epsilon^{-N-m}(\tilde{f}_{1}(\hat{w})+\mathcal{O}(\epsilon)),\\\
&\tilde{F}_{2}(w)=\epsilon^{-M+m}((\hat{w}+1)^{-M+m}e^{(s_{1}+s_{2})\hat{w}}+\mathcal{O}(\epsilon))=\epsilon^{-M+m}(\tilde{f}_{2}(\hat{w})+\mathcal{O}(\epsilon)),\\\
&\left(w+1\right)^{x}w^{n}=\epsilon^{n}(\hat{w}^{n}e^{s_{1}\hat{w}}+\mathcal{O}(\epsilon)),\quad\left(w+1\right)^{x}w^{n-1}=\epsilon^{n-1}(\hat{w}^{n-1}e^{s_{1}\hat{w}}+\mathcal{O}(\epsilon)),\\\
&\mathcal{C}_{\mathrm{z}}(w_{1},w_{2})=\frac{\mathrm{z}}{\epsilon(\hat{w}_{1}-\hat{w}_{2})}\cdot\frac{\hat{w}_{1}^{n-1}e^{s_{1}\hat{w}_{1}}}{\hat{w}_{2}^{n-1}e^{s_{1}\hat{w}_{2}}}+\frac{1}{\epsilon(-\hat{w}_{1}+\hat{w}_{2})}\cdot\frac{\hat{w}_{1}^{n+1}e^{s_{1}\hat{w}_{1}}}{\hat{w}_{2}^{n+1}e^{s_{1}\hat{w}_{2}}}+\mathcal{O}(1)=\epsilon^{-1}(C_{\mathrm{z}}(\hat{w}_{1},\hat{w}_{2})+\mathcal{O}(\epsilon)),\\\
&\mathcal{D}_{\mathrm{z}}(w_{1},w_{2})=\frac{\mathrm{z}}{\epsilon(-\hat{w}_{1}+\hat{w}_{2})}\cdot\frac{\hat{w}_{1}}{\hat{w}_{2}}+\frac{1}{\epsilon(\hat{w}_{1}-\hat{w}_{2})}\cdot\frac{\hat{w}_{1}^{N}e^{(s_{1}+s_{2})\hat{w}_{1}}}{\hat{w}_{2}^{N}e^{(s_{1}+s_{2})\hat{w}_{2}}}+\mathcal{O}(1)=\epsilon^{-1}(D_{\mathrm{z}}(\hat{w}_{1},\hat{w}_{2})+\mathcal{O}(\epsilon)).\end{split}$
(2.18)

We remind that $\mathrm{d}w=\epsilon\mathrm{d}\hat{w}$. Therefore by inserting
these leading terms, we heuristically obtain that

$\lim_{\epsilon\to
0}\frac{1}{\epsilon^{2}}\mathbb{P}(A(s_{1}/\epsilon,s_{2}/\epsilon))=\text{
the right hand side of~{}\eqref{eq:finite_time_formula01}}.$ (2.19)

On the other hand, since all other parameters are fixed and the contours
$|\hat{w}_{i_{1}}^{(1)}|=\hat{R}_{1}$ and
$|\hat{w}_{i_{2}}^{(2)}|=\hat{R}_{2}$ are of finite size, if we insert the
above estimates (2.18) with the error terms into (2.11), all the terms
involving $\mathcal{O}(\epsilon)$ are uniformly bounded by $C\epsilon$ for
some constant $C$, and there are only finitely many such terms. Therefore the
equation (2.19) holds uniformly. This proves (2.17).

The remaining part of this subsection is to prove Lemma 2.4.

Denote

$G(m)=\left(G_{(1,1)}(m,1),\cdots,G_{(1,1)}(m,N)\right)$ (2.20)

the vector of the last passage times from the site $(1,1)$ to $(m,i)$, $1\leq
i\leq N$.

Our starting point is the following remarkable formula for the distribution of
$G(m)$.

######  Theorem 2.5.

[Joh10, Theorem 2.1] Suppose $X=(x_{1},\cdots,x_{N})\in\mathbb{Z}_{\geq
0}^{N}$ satisfies $x_{1}\leq x_{2}\leq\cdots\leq x_{N}$, then

$\mathbb{P}\left(G(m)=X\right)=\det\left[(1-q)^{m}\int_{|w|=R}(w+1)^{x_{j}+m-1}w^{j-i}\frac{{\mathrm{d}}w}{2\pi{\mathrm{i}}(w+1-q)^{m}}\right]_{i,j=1}^{N},$

where $R>1$ is any constant.

Note that the contour is of radius $R>1$ in the above theorem. This
restriction will be kept throughout the proof of Lemma 2.4 and finally lead to
the requirements $R_{1}>1$ and $R_{2}>1$.

The original theorem of [Joh10, Theorem 2.1] considered the finite-step
transition probabilities from any column to another, and for any
$x_{1}\leq\cdots\leq x_{N}$ without assuming $x_{1}\geq 0$. For our purpose we
only need this simpler version. The assumption that $x_{1}\geq 0$ comes from
the fact that all random variables $\tilde{w}(\mathbf{p})$ are nonnegative.
Moreover, we use the contour integral formula in the above determinant for
later computations. This formula is equivalent to the original version by
combining the equations (9) and (25) in [Joh10].

Denote

$\tilde{G}(m+1)=\left(G_{(m+1,1)}(M,N),\cdots,G_{(m+1,N)}(M,N)\right).$

Note that, by flipping the sites $(i,j)\to(-i,-j)$ and shifting the site
$(-M,-N)$ to $(1,1)$, $\tilde{G}(m+1)$ has the same distribution as

$\left(G_{(1,1)}(M-m,N),G_{(1,1)}(M-m,N-1),\cdots,G_{(1,1)}(M-m,1)\right).$

Therefore, by applying Theorem 2.5 we have

$\mathbb{P}\left(\tilde{G}(m+1)=Y\right)=\det\left[(1-q)^{M-m}\int_{|w|=R}(w+1)^{y_{N+1-j}+M-m-1}w^{j-i}\frac{{\mathrm{d}}w}{2\pi{\mathrm{i}}(w+1-q)^{M-m}}\right]_{i,j=1}^{N}$

for any $Y=(y_{1},\cdots,y_{N})\in\mathbb{Z}^{N}$ satisfying $y_{1}\geq
y_{2}\geq\cdots\geq y_{N}\geq 0$.

Note that $G(M)$ and $\tilde{G}(m+1)$ are independent since they are defined
on the lattices $\mathbb{Z}_{\leq m}\times\mathbb{Z}$ and $\mathbb{Z}_{\geq
m+1}\times\mathbb{Z}$ respectively. Also note the event $A$ is equivalent to
the event that $G_{(1,1)}(m,n)=x$, $G_{(m+1,n)(M,N)}=y$, and
$G_{(1,1)}(m,i)+G_{(m+1,i)}(M,N)\leq G_{(1,1)}(M,N)=x+y$ for all other $i$’s.
Thus by combining Theorem 2.5 and the above formula for $\tilde{G}(m+1)$, we
obtain

$\begin{split}\mathbb{P}(A)&=\sum\mathbb{P}\left(G(m)=X\right)\mathbb{P}\left(\tilde{G}(m+1)=Y\right)\\\
&=(1-q)^{MN}\sum\det\left[\int_{|w|=R}(w+1)^{x_{j}+m-1}w^{j-i}(w+1-q)^{-m}\frac{{\mathrm{d}}w}{2\pi{\mathrm{i}}}\right]_{i,j=1}^{N}\\\
&\qquad\qquad\qquad\quad\cdot\det\left[\int_{|w|=R}(w+1)^{y_{N+1-j}+M-m-1}w^{j-i}(w+1-q)^{-M+m}\frac{{\mathrm{d}}w}{2\pi{\mathrm{i}}}\right]_{i,j=1}^{N},\end{split}$
(2.21)

where the summation is running over all possible
$X=(x_{1},\cdots,x_{N})\in\mathbb{Z}^{N}$ and
$Y=(y_{1},\cdots,y_{N})\in\mathbb{Z}^{N}$ satisfying

$\begin{split}&0\leq x_{1}\leq\cdots\leq x_{N},\quad y_{1}\geq\cdots\geq
y_{N}\geq 0,\\\ &x_{i}+y_{i}\leq x+y,\quad\text{for all }i=1,\cdots,N,\\\
&\text{and }x_{n}=x,\ y_{n}=y.\end{split}$ (2.22)

We will consider the above summation in two steps. First, we fix $X$
satisfying $0\leq x_{1}\leq\cdots\leq x_{N}\leq x+y$ and $x_{n}=x$, and take
the sum over $Y$ satisfying (2.22). Note that only the last determinant in
(2.21) contains $Y$. We formulate such a summation in the following lemma.

######  Lemma 2.6.

Suppose $0\leq x_{1}\leq\cdots\leq x_{N}\leq x+y$, $x_{n}=x$ and $y_{n}=y$.
Assume that $F(w)$ is a function which is analytic on $|w|\geq R$ and
satisfies $|F(w)|\to 0$ uniformly as $|w|\to\infty$. Then

$\begin{split}&\sum_{y_{N}=0}^{x+y-x_{N}}\sum_{y_{N-1}=y_{N}}^{x+y-x_{N-1}}\cdots\sum_{y_{n+1}=y_{n+2}}^{x+y-x_{n+1}}\sum_{y_{n-1}=y}^{x+y-x_{n-1}}\cdots\sum_{y_{1}=y_{2}}^{x+y-x_{1}}\det\left[\int_{|w|=R}(w+1)^{y_{j}}w^{-j+i}F(w)\frac{{\mathrm{d}}w}{2\pi{\mathrm{i}}}\right]_{i,j=1}^{N}\\\
&=\det\left[\int_{|w|=R}(w+1)^{x+y-x_{j}+1_{j\neq n}}w^{-j+i-1_{j\neq
n}}F(w)\frac{{\mathrm{d}}w}{2\pi{\mathrm{i}}}\right]_{i,j=1}^{N}.\end{split}$

###### Proof of Lemma 2.6.

Due to the linearity of determinant, we can take the summation of the columns
inside the determinant. For each $j=1,\cdots,n-1,n+1,\cdots,N-1$, we have

$\begin{split}&\sum_{y_{j}=y_{j+1}}^{x+y-x_{j}}\int_{|w|=R}(w+1)^{y_{j}}w^{-j+i}F(w)\frac{{\mathrm{d}}w}{2\pi{\mathrm{i}}}\\\
&=\int_{|w|=R}(w+1)^{x+y-x_{j}+1}w^{-j-1+i}F(w)\frac{{\mathrm{d}}w}{2\pi{\mathrm{i}}}-\int_{|w|=R}(w+1)^{y_{j+1}}w^{-j-1+i}F(w)\frac{{\mathrm{d}}w}{2\pi{\mathrm{i}}},\end{split}$

where the second term matches the corresponding entry in the $(j+1)$-th
column. Therefore we can remove this term without changing the determinant.
For the summation over $y_{N}$, we have a similar identity where the second
term becomes

$\int_{|w|=R}w^{-N-1+i}F(w)\frac{{\mathrm{d}}w}{2\pi{\mathrm{i}}}=0$

by deforming the contour to infinity. We complete the proof by combining the
above summations. ∎

Now we come back to (2.21). We reorder the rows and columns in the second
determinant by replacing $i\to N+1-i$ and $j\to N+1-j$, and apply Lemma 2.6
with $F(w)=(w+1)^{M-m-1}(w+1-q)^{-M+m}$. We have

$\begin{split}\mathbb{P}(A)&=(1-q)^{MN}\sum\det\left[\int_{|w|=R}(w+1)^{x_{j}+m-1}w^{j-i}(w+1-q)^{-m}\frac{{\mathrm{d}}w}{2\pi{\mathrm{i}}}\right]_{i,j=1}^{N}\\\
&\qquad\qquad\qquad\quad\cdot\det\left[\int_{|w|=R}(w+1)^{-x_{j}+x+y+M-m-1_{j=N}}w^{-j-1+i+1_{j=n}}(w+1-q)^{-M+m}\frac{{\mathrm{d}}w}{2\pi{\mathrm{i}}}\right]_{i,j=1}^{N},\end{split}$
(2.23)

where the summation is over all $0\leq x_{1}\leq\cdots\leq x_{N}\leq x+y$ with
$x_{n}=x$.

In the next step, we consider the sum over $X$ in (2.23). We first apply the
following Cauchy-Binet/Andreief’s formula in (2.23)

$\det\left[\int
f_{i}(z)g_{j}(z)\mathrm{d}\mu(z)\right]_{i,j=1}^{N}=\frac{1}{N!}\int\cdots\int\det\left[f_{i}(z_{j})\right]_{i,j=1}^{N}\det\left[g_{i}(z_{j})\right]_{i,j=1}^{N}\mathrm{d}\mu(z_{1})\cdots\mathrm{d}\mu(z_{N}).$

We also relabel the variables to avoid confusions. Recall the functions
$\tilde{F}_{1}$ and $\tilde{F}_{2}$ defined in (2.12). We have

$\begin{split}&\det\left[\int_{|w|=R}(w+1)^{x_{j}+m-1}w^{j-i}(w+1-q)^{-m}\frac{{\mathrm{d}}w}{2\pi{\mathrm{i}}}\right]_{i,j=1}^{N}\\\
&=\frac{1}{N!}\prod_{i=1}^{N}\int_{|w_{i}^{(1)}|=R_{1}}\frac{{\mathrm{d}}w_{i}^{(1)}}{2\pi{\mathrm{i}}}\tilde{F}_{1}\left(W^{(1)}\right)\det\left[\left(w_{i}^{(1)}+1\right)^{x_{j}}\left(w_{i}^{(1)}\right)^{j}\right]_{i,j=1}^{N}\det\left[\left(w_{i}^{(1)}\right)^{N-j}\right]_{i,j=1}^{N}\end{split}$

and

$\begin{split}&\det\left[\int_{|w|=R}(w+1)^{-x_{j}+x+y+M-m-1_{j=N}}w^{-j-1+i+1_{j=n}}(w+1-q)^{-M+m}\frac{{\mathrm{d}}w}{2\pi{\mathrm{i}}}\right]_{i,j=1}^{N}\\\
&=\frac{1}{N!}\prod_{i=1}^{N}\int_{|w_{i}^{(2)}|=R_{2}}\frac{{\mathrm{d}}w_{i}^{(2)}}{2\pi{\mathrm{i}}}\tilde{F}_{2}\left(W^{(2)}\right)\det\left[\left(w_{i}^{(1)}+1\right)^{-x_{j}-1_{j=N}}\left(w_{i}^{(1)}\right)^{-j+1_{j=N}}\right]_{i,j=1}^{N}\det\left[\left(w_{i}^{(2)}\right)^{j-1}\right]_{i,j=1}^{N}.\end{split}$

Thus we write

$\begin{split}\mathbb{P}(A)&=(-1)^{N(N-1)/2}\frac{(1-q)^{MN}}{(N!)^{2}}\prod_{i=1}^{N}\int_{|w_{i}^{(1)}|=R_{1}}\frac{{\mathrm{d}}w_{i}^{(1)}}{2\pi{\mathrm{i}}}\int_{|w_{i}^{(2)}|=R_{2}}\frac{{\mathrm{d}}w_{i}^{(2)}}{2\pi{\mathrm{i}}}\tilde{F}_{1}\left(W^{(1)}\right)\tilde{F}_{2}\left(W^{(2)}\right)\\\
&\quad\cdot\Delta\left(W^{(1)}\right)\Delta\left(W^{(2)}\right)\cdot
S\left(W^{(1)},W^{(2)}\right),\end{split}$ (2.24)

where $W^{(1)}=\left(w_{1}^{(1)},\cdots,w_{N}^{(1)}\right)$,
$W^{(2)}=\left(w_{1}^{(2)},\cdots,w_{N}^{(2)}\right)$. We also rewrote
$\det\left[\left(w_{i}^{(\ell)}\right)^{j-1}\right]_{i,j=1}^{N}=\Delta\left(W^{(\ell)}\right)$
for both $\ell=1,2$. Finally, the function

$S\left(W^{(1)};W^{(2)}\right):=\sum_{X}\det\left[\left(w_{i}^{(1)}+1\right)^{x_{j}}\left(w_{i}^{(1)}\right)^{j}\right]_{i,j=1}^{N}\det\left[\left(w_{i}^{(2)}+1\right)^{-x_{j}-1_{j=n}}\left(w_{i}^{(2)}\right)^{-j+1_{j=n}}\right]_{i,j=1}^{N},$
(2.25)

where the summation is over all $0\leq x_{1}\leq\cdots\leq x_{N}\leq x+y$ with
fixed $x_{n}=x$.

Note that the summation over $X$ only appears in the function
$S\left(W^{(1)};W^{(2)}\right)$. Our goal in this step is to evaluate this
summation explicitly. We remark that this summation without the extra
$1_{j=n}$ in the exponents can be simplified to a compact formula if all the
coordinates of $W^{(\ell)}$ satisfy a so-called Bethe equation, see [BL19,
Proposition 5.2]. However, here we do not have the Bethe roots structure for
the coordinates and the resulting formulas are more complicated.

To proceed, we need an identity to expand the determinants in (2.25). By using
the Laplace expansion of the determinant along the $n$-th column and the
Cauchy-Binet formula for the cofactors, we have the identity

$\det\left[A_{i,j}\right]_{i,j=1}^{N}=\sum_{\ell}(-1)^{\ell+n}A_{\ell,n}\sum_{\begin{subarray}{c}I_{1}\cup
I_{2}=\\{1,\cdots,N\\}\setminus\\{\ell\\}\\\
|I_{1}|=n-1,|I_{2}|=N-n\end{subarray}}(-1)^{\\#(I,J)}\det\left[A_{i,j}\right]_{\begin{subarray}{c}i\in
I_{1}\\\ 1\leq j\leq
n-1\end{subarray}}\det\left[A_{i,j}\right]_{\begin{subarray}{c}i\in I_{2}\\\
n+1\leq j\leq N\end{subarray}},$

where

$\\#(I_{1},I_{2}):=\text{ the number of pairs $(i_{1},i_{2})\in I_{1}\times
I_{2}$ such that $i_{1}>i_{2}$}.$ (2.26)

We apply the above identity in (2.25) and change the order of summations. This
leads to

$\begin{split}&S\left(W^{(1)};W^{(2)}\right)\\\ &=\sum_{\ell_{1},\ell_{2}\geq
1}(-1)^{\ell_{1}+\ell_{2}}\frac{\left(w_{\ell_{1}}^{(1)}+1\right)^{x}\left(w_{\ell_{1}}^{(1)}\right)^{n}}{\left(w_{\ell_{2}}^{(2)}+1\right)^{x+1}\left(w_{\ell_{2}}^{(2)}\right)^{n-1}}\sum_{\begin{subarray}{c}I_{1}^{(1)}\cup
I_{2}^{(1)}=\\{1,\cdots,N\\}\setminus\\{\ell_{1}\\}\\\ I_{1}^{(2)}\cup
I_{2}^{(2)}=\\{1,\cdots,N\\}\setminus\\{\ell_{2}\\}\\\
|I_{1}^{(1)}|=|I_{1}^{(2)}|=n-1\\\
|I_{2}^{(1)}|=|I_{2}^{(2)}|=N-n\end{subarray}}(-1)^{\\#(I_{1}^{(1)},I_{2}^{(1)})+\\#(I_{1}^{(2)},I_{2}^{(2)})}\\\
&\quad\frac{\prod_{i\in I_{2}^{(1)}}\left(w_{i}^{(1)}\right)^{n}}{\prod_{i\in
I_{2}^{(2)}}\left(w_{i}^{(2)}\right)^{n}}\hat{S}_{0,x}\left(W^{(1)}_{I_{1}^{(1)}},W^{(2)}_{I_{1}^{(2)}}\right)\hat{S}_{x,x+y}\left(W^{(1)}_{I_{2}^{(1)}},W^{(2)}_{I_{2}^{(2)}}\right),\end{split}$
(2.27)

where for simplification we use the notation $W_{I}$ for the vector with
coordinates $w_{i}$’s satisfying $i\in I$. More explicitly,
$W_{I}=(w_{i_{1}},w_{i_{2}},\cdots,w_{i_{k}})$ for any
$I=(i_{1},\cdots,i_{k})$. The function

$\hat{S}_{a,b}\left(W,W^{\prime}\right):=\sum_{a\leq x_{1}\leq\cdots\leq
x_{k}\leq b}\det\left[(w_{i}+1)^{x_{j}}w_{i}^{j}\right]_{1\leq i,j\leq
k}\det\left[(w^{\prime}_{i}+1)^{-x_{j}}(w^{\prime}_{i})^{-j}\right]_{1\leq
i,j\leq k}$

for any $a\leq b$ and vectors $W$ and $W^{\prime}$ of the same size. Here $k$
is the size of $W$ and $W^{\prime}$, $w_{i}$’s and $w^{\prime}_{i}$’s are the
coordinates of $W$ and $W^{\prime}$ respectively.

We have the following identity to simplify
$\hat{S}_{a,b}\left(W,W^{\prime}\right)$.

######  Lemma 2.7.

[BL19] We have

$\hat{S}_{a,b}(W,W^{\prime})=\det\left[\frac{1}{-w_{i}+w^{\prime}_{i^{\prime}}}\cdot\frac{w_{i}(w_{i}+1)^{a}}{w^{\prime}_{i^{\prime}}(w^{\prime}_{i^{\prime}}+1)^{a-1}}+\frac{1}{w_{i}-w^{\prime}_{i^{\prime}}}\cdot\frac{w_{i}^{k}(w_{i}+1)^{b+1}}{(w^{\prime}_{i^{\prime}})^{k}(w^{\prime}_{i^{\prime}}+1)^{b}}\right]_{i,i^{\prime}=1}^{k}.$
(2.28)

###### Proof of Lemma 2.7.

The main technical part of the summation was included in [BL19]. Here we
simply mention how to arrive (2.28) using the known results in [BL19].

In [BL19], the authors introduced a similar sum $H_{a}(W;W^{\prime})$, where
$W$ and $W^{\prime}$ both are of size $N$. See equation (5.6) in [BL19]. It
reads

$H_{a}(W;W^{\prime})=\sum_{a-1=x_{1}\leq\cdots\leq x_{N}\leq
a+L-N-1}\det\left[(w^{\prime}_{i}+1)^{x_{j}}(w^{\prime}_{i})^{j}\right]_{1\leq
i,j\leq N}\det\left[(w_{i}+1)^{-x_{j}}w_{i}^{-j}\right]_{1\leq i,j\leq N}.$

Here we emphasize that $x_{1}=a-1$ is fixed in this summation. We also remark
that the original definition of $H_{a}(W;W^{\prime})$ assumes that the
coordinates of $W$ and $W^{\prime}$ are roots of the so-called Bethe equation,
but we will only cite the identities in §5.1-5.3 in [BL19] where the Bethe
roots properties are not used.

The equation (5.44) in [BL19] can be viewed as a difference of two terms. We
apply Lemma 5.9 of [BL19] for each term and rewrite the equation as

$\begin{split}H_{a}(W,W^{\prime})&=\det\left[\frac{1}{w_{i}-w^{\prime}_{i^{\prime}}}\cdot\frac{w^{\prime}_{i^{\prime}}(w^{\prime}_{i^{\prime}}+1)^{a-1}}{w_{i}(w_{i}+1)^{a-2}}+\frac{1}{-w_{i}+w^{\prime}_{i^{\prime}}}\cdot\frac{(w^{\prime}_{i^{\prime}})^{N}(w^{\prime}_{i^{\prime}}+1)^{a+L-N}}{w_{i}^{N}(w_{i}+1)^{a+L-N-1}}\right]_{i,i^{\prime}=1}^{N}\\\
&\quad-\det\left[\frac{1}{w_{i}-w^{\prime}_{i^{\prime}}}\cdot\frac{w^{\prime}_{i^{\prime}}(w^{\prime}_{i^{\prime}}+1)^{a}}{w_{i}(w_{i}+1)^{a-1}}+\frac{1}{-w_{i}+w^{\prime}_{i^{\prime}}}\cdot\frac{(w^{\prime}_{i^{\prime}})^{N}(w^{\prime}_{i^{\prime}}+1)^{a+L-N}}{w_{i}^{N}(w_{i}+1)^{a+L-N-1}}\right]_{i,i^{\prime}=1}^{N}.\end{split}$

We replace $a+L-N-1$ by $b$, and then $a-1$ by $a$, and get

$\begin{split}&\sum_{a=x_{1}\leq\cdots\leq x_{N}\leq
b}\det\left[(w^{\prime}_{i}+1)^{x_{j}}(w^{\prime}_{i})^{j}\right]_{1\leq
i,j\leq N}\det\left[(w_{i}+1)^{-x_{j}}w_{i}^{-j}\right]_{1\leq i,j\leq N}\\\
&=\det\left[\frac{1}{w_{i}-w^{\prime}_{i^{\prime}}}\cdot\frac{w^{\prime}_{i^{\prime}}(w^{\prime}_{i^{\prime}}+1)^{a}}{w_{i}(w_{i}+1)^{a-1}}+\frac{1}{-w_{i}+w^{\prime}_{i^{\prime}}}\cdot\frac{(w^{\prime}_{i^{\prime}})^{N}(w^{\prime}_{i^{\prime}}+1)^{b+1}}{w_{i}^{N}(w_{i}+1)^{b}}\right]_{i,i^{\prime}=1}^{N}\\\
&\quad-\det\left[\frac{1}{w_{i}-w^{\prime}_{i^{\prime}}}\cdot\frac{w^{\prime}_{i^{\prime}}(w^{\prime}_{i^{\prime}}+1)^{a+1}}{w_{i}(w_{i}+1)^{a}}+\frac{1}{-w_{i}+w^{\prime}_{i^{\prime}}}\cdot\frac{(w^{\prime}_{i^{\prime}})^{N}(w^{\prime}_{i^{\prime}}+1)^{b+1}}{w_{i}^{N}(w_{i}+1)^{b}}\right]_{i,i^{\prime}=1}^{N}.\end{split}$

So far $x_{1}=a$ is fixed. Now by summing the above identity for all $x_{1}$
from $a$ to $b$, we get

$\begin{split}\hat{S}_{a,b}(W^{\prime},W)&=\det\left[\frac{1}{w_{i}-w^{\prime}_{i^{\prime}}}\cdot\frac{w^{\prime}_{i^{\prime}}(w^{\prime}_{i^{\prime}}+1)^{a}}{w_{i}(w_{i}+1)^{a-1}}+\frac{1}{-w_{i}+w^{\prime}_{i^{\prime}}}\cdot\frac{(w^{\prime}_{i^{\prime}})^{N}(w^{\prime}_{i^{\prime}}+1)^{b+1}}{w_{i}^{N}(w_{i}+1)^{b}}\right]_{i,i^{\prime}=1}^{N}\\\
&\quad-\det\left[\frac{1}{w_{i}-w^{\prime}_{i^{\prime}}}\cdot\frac{w^{\prime}_{i^{\prime}}(w^{\prime}_{i^{\prime}}+1)^{b+1}}{w_{i}(w_{i}+1)^{b}}+\frac{1}{-w_{i}+w^{\prime}_{i^{\prime}}}\cdot\frac{(w^{\prime}_{i^{\prime}})^{N}(w^{\prime}_{i^{\prime}}+1)^{b+1}}{w_{i}^{N}(w_{i}+1)^{b}}\right]_{i,i^{\prime}=1}^{N}.\end{split}$

It is easy to see that the second determinant is zero. Therefore we obtain a
formula for $\hat{S}_{a,b}(W^{\prime},W)$ with a single determinant. By
switching $W$ and $W^{\prime}$, and replace the size $N$ by $k$, we obtain
(2.28). ∎

Now we apply Lemma 2.7 to (2.27). We also use the identity

$\begin{split}&\sum_{\begin{subarray}{c}I_{1}\cup J_{1}=\\{1,\cdots,L\\}\\\
J_{1}\cup J_{2}=\\{1,\cdots,L\\}\\\ |I_{1}|=|I_{2}|=n-1\\\
|J_{1}|=|J_{2}|=L-n+1\end{subarray}}(-1)^{\\#(I_{1},J_{1})+\\#(I_{2},J_{2})}\det\left[A(i_{1},j_{1})\right]_{\begin{subarray}{c}i_{1}\in
I_{1}\\\ j_{1}\in
J_{1}\end{subarray}}\det\left[B(i_{2},j_{2})\right]_{\begin{subarray}{c}i_{2}\in
I_{2}\\\ j_{2}\in J_{2}\end{subarray}}\\\
&=\oint_{0}\det\left[\mathrm{z}A(i,j)+B(i,j)\right]_{i,j=1}^{L}\frac{{\mathrm{d}}\mathrm{z}}{2\pi{\mathrm{i}}\mathrm{z}^{n}},\end{split}$

which follows from the multilinearity of the determinant on the rows and the
Cauchy-Binet formula. It can also be derived from Lemma 5.9 of [BL19]. Then we
arrive at

$\begin{split}&S\left(W^{(1)};W^{(2)}\right)=\sum_{\ell_{1},\ell_{2}\geq
1}(-1)^{\ell_{1}+\ell_{2}}\frac{\left(w_{\ell_{1}}^{(1)}+1\right)^{x}\left(w_{\ell_{1}}^{(1)}\right)^{n}}{\left(w_{\ell_{2}}^{(2)}+1\right)^{x+1}\left(w_{\ell_{2}}^{(2)}\right)^{n-1}}\\\
&\oint_{0}\frac{\mathrm{dz}}{2\pi\mathrm{i}\mathrm{z}^{n}}\det\left[\frac{\mathrm{z}}{-w_{i_{1}}^{(1)}+w_{i_{2}}^{(2)}}\cdot\frac{w_{i_{1}}^{(1)}\left(w_{i_{2}}^{(2)}+1\right)}{w_{i_{2}}^{(2)}}+\frac{\mathrm{z}}{w_{i_{1}}^{(1)}-w_{i_{2}}^{(2)}}\cdot\frac{\left(w_{i_{1}}^{(1)}\right)^{n-1}\left(w_{i_{1}}^{(1)}+1\right)^{x+1}}{\left(w_{i_{2}}^{(2)}\right)^{n-1}\left(w_{i_{2}}^{(2)}+1\right)^{x}}\right.\\\
&\qquad\quad\left.+\frac{1}{-w_{i_{1}}^{(1)}+w_{i_{2}}^{(2)}}\cdot\frac{\left(w_{i_{1}}^{(1)}\right)^{n+1}\left(w_{i_{1}}^{(1)}+1\right)^{x}}{\left(w_{i_{2}}^{(2)}\right)^{n+1}\left(w_{i_{2}}^{(2)}+1\right)^{x-1}}+\frac{1}{w_{i_{1}}^{(1)}-w_{i_{2}}^{(2)}}\cdot\frac{\left(w_{i_{1}}^{(1)}\right)^{N}\left(w_{i_{1}}^{(1)}+1\right)^{x+y+1}}{\left(w_{i_{2}}^{(2)}\right)^{N}\left(w_{i_{2}}^{(2)}+1\right)^{x+y}}\right]_{\begin{subarray}{c}i_{1}\neq\ell_{1}\\\
i_{2}\neq\ell_{2}\end{subarray}}.\end{split}$ (2.29)

By inserting this formula to (2.24), we obtain Lemma 2.4.

###  2.3 Proof of Proposition 2.2

In this subsection, we prove Proposition 2.2. There are two main steps in the
proof. In the first step we will deform the contours and get rid of term
$D_{\mathrm{z}}$ in (2.1). In the second step we will evaluate the summation
over $\ell_{1}$ and $\ell_{2}$.

####  2.3.1 Step 1: Deforming the contours

We first realize that

$C_{\mathrm{z}}(w_{1},w_{2})+D_{\mathrm{z}}(w_{1},w_{2})=\frac{\mathrm{z}}{w_{1}-w_{2}}\frac{w_{1}^{n-1}e^{s_{1}w_{1}}}{w_{2}^{n-1}e^{s_{1}w_{2}}}+\frac{1}{-w_{1}+w_{2}}\frac{w_{1}^{n+1}e^{s_{1}w_{1}}}{w_{2}^{n+1}e^{s_{1}w_{2}}}+\frac{\mathrm{z}}{-w_{1}+w_{2}}\frac{w_{1}}{w_{2}}+\frac{1}{w_{1}-w_{2}}\frac{w_{1}^{N}e^{(s_{1}+s_{2})w_{1}}}{w_{2}^{N}e^{(s_{1}+s_{2})w_{2}}}$

does not have a pole at $w_{1}=w_{2}$. Hence the integrand in (2.1) only has
poles at $0$ and $-1$. Furthermore, we can rewrite the $w_{i_{1}}^{(1)}$
integrals as

$\int_{|w_{i_{1}}^{(1)}|=R_{1}}\frac{{\mathrm{d}}w_{i_{1}}^{(1)}}{2\pi{\mathrm{i}}}=\frac{-\mathrm{z}}{1-\mathrm{z}}\int_{\Sigma_{\mathrm{out}}}\frac{{\mathrm{d}}w_{i_{1}}^{(1)}}{2\pi{\mathrm{i}}}+\frac{1}{1-\mathrm{z}}\int_{\Sigma_{\mathrm{in}}}\frac{{\mathrm{d}}w_{i_{1}}^{(1)}}{2\pi{\mathrm{i}}}$
(2.30)

and the $w_{i_{2}}^{(2)}$ integrals as

$\int_{|w_{i_{2}}^{(2)}|=R_{2}}\frac{{\mathrm{d}}w_{i_{2}}^{(2)}}{2\pi{\mathrm{i}}}=\int_{\Sigma}\frac{{\mathrm{d}}w_{i_{2}}^{(2)}}{2\pi{\mathrm{i}}}$
(2.31)

without changing the value of (2.1). After we change the order of summation
and integrals, we have

$\begin{split}&p(s_{1},s_{2};m,n,M,N)\\\
&=\frac{(-1)^{N(N-1)/2}}{(N!)^{2}}\oint_{0}\frac{{\mathrm{d}}\mathrm{z}}{2\pi{\mathrm{i}}\mathrm{z}^{n}}\sum_{\ell_{1},\ell_{2}=1}^{N}(-1)^{\ell_{1}+\ell_{2}}\left(\frac{-\mathrm{z}}{1-\mathrm{z}}\int_{\Sigma_{\mathrm{out}}}\frac{{\mathrm{d}}w_{\ell_{1}}^{(1)}}{2\pi{\mathrm{i}}}+\frac{1}{1-\mathrm{z}}\int_{\Sigma_{\mathrm{in}}}\frac{{\mathrm{d}}w_{\ell_{1}}^{(1)}}{2\pi{\mathrm{i}}}\right)\int_{\Sigma}\frac{{\mathrm{d}}w_{\ell_{2}}^{(2)}}{2\pi{\mathrm{i}}}\frac{\left(w_{\ell_{1}}^{(1)}\right)^{n}e^{s_{1}w_{\ell_{1}}^{(1)}}}{\left(w_{\ell_{2}}^{(2)}\right)^{n-1}e^{s_{1}w_{\ell_{2}}^{(2)}}}\\\
&\prod_{i_{1}\neq\ell_{1}}\left(\frac{-\mathrm{z}}{1-\mathrm{z}}\int_{\Sigma_{\mathrm{out}}}\frac{{\mathrm{d}}w_{i_{1}}^{(1)}}{2\pi{\mathrm{i}}}+\frac{1}{1-\mathrm{z}}\int_{\Sigma_{\mathrm{in}}}\frac{{\mathrm{d}}w_{i_{1}}^{(1)}}{2\pi{\mathrm{i}}}\right)\prod_{i_{2}\neq\ell_{2}}\int_{\Sigma}\frac{{\mathrm{d}}w_{i_{2}}^{(2)}}{2\pi{\mathrm{i}}}\\\
&\quad\cdot\Delta\left(W^{(1)}\right)\Delta\left(W^{(2)}\right)\tilde{f}_{1}\left(W^{(1)}\right)\tilde{f}_{2}\left(W^{(2)}\right)\det\left[C_{\mathrm{z}}\left(w_{i_{1}}^{(1)},w_{i_{2}}^{(2)}\right)+D_{\mathrm{z}}\left(w_{i_{1}}^{(1)},w_{i_{2}}^{(2)}\right)\right]_{\begin{subarray}{c}i_{1}\neq\ell_{1},\\\
i_{2}\neq\ell_{2}\end{subarray}}.\end{split}$ (2.32)

Although this rewriting seems simple, it turns out with these changes, we can
drop the term $D_{\mathrm{z}}$ in the integrand, following from the lemma
below.

######  Lemma 2.8.

Suppose $\Sigma$ and $\Sigma^{\prime}$ are contours on the complex plane,
$\mathrm{d}\mu(w)$ and $\mathrm{d}\mu^{\prime}(w^{\prime})$ are two measures
on these contours respectively. Suppose $C(w,w^{\prime})$ and
$D(w,w^{\prime})$ are two complex-valued functions on
$\Sigma\times\Sigma^{\prime}$, and
$B(w_{1},\cdots,w_{N};w_{1}^{\prime},\cdots,w^{\prime}_{N})$ is a complex-
valued function defined on $\Sigma^{N}\times(\Sigma^{\prime})^{N}$. Assume
that

$\int_{\Sigma^{N}}\int_{(\Sigma)^{N}}\left|B(w_{1},\cdots,w_{N};w_{1}^{\prime},\cdots,w^{\prime}_{N})\right|\cdot\prod_{i=1}^{N}\left(\left|C(w_{i},w^{\prime}_{\sigma(i)})\right|+\left|D(w_{i},w^{\prime}_{\sigma(i)})\right|\right)\prod_{i=1}^{N}\left|\mathrm{d}\mu(w_{i})\right|\prod_{i^{\prime}=1}^{N}\left|\mathrm{d}\mu^{\prime}(w^{\prime}_{i^{\prime}})\right|<\infty$

for each permutation $\sigma\in S_{N}$. We further assume that

$\int_{\Sigma}\int_{\Sigma^{\prime}}B(w_{1},\cdots,w_{N};w_{1}^{\prime},\cdots,w^{\prime}_{N})D(w_{i},w^{\prime}_{i^{\prime}})\mathrm{d}\mu(w_{i})\mathrm{d}\mu^{\prime}(w^{\prime}_{i^{\prime}})=0$
(2.33)

for any $1\leq i,i^{\prime}\leq N$, and any $w_{\ell}\in\Sigma,\ell\neq i$,
any $w^{\prime}_{\ell^{\prime}}\in\Sigma^{\prime},\ell^{\prime}\neq
i^{\prime}$. Then we have

$\begin{split}&\int_{\Sigma^{N}}\int_{(\Sigma)^{N}}B(w_{1},\cdots,w_{N};w_{1}^{\prime},\cdots,w^{\prime}_{N})\cdot\det\left[C(w_{i},w^{\prime}_{i^{\prime}})+D(w_{i},w^{\prime}_{i^{\prime}})\right]_{i,i^{\prime}=1}^{N}\prod_{i=1}^{N}\mathrm{d}\mu(w_{i})\prod_{i^{\prime}=1}^{N}\mathrm{d}\mu^{\prime}(w^{\prime}_{i})\\\
&=\int_{\Sigma^{N}}\int_{(\Sigma)^{N}}B(w_{1},\cdots,w_{N};w_{1}^{\prime},\cdots,w^{\prime}_{N})\cdot\det\left[C(w_{i},w^{\prime}_{i^{\prime}})\right]_{i,i^{\prime}=1}^{N}\prod_{i=1}^{N}\mathrm{d}\mu(w_{i})\prod_{i^{\prime}=1}^{N}\mathrm{d}\mu^{\prime}(w^{\prime}_{i}).\end{split}$
(2.34)

###### Proof of Lemma 2.8.

We expand the determinants on both sides of (2.34). It turns out all the terms
that appear on the left side but not the right side have some factor
$D(w_{i},w^{\prime}_{i^{\prime}})$ in the integrand and hence these terms are
zero by the assumption (2.33). This proves the identity. ∎

In order to apply Lemma 2.8 in (2.32), we need to check the assumptions. All
of these assumptions are obvious except for the assumption (2.33), which we
verify below. We need to show

$\begin{split}\left(\frac{-\mathrm{z}}{1-\mathrm{z}}\int_{\Sigma_{\mathrm{out}}}\frac{{\mathrm{d}}w_{i_{1}}^{(1)}}{2\pi{\mathrm{i}}}+\frac{1}{1-\mathrm{z}}\int_{\Sigma_{\mathrm{in}}}\frac{{\mathrm{d}}w_{i_{1}}^{(1)}}{2\pi{\mathrm{i}}}\right)\int_{\Sigma}\frac{{\mathrm{d}}w_{i_{2}}^{(2)}}{2\pi{\mathrm{i}}}\Delta\left(W^{(1)}\right)\Delta\left(W^{(2)}\right)\tilde{f}_{1}\left(W^{(1)}\right)\tilde{f}_{2}\left(W^{(2)}\right)D_{\mathrm{z}}\left(w_{i_{1}}^{(1)},w_{i_{2}}^{(2)}\right)\end{split}$

equals to zero. If we insert the formulas of $\tilde{f}_{1}$ and
$\tilde{f}_{2}$ (see (2.2)) and $D_{\mathrm{z}}$ (see (2.4)) in the above
formula, we only need to prove

$\begin{split}&\left(\frac{-\mathrm{z}}{1-\mathrm{z}}\int_{\Sigma_{\mathrm{out}}}\frac{{\mathrm{d}}w_{1}}{2\pi{\mathrm{i}}}+\frac{1}{1-\mathrm{z}}\int_{\Sigma_{\mathrm{in}}}\frac{{\mathrm{d}}w_{1}}{2\pi{\mathrm{i}}}\right)\int_{\Sigma}\frac{{\mathrm{d}}w_{2}}{2\pi{\mathrm{i}}}\\\
&\quad\tilde{G}_{1}(w_{1})\tilde{G}_{2}(w_{2})w_{1}^{-N}(w_{1}+1)^{-m}(w_{2}+1)^{-M+m}e^{(s_{1}+s_{2})w_{2}}\left(\frac{\mathrm{z}}{-w_{1}+w_{2}}\frac{w_{1}}{w_{2}}+\frac{1}{w_{1}-w_{2}}\frac{w_{1}^{N}e^{(s_{1}+s_{2})w_{1}}}{w_{2}^{N}e^{(s_{1}+s_{2})w_{2}}}\right)=0\end{split}$
(2.35)

for some polynomials $\tilde{G}_{1}$ and $\tilde{G}_{2}$ of degree $N-1$.
Using a simple residue computation, we have

$\begin{split}&\int_{\Sigma_{\mathrm{out}}}\frac{{\mathrm{d}}w_{1}}{2\pi{\mathrm{i}}}\int_{\Sigma}\frac{{\mathrm{d}}w_{2}}{2\pi{\mathrm{i}}}\tilde{G}_{1}(w_{1})\tilde{G}_{2}(w_{2})w_{1}^{-N}(w_{1}+1)^{-m}(w_{2}+1)^{-M+m}e^{(s_{1}+s_{2})w_{2}}\frac{\mathrm{z}}{-w_{1}+w_{2}}\frac{w_{1}}{w_{2}}=0,\\\
&\int_{\Sigma_{\mathrm{in}}}\frac{{\mathrm{d}}w_{1}}{2\pi{\mathrm{i}}}\int_{\Sigma}\frac{{\mathrm{d}}w_{2}}{2\pi{\mathrm{i}}}\tilde{G}_{1}(w_{1})\tilde{G}_{2}(w_{2})w_{1}^{-N}(w_{1}+1)^{-m}(w_{2}+1)^{-M+m}e^{(s_{1}+s_{2})w_{2}}\frac{\mathrm{z}}{-w_{1}+w_{2}}\frac{w_{1}}{w_{2}}\\\
&\quad=\mathrm{z}\int_{\Sigma}\frac{{\mathrm{d}}w}{2\pi{\mathrm{i}}}\tilde{G}_{1}(w)\tilde{G}_{2}(w)w^{-N}(w+1)^{-M}e^{(s_{1}+s_{2})w},\\\
&\int_{\Sigma_{\mathrm{out}}}\frac{{\mathrm{d}}w_{1}}{2\pi{\mathrm{i}}}\int_{\Sigma}\frac{{\mathrm{d}}w_{2}}{2\pi{\mathrm{i}}}\tilde{G}_{1}(w_{1})\tilde{G}_{2}(w_{2})w_{1}^{-N}(w_{1}+1)^{-m}(w_{2}+1)^{-M+m}e^{(s_{1}+s_{2})w_{2}}\frac{1}{w_{1}-w_{2}}\frac{w_{1}^{N}e^{(s_{1}+s_{2})w_{1}}}{w_{2}^{N}e^{(s_{1}+s_{2})w_{2}}}\\\
&\quad=\int_{\Sigma}\frac{{\mathrm{d}}w}{2\pi{\mathrm{i}}}\tilde{G}_{1}(w)\tilde{G}_{2}(w)w^{-N}(w+1)^{-M}e^{(s_{1}+s_{2})w},\\\
&\int_{\Sigma_{\mathrm{in}}}\frac{{\mathrm{d}}w_{1}}{2\pi{\mathrm{i}}}\int_{\Sigma}\frac{{\mathrm{d}}w_{2}}{2\pi{\mathrm{i}}}\tilde{G}_{1}(w_{1})\tilde{G}_{2}(w_{2})w_{1}^{-N}(w_{1}+1)^{-m}(w_{2}+1)^{-M+m}e^{(s_{1}+s_{2})w_{2}}\frac{1}{w_{1}-w_{2}}\frac{w_{1}^{N}e^{(s_{1}+s_{2})w_{1}}}{w_{2}^{N}e^{(s_{1}+s_{2})w_{2}}}=0.\end{split}$

(2.35) follows immediately.

Thus we can apply Lemma 2.8 in (2.32). After we remove the term
$D_{\mathrm{z}}$, we exchange the integral and summation again and obtain

$\begin{split}&p(s_{1},s_{2};m,n,M,N)\\\
&=\frac{(-1)^{N(N-1)/2}}{(N!)^{2}}\oint_{0}\frac{{\mathrm{d}}\mathrm{z}}{2\pi{\mathrm{i}}\mathrm{z}^{n}}\prod_{i_{1}=1}^{N}\left(\frac{-\mathrm{z}}{1-\mathrm{z}}\int_{\Sigma_{\mathrm{out}}}\frac{{\mathrm{d}}w_{i_{1}}^{(1)}}{2\pi{\mathrm{i}}}+\frac{1}{1-\mathrm{z}}\int_{\Sigma_{\mathrm{in}}}\frac{{\mathrm{d}}w_{i_{1}}^{(1)}}{2\pi{\mathrm{i}}}\right)\prod_{i_{2}=1}^{N}\int_{\Sigma}\frac{{\mathrm{d}}w_{i_{2}}^{(2)}}{2\pi{\mathrm{i}}}\\\
&\quad\Delta\left(W^{(1)}\right)\Delta\left(W^{(2)}\right)\tilde{f}_{1}\left(W^{(1)}\right)\tilde{f}_{2}\left(W^{(2)}\right)\sum_{\ell_{1},\ell_{2}=1}^{N}(-1)^{\ell_{1}+\ell_{2}}\frac{\left(w_{\ell_{1}}^{(1)}\right)^{n}e^{s_{1}w_{\ell_{1}}^{(1)}}}{\left(w_{\ell_{2}}^{(2)}\right)^{n-1}e^{s_{1}w_{\ell_{2}}^{(2)}}}\det\left[C_{\mathrm{z}}\left(w_{i_{1}}^{(1)},w_{i_{2}}^{(2)}\right)\right]_{\begin{subarray}{c}i_{1}\neq\ell_{1},\\\
i_{2}\neq\ell_{2}\end{subarray}}.\end{split}$ (2.36)

####  2.3.2 Step 2: Evaluating the summation

Recall the formula of $C_{\mathrm{z}}$ in (2.3). We can write

$C_{\mathrm{z}}(w_{1},w_{2})=\frac{w_{1}^{n}e^{s_{1}w_{1}}}{w_{2}^{n}e^{s_{1}w_{2}}}\cdot\left(\frac{\mathrm{z}}{w_{1}-w_{2}}\frac{w_{2}}{w_{1}}+\frac{1}{-w_{1}+w_{2}}\frac{w_{1}}{w_{2}}\right).$

We insert this formula in (2.36). Recall the formulas of
$\tilde{f}_{1},\tilde{f}_{2}$ in (2.2), and $\hat{f}_{1},\hat{f}_{2}$ in
(2.6). We arrive at

$\begin{split}&p(s_{1},s_{2};m,n,M,N)\\\
&=\frac{(-1)^{N(N-1)/2}}{(N!)^{2}}\oint_{0}\frac{{\mathrm{d}}\mathrm{z}}{2\pi{\mathrm{i}}\mathrm{z}^{n}}\prod_{i_{1}=1}^{N}\left(\frac{-\mathrm{z}}{1-\mathrm{z}}\int_{\Sigma_{\mathrm{out}}}\frac{{\mathrm{d}}w_{i_{1}}^{(1)}}{2\pi{\mathrm{i}}}+\frac{1}{1-\mathrm{z}}\int_{\Sigma_{\mathrm{in}}}\frac{{\mathrm{d}}w_{i_{1}}^{(1)}}{2\pi{\mathrm{i}}}\right)\prod_{i_{2}=1}^{N}\int_{\Sigma}\frac{{\mathrm{d}}w_{i_{2}}^{(2)}}{2\pi{\mathrm{i}}}\Delta\left(W^{(1)}\right)\Delta\left(W^{(2)}\right)\\\
&\quad\hat{f}_{1}\left(W^{(1)}\right)\hat{f}_{2}\left(W^{(2)}\right)\sum_{\ell_{1},\ell_{2}=1}^{N}(-1)^{\ell_{1}+\ell_{2}}w_{\ell_{2}}^{(2)}\det\left[\frac{\mathrm{z}}{w_{i_{1}}^{(1)}-w_{i_{2}}^{(2)}}\frac{w_{i_{2}}^{(2)}}{w_{i_{1}}^{(1)}}+\frac{1}{-w_{i_{1}}^{(1)}+w_{i_{2}}^{(2)}}\frac{w_{i_{1}}^{(1)}}{w_{i_{2}}^{(2)}}\right]_{\begin{subarray}{c}i_{1}\neq\ell_{1},\\\
i_{2}\neq\ell_{2}\end{subarray}}.\end{split}$ (2.37)

Compare the above formula with (2.5). Note the following Cauchy determinant
formula

$\frac{\Delta\left(W^{(1)}\right)\Delta\left(W^{(2)}\right)}{\Delta\left(W^{(2)};W^{(1)}\right)}=(-1)^{N(N-1)/2}\det\left[\frac{1}{w_{i_{2}}^{(2)}-w_{i_{1}}^{(1)}}\right]_{i_{1},i_{2}=1}^{N}.$

We see that (2.5) follows from (2.37) and Lemma 2.9 below. This completes the
proof of Proposition 2.2.

The remaining part of this subsection is the next lemma and its proof.

######  Lemma 2.9.

Suppose $X=(x_{1},\cdots,x_{N})$ and $Y=(y_{1},\cdots,y_{N})$ are two vectors
in $\mathbb{C}^{N}$ satisfying $x_{i}\neq y_{j}$ for all $1\leq i,j\leq N$.
Suppose $\mathrm{z}$ is an arbitrary complex number. Then we have the
following identity

$\begin{split}&\sum_{a,b=1}^{N}(-1)^{a+b}y_{b}\det\left[\frac{\mathrm{z}}{x_{i}-y_{j}}\frac{y_{j}}{x_{i}}+\frac{1}{-x_{i}+y_{j}}\frac{x_{i}}{y_{j}}\right]_{\begin{subarray}{c}i\neq
a\\\ j\neq b\end{subarray}}\\\
&=(1-\mathrm{z})^{N-2}\left(\hat{H}(X;Y)+\mathrm{z}\prod_{i=1}^{N}\frac{y_{i}}{x_{i}}\hat{H}(Y;X)\right)\det\left[\frac{1}{y_{j}-x_{i}}\right]_{i,j=1}^{N},\end{split}$
(2.38)

where $\hat{H}$ is defined in (2.7).

###### Proof of Lemma 2.9.

We first use the identity

$\frac{\mathrm{z}}{x-y}\frac{y}{x}+\frac{1}{-x+y}\frac{x}{y}=(1-\mathrm{z})\frac{x}{y}\cdot\left(\frac{1}{-x+y}-\frac{\mathrm{z}}{1-\mathrm{z}}\frac{1}{x}-\frac{\mathrm{z}}{1-\mathrm{z}}\frac{y}{x^{2}}\right)$

and write the left hand side of (2.38) as

$(1-\mathrm{z})^{N-1}\prod_{i=1}^{N}\frac{x_{i}}{y_{i}}\sum_{a,b=1}^{N}(-1)^{a+b}\frac{y_{b}^{2}}{x_{a}}\det\left[\frac{1}{-x_{i}+y_{j}}-\frac{\mathrm{z}}{1-\mathrm{z}}\frac{1}{x_{i}}-\frac{\mathrm{z}}{1-\mathrm{z}}\frac{y_{j}}{x_{i}^{2}}\right]_{\begin{subarray}{c}i\neq
a\\\ j\neq b\end{subarray}}.$ (2.39)

Thus the equation (2.38) is equivalent to, by setting
$u=-\mathrm{z}/(1-\mathrm{z})$,

$\begin{split}&\sum_{a,b=1}^{N}(-1)^{a+b}\frac{y_{b}^{2}}{x_{a}}\det\left[\frac{1}{-x_{i}+y_{j}}+u\frac{1}{x_{i}}+u\frac{y_{j}}{x_{i}^{2}}\right]_{\begin{subarray}{c}i\neq
a\\\ j\neq b\end{subarray}}\\\
&=-\left((u-1)\prod_{i=1}^{N}\frac{y_{i}}{x_{i}}\hat{H}(X;Y)+u\prod_{i=1}^{N}\frac{y_{i}^{2}}{x_{i}^{2}}\hat{H}(Y;X)\right)\cdot\det\left[\frac{1}{y_{j}-x_{i}}\right]_{i,j=1}^{N}.\end{split}$
(2.40)

The proof of (2.40) is tedious while the strategy is quite straightforward.
Below we will show the proof but omit some details which are direct to check.
We remark that the strategy was applied to a much simpler identity in [BL19,
Lemma 5.5], but this identity (2.40) is much more complicated.

Before we prove (2.40), we need to prepare some easier identities. We denote

$X(w):=\prod_{i=1}^{N}(w-x_{i}),\quad Y(w):=\prod_{i=1}^{N}(w-y_{i}),$

and introduce

$C_{p,q}=\sum_{a,b=1}^{n}x_{a}^{p}y_{b}^{q}\frac{Y(x_{a})X(y_{b})}{(x_{a}-y_{b})X^{\prime}(x_{a})Y^{\prime}(y_{b})},$

where $p,q$ are both integers. It is not hard to verify, by using the Cauchy
determinant formula, that

$C_{p,q}=\sum_{a,b=1}^{N}(-1)^{a+b}x_{a}^{p}y_{b}^{q}\det\left[\frac{1}{-x_{i}+y_{j}}\right]_{\begin{subarray}{c}i\neq
a\\\ j\neq
b\end{subarray}}/\det\left[\frac{1}{-x_{i}+y_{j}}\right]_{i,j=1}^{N}.$ (2.41)

One can evaluate $C_{p,q}$ by converting the sum as a residue computation of
an integral on the complex plane. As an illustration, we show how to obtain
$C_{-1,2}$, then we will list all the $C_{p,q}$ values we will use later
without providing proofs, see Table 1.

We consider a double integral

$\int_{|y|=R_{2}}\int_{|x|=R_{1}}\frac{y^{2}}{x}\frac{Y(x)X(y)}{(x-y)X(x)Y(y)}\frac{{\mathrm{d}}x}{2\pi{\mathrm{i}}}\frac{{\mathrm{d}}y}{2\pi{\mathrm{i}}},$

where $R_{1}>R_{2}>\max_{i}\\{|x_{i}|+|y_{i}|\\}$. Note that we can deform the
$x$-contour to infinity and the integral becomes zero. Hence the above double
integral is zero. On the other hand, we can change the order of integrals and
evaluate the $y$-integral first. It gives a sum over all roots of $Y(y)$:

$0=\int_{|x|=R_{1}}\sum_{b=1}^{N}\frac{y_{b}^{2}}{x}\frac{Y(x)X(y_{b})}{(x-y_{b})X(x)Y^{\prime}(y_{b})}\frac{{\mathrm{d}}x}{2\pi{\mathrm{i}}}.$

Then we exchange the summation and integral, and evaluate the $x$-integral by
computing the residues within the contour. Note that $x=y_{b}$ is not a pole.
We get

$0=C_{-1,2}-\frac{Y(0)}{X(0)}\sum_{b=1}^{N}y_{b}\frac{X(y_{b})}{Y^{\prime}(y_{b})}.$
(2.42)

We need to continue to evaluate the summation in (2.42). We have, by a residue
computation,

$\begin{split}\sum_{b=1}^{N}y_{b}\frac{X(y_{b})}{Y^{\prime}(y_{b})}&=\int_{|y|=R_{2}}y\frac{X(y)}{Y(y)}\frac{{\mathrm{d}}y}{2\pi{\mathrm{i}}}\\\
&=\int_{|y|=R_{2}}y\left(1+\frac{1}{y}\sum_{i=1}^{N}(y_{i}-x_{i})+\frac{1}{y^{2}}\hat{H}(X;Y)+O(y^{-3})\right)\frac{{\mathrm{d}}y}{2\pi{\mathrm{i}}}=\hat{H}(X;Y),\end{split}$

where we evaluated the integral by expanding the integrand for large $y$. Here
the function $\hat{H}$ is defined in (2.7).

By inserting the above formula to (2.42), we obtain

$C_{-1,2}=\prod_{i}\frac{y_{i}}{x_{i}}\hat{H}(X;Y).$

Using similar calculations, we can find all $C_{p,q}$ for small $p,q$ values.
In Table 1 we list some $C_{p,q}$ identities we will use in the proof of
(2.40). We remark that the proof of these identities are analogous to that of
$C_{-1,2}$ without adding extra difficulties.

Expression |  Value |  Expression |  Value
---|---|---|---
$C_{0,-1}$ |  $\displaystyle 1-\prod_{i}\frac{x_{i}}{y_{i}}$ |  $C_{-1,2}$ |  $\displaystyle\prod_{i}\frac{y_{i}}{x_{i}}\hat{H}(X;Y)$
$C_{-1,0}$ |  $\displaystyle-1+\prod_{i}\frac{y_{i}}{x_{i}}$ |  $C_{-1,1}-C_{0,0}$ |  $\displaystyle\left(1-\prod_{i}\frac{y_{i}}{x_{i}}\right)\sum_{i}(x_{i}-y_{i})$
$C_{1,0}$ |  $\displaystyle-\hat{H}(Y;X)$ |  $C_{0,2}-C_{1,1}$ |  $\displaystyle-\sum_{i}(x_{i}-y_{i})\hat{H}(X;Y)$
$C_{0,1}$ |  $\displaystyle\hat{H}(X;Y)$ |  $C_{-2,1}$ |  $\displaystyle-1+\prod_{i}\frac{y_{i}}{x_{i}}\left(1-\sum_{i}\left(\frac{1}{x_{i}}-\frac{1}{y_{i}}\right)\sum_{i}\left(x_{i}-y_{i}\right)\right)$
Table 1: Values of some $C_{p,q}$ expressions.

We need to evaluate

$\det\left[\frac{1}{-x_{i}+y_{j}}+u\frac{1}{x_{i}}\right]_{i,j=1}^{N}=\det\left[\frac{1}{-x_{i}+y_{j}}\right]_{i,j=1}^{N}+u\sum_{a,b=1}^{N}(-1)^{a+b}\frac{1}{x_{a}}\det\left[\frac{1}{-x_{i}+y_{j}}\right]_{\begin{subarray}{c}i\neq
a\\\ j\neq b\end{subarray}}.$

By applying (2.41) and finding the $C_{-1,0}$ value in Table 1, we get

$\begin{split}\det\left[\frac{1}{-x_{i}+y_{j}}+u\frac{1}{x_{i}}\right]_{i,j=1}^{N}&=\det\left[\frac{1}{-x_{i}+y_{j}}\right]_{i,j=1}^{N}\left(1+uC_{-1,0}\right)\\\
&=\det\left[\frac{1}{-x_{i}+y_{j}}\right]_{i,j=1}^{N}\left(1+u\left(-1+\prod_{i=1}^{N}\frac{y_{i}}{x_{i}}\right)\right).\end{split}$
(2.43)

Then we evaluate

$\det\left[\frac{1}{-x_{i}+y_{j}}+u\frac{1}{x_{i}}+u\frac{y_{j}}{x_{i}^{2}}\right]_{i,j=1}^{N}=\det\left[\frac{1}{-x_{i}+y_{j}}+u\frac{1}{x_{i}}\right]_{i,j=1}^{N}+u\sum_{a,b=1}^{N}(-1)^{a+b}\frac{y_{b}}{x_{a}^{2}}\det\left[\frac{1}{-x_{i}+y_{j}}+u\frac{1}{x_{i}}\right]_{\begin{subarray}{c}i\neq
a\\\ j\neq b\end{subarray}}.$

We insert (2.43) in the above equation and obtain

$\begin{split}&\det\left[\frac{1}{-x_{i}+y_{j}}+u\frac{1}{x_{i}}+u\frac{y_{j}}{x_{i}^{2}}\right]_{i,j=1}^{N}\\\
&=\det\left[\frac{1}{-x_{i}+y_{j}}\right]_{i,j=1}^{N}\left(1-u+u\prod_{i=1}^{N}\frac{y_{i}}{x_{i}}\right)+u(1-u)\sum_{a,b=1}^{N}(-1)^{a+b}\frac{y_{b}}{x_{a}^{2}}\det\left[\frac{1}{-x_{i}+y_{j}}\right]_{\begin{subarray}{c}i\neq
a\\\ j\neq b\end{subarray}}\\\
&\quad+u^{2}\prod_{i=1}^{N}\frac{y_{i}}{x_{i}}\sum_{a,b=1}^{N}(-1)^{a+b}\frac{1}{x_{a}}\det\left[\frac{1}{-x_{i}+y_{j}}\right]_{\begin{subarray}{c}i\neq
a\\\ j\neq b\end{subarray}}\\\
&=\det\left[\frac{1}{-x_{i}+y_{j}}\right]_{i,j=1}^{N}\left(1-u+u\prod_{i=1}^{N}\frac{y_{i}}{x_{i}}+u(1-u)C_{-2,1}+u^{2}\prod_{i=1}^{N}\frac{y_{i}}{x_{i}}C_{-1,0}\right).\end{split}$
(2.44)

By inserting the values of $C_{-2,1}$ and $C_{-1,0}$ and simplifying the
expression, we obtain

$\begin{split}&\det\left[\frac{1}{-x_{i}+y_{j}}+u\frac{1}{x_{i}}+u\frac{y_{j}}{x_{i}^{2}}\right]_{i,j=1}^{N}=\det\left[\frac{1}{-x_{i}+y_{j}}\right]_{i,j=1}^{N}\\\
&\quad\cdot\left[1-2u\left(1-\prod_{i=1}^{N}\frac{y_{i}}{x_{i}}\right)-(u-u^{2})\prod_{i=1}^{N}\frac{y_{i}}{x_{i}}\sum_{i=1}^{N}\left(\frac{1}{x_{i}}-\frac{1}{y_{i}}\right)\sum_{i=1}^{N}(x_{i}-y_{i})+u^{2}\left(\prod_{i=1}^{N}\frac{y_{i}}{x_{i}}-1\right)^{2}\right].\end{split}$
(2.45)

Finally we are ready to prove (2.40). Inserting (2.45), we can write

$\begin{split}&\sum_{a,b=1}^{N}(-1)^{a+b}\frac{y_{b}^{2}}{x_{a}}\det\left[\frac{1}{-x_{i}+y_{j}}+u\frac{1}{x_{i}}+u\frac{y_{j}}{x_{i}^{2}}\right]_{\begin{subarray}{c}i\neq
a\\\ j\neq b\end{subarray}}\\\
&=\sum_{a,b=1}^{N}(-1)^{a+b}\frac{y_{b}^{2}}{x_{a}}\det\left[\frac{1}{-x_{i}+y_{j}}\right]_{\begin{subarray}{c}i\neq
a\\\ j\neq
b\end{subarray}}\cdot\left[1-2u\left(1-\frac{x_{a}}{y_{b}}\prod_{i=1}^{N}\frac{y_{i}}{x_{i}}\right)+u^{2}\left(\frac{x_{a}}{y_{b}}\prod_{i=1}^{N}\frac{y_{i}}{x_{i}}-1\right)^{2}\right.\\\
&\quad\left.-(u-u^{2})\frac{x_{a}}{y_{b}}\prod_{i=1}^{N}\frac{y_{i}}{x_{i}}\left(-\frac{1}{x_{a}}+\frac{1}{y_{b}}+\sum_{i=1}^{N}\left(\frac{1}{x_{i}}-\frac{1}{y_{i}}\right)\right)\left(-x_{a}+y_{b}+\sum_{i=1}^{N}(x_{i}-y_{i})\right)\right].\end{split}$

We apply (2.41) and rewrite the above equation as

$\begin{split}&\sum_{a,b=1}^{N}(-1)^{a+b}\frac{y_{b}^{2}}{x_{a}}\det\left[\frac{1}{-x_{i}+y_{j}}+u\frac{1}{x_{i}}+u\frac{y_{j}}{x_{i}^{2}}\right]_{\begin{subarray}{c}i\neq
a\\\ j\neq
b\end{subarray}}/\det\left[\frac{1}{-x_{i}+y_{j}}+u\frac{1}{x_{i}}+u\frac{y_{j}}{x_{i}^{2}}\right]_{i,j=1}^{N}\\\
&=\left((1-u)^{2}+u(1-u)\prod_{i}\frac{y_{i}}{x_{i}}\right)C_{-1,2}+u\prod_{i}\frac{y_{i}}{x_{i}}\left(1-u+u\cdot\prod_{i}\frac{y_{i}}{x_{i}}\right)C_{1,0}\\\
&\quad-u(1-u)\prod_{i}\frac{y_{i}}{x_{i}}\left(\sum_{i}\frac{1}{x_{i}}-\sum_{i}\frac{1}{y_{i}}\right)\left(\sum_{i}x_{i}-\sum_{i}y_{i}\right)C_{0,1}\\\
&\quad-u(1-u)\prod_{i}\frac{y_{i}}{x_{i}}\left(\sum_{i}x_{i}-\sum_{i}y_{i}\right)C_{0,0}+u(1-u)\prod_{i}\frac{y_{i}}{x_{i}}\left(\sum_{i}x_{i}-\sum_{i}y_{i}\right)C_{-1,1}\\\
&\quad-u(1-u)\prod_{i}\frac{y_{i}}{x_{i}}\left(\sum_{i}\frac{1}{x_{i}}-\sum_{i}\frac{1}{y_{i}}\right)C_{0,2}+u(1-u)\prod_{i}\frac{y_{i}}{x_{i}}\left(\sum_{i}\frac{1}{x_{i}}-\sum_{i}\frac{1}{y_{i}}\right)C_{1,1}.\end{split}$
(2.46)

By checking the values of Table 1, and noting that
$\left(\sum_{i}(x_{i}-y_{i})\right)^{2}=\hat{H}(X;Y)+\hat{H}(Y;X)$, we can
simplify the above expression. It turns out, after a careful but
straightforward calculation, the $u^{2}$ term vanishes, and the remaining
terms match the right hand side of (2.40). We hence complete the proof. ∎

###  2.4 Proof of Proposition 2.3

In this subsection, we prove Proposition 2.3. Note that the equation (2.5)
involves a Cauchy determinant factor

$\frac{\Delta\left(W^{(1)}\right)\Delta\left(W^{(2)}\right)}{\Delta\left(W^{(2)};W^{(1)}\right)}=(-1)^{N(N-1)/2}\det\left[\frac{1}{w_{i_{2}}^{(2)}-w_{i_{1}}^{(1)}}\right]_{i_{1},i_{2}=1}^{N},$

which is of size N, while the formula (1.7) is analogous to a Fredholm
determinant expansion. So Proposition 2.3 can be interpreted as an identity
between a Cauchy determinant of large size and a Fredholm-determinant-like
expansion. Our strategy contains three steps. First, we rewrite the formula
(2.5) to a summation on discrete spaces with summand having similar Cauchy
determinant structures. This rewriting involves a generalized version of an
identity in [Liu19]. In the second step, we reformulate the summation to a
Fredholm-determinant-like expansion on the same discrete space. We remark that
similar calculation were considered in [BL18, BL19] but our summand is more
involved. Finally, we verify that the expansion indeed matches (1.7) using the
identity obtained in the first step.

Below we will first introduce a generalized version of an identity in [Liu19],
the Proposition 4.3 of [Liu19]. Then we prove Proposition 2.3 using the above
strategy.

####  2.4.1 A Cauchy-type summation identity

We introduce a few concepts before we state the results. We will mainly follow
[Liu19, Section 4] but add a small generalization.

Suppose $W=(w_{1},\cdots,w_{n})\in\mathbb{C}^{n}$ and
$W^{\prime}=(w^{\prime}_{1},\cdots,w^{\prime}_{n^{\prime}})\in\mathbb{C}^{n^{\prime}}$
are two vectors without overlapping coordinates, i.e., they satisfy $w_{i}\neq
w^{\prime}_{i^{\prime}}$ for all $i,i^{\prime}$. We define

$\mathcal{C}(W;W^{\prime})=\frac{\Delta(W)\Delta(W^{\prime})}{\Delta(W;W^{\prime})}$
(2.47)

and call it a _Cauchy-type factor_. Note that when $n=n^{\prime}$,
$\mathcal{C}(W;W^{\prime})$ equals to a Cauchy determinant
$\det\left[1/(w_{i}-w^{\prime}_{i^{\prime}})\right]_{i,i^{\prime}=1}^{n}$
multiplied by a sign factor $(-1)^{n(n-1)/2}$. We remark that we allow empty
product and view it as $1$ in the above definition. For example, when
$n^{\prime}=0$, we have $\mathcal{C}(W;W^{\prime})=\Delta(W)$.

Similar as in (2.27), we use the convention that
$W_{I}=(w_{i_{1}},\cdots,w_{i_{k}})$ for any index set
$I=\\{i_{1},\cdots,i_{k}\\}$ where $1\leq i_{1}<\cdots<i_{k}\leq n$. In other
words, $W_{I}$ is the vector formed by the coordinates with indices in $I$.

We denote

$\mathbb{D}(r):=\\{z:|z|<r\\},\text{ and }\mathbb{D}_{0}(r)=\\{z:0<|z|<r\\}.$

And we omit $r$ when $r=1$, i.e., $\mathbb{D}=\mathbb{D}(1)$ and
$\mathbb{D}_{0}=\mathbb{D}_{0}(1)$.

Suppose $q(w)$ is a function which is analytic in a certain bounded region
$\mathcal{D}$. Denote

$\mathcal{R}_{z}=\left\\{w\in\mathcal{D}:q(w)=z\right\\}.$ (2.48)

Assume that $\mathcal{R}_{0}\neq\emptyset$. In other words, there is at least
one root of $q(w)$ within $\mathcal{D}$. We also assume that
$\mathrm{r_{max}}$ is a positive constant such that
$\cup_{z\in\mathbb{D}(\mathrm{r_{max}})}\mathcal{R}_{z}=\\{w\in\mathcal{D}:|q(w)|\leq\mathrm{r_{max}}\\}$
lies within a compact subset of $\mathcal{D}$, and
$\\{w\in\mathcal{D}:|q(w)|=r\\}$ for all $0<|r|<\mathrm{r_{max}}$ consists of
$|\mathcal{R}_{0}|$ non-intersecting simply connected contours around the
points in $\mathcal{R}_{0}$. It is easy to see that with these assumptions
$q^{\prime}(w)\neq 0$ for all
$w\in\\{w\in\mathcal{D}:|q(w)|<\mathrm{r_{max}}\\}$. We remark that in the
original setting of [Liu19], they assumed $\mathcal{R}_{0}=\\{0\\}$ or
$\\{-1\\}$. Here we drop this assumption.

We will consider a Cauchy-type summation, which involves an expression

$\mathcal{H}\left(W^{(1)},\cdots,W^{(\ell)};z_{0},\cdots,z_{\ell-1}\right):=\left[\prod_{k=1}^{\ell-1}\mathcal{C}\left(W_{I^{(k)}}^{(k)};W_{J^{(k+1)}}^{(k+1)}\right)\right]\cdot\mathcal{A}\left(W^{(1)},\cdots,W^{(\ell)};z_{0},\cdots,z_{\ell-1}\right),$
(2.49)

where
$W^{(k)}=\left(w^{(k)}_{1},\cdots,w_{n_{k}}^{(k)}\right)\in\mathbb{C}^{n_{k}}$,
$1\leq k\leq\ell$, such that $W^{(k)}$ and $W^{(k+1)}$ do not have overlapping
coordinates for $1\leq k\leq\ell-1$. $I^{(k)}$ and $J^{(k)}$ are arbitrary
subsets of $\\{1,\cdots,n_{k}\\}$ for $1\leq k\leq\ell-1$ and $2\leq
k\leq\ell$ respectively. The function $\mathcal{A}$ is analytic for all
$w_{j_{k}}^{(k)}\in\mathcal{D}\setminus\mathcal{R}_{0}$, $1\leq j_{k}\leq
n_{k},1\leq k\leq\ell$, and for all
$(z_{0},\cdots,z_{\ell-1})\in\mathbb{D}(\mathrm{r_{max}})\times\mathbb{D}^{\ell-1}$.
Hence $\mathcal{H}$ is also analytic on
$(\mathcal{D}\setminus\mathcal{R}_{0})^{n_{1}+\cdots+n_{\ell}}\times\mathbb{D}(\mathrm{r_{max}})\times\mathbb{D}^{\ell-1}$,
except for having possible poles at $w_{i_{k}}^{(k)}=w_{i_{k+1}}^{(k+1)}$ for
some $i_{k}\in I^{(k)}$ and $i_{k+1}\in I^{(k+1)}$, which comes from the
Cauchy-type factors. We remark that the function $\mathcal{H}$ also depends on
the index sets $I^{(k)},J^{(k+1)}$, $1\leq k\leq\ell-1$.

Now we introduce the summation. We consider

$\mathcal{G}(z_{0},\cdots,z_{\ell-1})=\sum_{W^{(1)}\in\mathcal{R}_{\hat{z}_{1}}^{n_{1}}}\cdots\sum_{W^{(\ell)}\in\mathcal{R}_{\hat{z}_{\ell}}^{n_{\ell}}}\left[\prod_{k=1}^{\ell}J(W^{(k)})\right]\mathcal{H}\left(W^{(1)},\cdots,W^{(\ell)};z_{0},\cdots,z_{\ell-1}\right)$
(2.50)

for
$(z_{0},\cdots,z_{\ell})\in\mathbb{D}_{0}(\mathrm{r_{max}})\times\mathbb{D}_{0}^{\ell-1}$,
where the function

$J(w):=\frac{q(w)}{q^{\prime}(w)}.$ (2.51)

Recall our convention $J(W^{(k)})=\prod_{a=1}^{n_{k}}J(w_{a}^{(k)})$. The
variables $\hat{z}_{k}$’s are defined by

$\hat{z}_{k}=z_{0}z_{1}\cdots z_{k-1},\quad k=1,\cdots,\ell.$ (2.52)

Note the identity

$\sum_{w\in\mathcal{R}_{z}}f(w)H(w)=\left(\int_{|q(w)|=C_{1}}-\int_{|q(w)|=C_{2}}\right)\frac{f(w)q(w)}{q(w)-z}\frac{{\mathrm{d}}w}{2\pi{\mathrm{i}}},$
(2.53)

where $C_{1}$ and $C_{2}$ are two positive constants satisfying
$C_{2}<|z|<C_{1}$ such that the function $f(w)$ is analytic in
$\\{w:C_{2}<|q(w)|<C_{1}\\}$. The right hand side is analytic as a function of
$z$ within $C_{2}<|z|<|C_{1}|$. This identity implies that
$\sum_{w\in\mathcal{R}_{z}}f(w)H(w)$ is also analytic as a function of $z$
within $C_{2}<|z|<|C_{1}|$. Using this fact we obtain that
$\mathcal{G}(z_{0},\cdots,z_{\ell-1})$ is analytic as a function of
$\hat{z}_{1},\cdots,\hat{z}_{\ell}$ within
$0<|\hat{z}_{\ell}|<\cdots<|\hat{z}_{1}|<\mathrm{r_{max}}$, and hence is
analytic as a function of $z_{0},\cdots,z_{\ell-1}$ in
$\mathbb{D}_{0}(\mathrm{r_{max}})\times\mathbb{D}_{0}^{\ell-1}$. We remark
that there are no poles from the Cauchy-type factor due to the order of
$|\hat{z}_{k}|$.

Our goal is to analytically extend the function $\mathcal{G}$ to
$\mathbb{D}(\mathrm{r_{max}})\times\mathbb{D}^{\ell-1}$ under certain
assumption. Below we introduce two more concepts related the assumption, then
we state the identity.

We call a sequence of variables
$w_{i_{k}}^{(k)},w_{i_{k+1}}^{(k+1)},\cdots,w_{i_{k^{\prime}}}^{(k^{\prime})}$
_a Cauchy chain_ with respect to the vectors $W^{(1)},\cdots,W^{(\ell)}$ and
index sets $I^{(1)},J^{(2)},I^{(2)},J^{(3)},\cdots,I^{(\ell-1)},J^{(\ell)}$,
if

$\left(w_{i_{k}}^{(k)}-w_{i_{k+1}}^{(k+1)}\right)\cdot\left(w_{i_{k+1}}^{(k+1)}-w_{i_{k+2}}^{(k+2)}\right)\cdot\cdots\cdot\left(w_{i_{k^{\prime}-1}}^{(k^{\prime}-1)}-w_{i_{k^{\prime}}}^{(k^{\prime})}\right)$

appears as a factor of the denominator in
$\prod_{k=1}^{\ell-1}\mathcal{C}\left(W_{I^{(k)}}^{(k)};W_{J^{(k+1)}}^{(k+1)}\right)$.
We allow any single variable $w_{i_{k}}^{(k)}$ to be a Cauchy chain as long as
it is a coordinate of $W^{(k)}$.

We say $q(w)$ _dominates_
$\mathcal{H}\left(W^{(1)},\cdots,W^{(\ell)};z_{0},\cdots,z_{\ell-1}\right)$ if
and only if the following function of $w$

$\left.q(w)\cdot\mathcal{A}\left(W^{(1)},\cdots,W^{(\ell)};z_{0},\cdots,z_{\ell-1}\right)\right|_{w_{i_{k}}^{(k)}=w_{i_{k+1}}^{(k+1)}=\cdots=w_{i_{k^{\prime}}}^{(k^{\prime})}=w}$
(2.54)

is analytic at any $w\in\mathcal{R}_{0}$ when all other variables are fixed,
here
$w_{i_{k}}^{(k)},w_{i_{k+1}}^{(k+1)},\cdots,w_{i_{k^{\prime}}}^{(k^{\prime})}$
is an arbitrary Cauchy chain with respect to $W^{(1)},\cdots,W^{(\ell)}$ and
$I^{(1)},J^{(2)},I^{(2)},J^{(3)},\cdots,I^{(\ell-1)},J^{(\ell)}$. We remark
that in [Liu19], this concept was only defined when $\mathcal{R}_{0}$ contains
one single point. Here we dropped this assumption.

######  Proposition 2.10.

If $q(w)$ dominates
$\mathcal{H}\left(W^{(1)},\cdots,W^{(\ell)};z_{0},\cdots,z_{\ell-1}\right)$,
then the function $\mathcal{G}(z_{0},\cdots,z_{\ell-1})$ can be analytically
extended to $\mathbb{D}(\mathrm{r_{max}})\times\mathbb{D}^{\ell-1}$. Moreover,
$\mathcal{G}(z_{0}=0,z_{1}\cdots,z_{\ell-1})$ is independent of $q(w)$, and it
equals to

$\prod_{k=2}^{\ell}\prod_{i_{k}=1}^{n_{k}}\left[\frac{1}{1-z_{k-1}}\int_{\Sigma_{\mathrm{in}}^{(k)}}\frac{{\mathrm{d}}w_{i_{k}}^{(k)}}{2\pi{\mathrm{i}}}-\frac{z_{k-1}}{1-z_{k-1}}\int_{\Sigma^{(k)}_{\mathrm{out}}}\frac{{\mathrm{d}}w_{i_{k}}^{(k)}}{2\pi{\mathrm{i}}}\right]\prod_{i_{1}=1}^{n_{1}}\int_{\Sigma^{(1)}}\frac{{\mathrm{d}}w_{i_{1}}^{(1)}}{2\pi{\mathrm{i}}}\mathcal{H}\left(W^{(1)},\cdots,W^{(\ell)};0,z_{1},\cdots,z_{\ell-1}\right),$

where
$\Sigma_{\mathrm{out}}^{(\ell)},\cdots,\Sigma_{\mathrm{out}}^{(2)},\Sigma^{(1)},\Sigma_{\mathrm{in}}^{(2)},\cdots,\Sigma_{\mathrm{in}}^{(\ell)}$
are $2\ell-1$ nested contours in $\mathcal{D}$ each of which encloses all the
points in $\mathcal{R}_{0}$.

###### Proof of Proposition 2.10.

When $\mathcal{R}_{0}=\\{0\\}$, this is exactly the same as [Liu19,
Proposition 4.3]. On the other hand, their proof does not use the fact
$\mathcal{R}_{0}=\\{0\\}$, see [Liu19, Section 6]. Hence Proposition 2.10
follows from the same argument. ∎

One can similarly consider a two-region version of the above result. Assume
that $\mathcal{D}_{\mathrm{L}}$ and $\mathcal{D}_{\mathrm{R}}$ are two
disjoint bounded regions on the complex plane. Let $q(w)$ be a function
analytic in $\mathcal{D}_{\mathrm{L}}\cup\mathcal{D}_{\mathrm{R}}$ and define

$\mathcal{R}_{z,\mathrm{L}}=\\{u\in\mathcal{D}_{\mathrm{L}}:q(u)=z\\},\quad\text{and
}\mathcal{R}_{z,\mathrm{R}}=\\{v\in\mathcal{D}_{\mathrm{R}}:q(v)=z\\}.$

Assume that both $\mathcal{R}_{0,\mathrm{L}}$ and $\mathcal{R}_{0,\mathrm{R}}$
are nonempty. The analog of (2.49) is

$\begin{split}&\mathcal{H}\left(U^{(1)},\cdots,U^{(\ell)};V^{(1)},\cdots,V^{(\ell)};z_{0},\cdots,z_{\ell-1}\right)\\\
&:=\left[\prod_{k=1}^{\ell-1}\mathcal{C}\left(U_{I_{\mathrm{L}}^{(k)}}^{(k)};U_{J_{\mathrm{L}}^{(k+1)}}^{(k+1)}\right)\mathcal{C}\left(V_{I_{\mathrm{R}}^{(k)}}^{(k)};V_{J_{\mathrm{R}}^{(k+1)}}^{(k+1)}\right)\right]\cdot\mathcal{A}\left(U^{(1)},\cdots,U^{(\ell)};V^{(1)},\cdots,V^{(\ell)};z_{0},\cdots,z_{\ell-1}\right),\end{split}$

where $\mathcal{A}$ is analytic in
$\mathcal{D}_{\mathrm{L}}\setminus\mathcal{R}_{0,\mathrm{L}}$ for each
coordinate of $U^{(k)}$, and in
$\mathcal{D}_{\mathrm{R}}\setminus\mathcal{R}_{0,\mathrm{R}}$ for each
coordinate of $V^{(k)}$, $1\leq k\leq\ell$, and analytic for all
$(z_{0},\cdots,z_{\ell})\in\mathbb{D}(\mathrm{r_{max}})\times\mathbb{D}^{\ell-1}$.
The analog of (2.50) is

$\mathcal{G}(z_{0},\cdots,z_{\ell-1})=\sum_{\begin{subarray}{c}U^{(1)}\in\mathcal{R}_{\hat{z}_{1},\mathrm{L}}^{n_{1,\mathrm{L}}}\\\
V^{(1)}\in\mathcal{R}_{\hat{z}_{1},\mathrm{R}}^{n_{1,\mathrm{R}}}\end{subarray}}\cdots\sum_{\begin{subarray}{c}U^{(\ell)}\in\mathcal{R}_{\hat{z}_{\ell},\mathrm{L}}^{n_{\ell,\mathrm{L}}}\\\
V^{(\ell)}\in\mathcal{R}_{\hat{z}_{\ell},\mathrm{R}}^{n_{\ell,\mathrm{R}}}\end{subarray}}\left[\prod_{k=1}^{\ell}J(U^{(k)})J(V^{(k)})\right]\mathcal{H}\left(U^{(1)},\cdots,U^{(\ell)};V^{(1)},\cdots,V^{(\ell)};z_{0},\cdots,z_{\ell-1}\right)$

for
$(z_{0},\cdots,z_{\ell})\in\mathbb{D}_{0}(\mathrm{r_{max}})\times\mathbb{D}_{0}^{\ell-1}$.
We can similarly define Cauchy chains in $\mathcal{D}_{\mathrm{L}}$ and in
$\mathcal{D}_{\mathrm{R}}$. We say $q(w)$ dominates
$\mathcal{H}\left(U^{(1)},\cdots,U^{(\ell)};V^{(1)},\cdots,V^{(\ell)};z_{0},\cdots,z_{\ell-1}\right)$
if

$\left.q(u)\cdot\mathcal{A}\left(U^{(1)},\cdots,U^{(\ell)};V^{(1)},\cdots,V^{(\ell)};z_{0},\cdots,z_{\ell-1}\right)\right|_{u_{i_{k}}^{(k)}=u_{i_{k+1}}^{(k+1)}=\cdots=u_{i_{k^{\prime}}}^{(k^{\prime})}=u}$

is analytic at any $u\in\mathcal{R}_{0,\mathrm{L}}$ for any Cauchy chain in
$\mathcal{D}_{\mathrm{L}}$, and

$\left.q(v)\cdot\mathcal{A}\left(U^{(1)},\cdots,U^{(\ell)};V^{(1)},\cdots,V^{(\ell)};z_{0},\cdots,z_{\ell-1}\right)\right|_{v_{i_{k}}^{(k)}=v_{i_{k+1}}^{(k+1)}=\cdots=v_{i_{k^{\prime}}}^{(k^{\prime})}=v}$

is analytic at any $v\in\mathcal{R}_{0,\mathrm{R}}$ for any Cauchy chain in
$\mathcal{D}_{\mathrm{R}}$. The analog of Proposition 2.10 is as follows.

######  Proposition 2.11.

If $q(w)$ dominates
$\mathcal{H}\left(U^{(1)},\cdots,U^{(\ell)};V^{(1)},\cdots,V^{(\ell)};z_{0},\cdots,z_{\ell-1}\right)$,
then the function $\mathcal{G}(z_{0},\cdots,z_{\ell-1})$ can be analytically
extended to $\mathbb{D}(\mathrm{r_{max}})\times\mathbb{D}^{\ell-1}$. Moreover,
$\mathcal{G}(z_{0}=0,z_{1}\cdots,z_{\ell-1})$ is independent of $q(w)$, and it
equals to

$\begin{split}&\prod_{k=2}^{\ell}\prod_{i_{k}=1}^{n_{k,\mathrm{L}}}\left[\frac{1}{1-z_{k-1}}\int_{\Sigma_{\mathrm{in},\mathrm{L}}^{(k)}}\frac{{\mathrm{d}}u_{i_{k}}^{(k)}}{2\pi{\mathrm{i}}}-\frac{z_{k-1}}{1-z_{k-1}}\int_{\Sigma^{(k)}_{\mathrm{out},\mathrm{L}}}\frac{{\mathrm{d}}u_{i_{k}}^{(k)}}{2\pi{\mathrm{i}}}\right]\prod_{i_{1}=1}^{n_{1,\mathrm{L}}}\int_{\Sigma^{(1)}_{\mathrm{L}}}\frac{{\mathrm{d}}u_{i_{1}}^{(1)}}{2\pi{\mathrm{i}}}\\\
&\prod_{k=2}^{\ell}\prod_{i_{k}=1}^{n_{k,\mathrm{R}}}\left[\frac{1}{1-z_{k-1}}\int_{\Sigma_{\mathrm{in},\mathrm{R}}^{(k)}}\frac{{\mathrm{d}}v_{i_{k}}^{(k)}}{2\pi{\mathrm{i}}}-\frac{z_{k-1}}{1-z_{k-1}}\int_{\Sigma^{(k)}_{\mathrm{out},\mathrm{R}}}\frac{{\mathrm{d}}v_{i_{k}}^{(k)}}{2\pi{\mathrm{i}}}\right]\prod_{i_{1}=1}^{n_{1,\mathrm{R}}}\int_{\Sigma^{(1)}_{\mathrm{R}}}\frac{{\mathrm{d}}v_{i_{1}}^{(1)}}{2\pi{\mathrm{i}}}\\\
&\mathcal{H}\left(U^{(1)},\cdots,U^{(\ell)};V^{(1)},\cdots,V^{(\ell)};0,z_{1}\cdots,z_{\ell-1}\right),\end{split}$

where
$\Sigma_{\mathrm{out},\mathrm{L}}^{(\ell)},\cdots,\Sigma_{\mathrm{out},\mathrm{L}}^{(2)},\Sigma^{(1)}_{\mathrm{L}},\Sigma_{\mathrm{in},\mathrm{L}}^{(2)},\cdots,\Sigma_{\mathrm{in},\mathrm{L}}^{(\ell)}$
are $2\ell-1$ nested contours in $\mathcal{D}_{\mathrm{L}}$ each of which
encloses all the points in $\mathcal{R}_{0,\mathrm{L}}$, and
$\Sigma_{\mathrm{out},\mathrm{R}}^{(\ell)},\cdots,\Sigma_{\mathrm{out},\mathrm{R}}^{(2)},\Sigma^{(1)}_{\mathrm{R}},\Sigma_{\mathrm{in},\mathrm{R}}^{(2)},\cdots,\Sigma_{\mathrm{in},\mathrm{R}}^{(\ell)}$
are $2\ell-1$ nested contours in $\mathcal{D}_{\mathrm{R}}$ each of which
encloses all the points in $\mathcal{R}_{0,\mathrm{R}}$.

###### Proof of Proposition 2.11.

The case when $\mathcal{R}_{0,\mathrm{L}}=\\{-1\\}$ and
$\mathcal{R}_{0,\mathrm{R}}=\\{0\\}$ was the same as [Liu19, Proposition 4.4].
The proof for the more general case is also the same as the proof of [Liu19,
Proposition 4.4], except that we apply Proposition 2.10 in this paper instead
of [Liu19, Proposition 4.2]. ∎

####  2.4.2 Rewriting (2.5)

Now we want to apply Proposition 2.10 to equation (2.5) and rewrite the
formula.

We first choose $q(w)=w^{N}(w+1)^{L-N}$, where $L$ is any fixed integer
satisfying $L\geq M+N$. Recall the formula (2.5). Let
$\mathcal{H}\left(W^{(2)},W^{(1)};z_{1},z_{0}=\mathrm{z}\right)$ be a slight
modification of the integrand in (2.5). More precisely, let

$\mathcal{H}\left(W^{(2)},W^{(1)};z_{1},z_{0}\right)=\mathcal{C}\left(W^{(2)};W^{(1)}\right)\mathcal{A}\left(W^{(2)},W^{(1)};z_{1},z_{0}\right),$
(2.55)

where

$\begin{split}&\mathcal{A}\left(W^{(2)},W^{(1)};z_{1},z_{0}\right)\\\
&:=\Delta\left(W^{(1)}\right)\Delta\left(W^{(2)}\right)\hat{f}_{1}\left(W^{(1)}\right)\hat{f}_{2}\left(W^{(2)}\right)\left[\hat{H}\left(W^{(1)};W^{(2)}\right)+z_{0}\prod_{i=1}^{N}\frac{w_{i}^{(2)}}{w_{i}^{(1)}}\hat{H}\left(W^{(1)};W^{(2)}\right)\right].\end{split}$
(2.56)

Note that when $z_{0}=\mathrm{z}$,
$\mathcal{H}\left(W^{(2)},W^{(1)};z_{1},z_{0}\right)$ is exactly the integrand
of (2.5). Assume $\mathcal{D}$ is a bounded region enclosing both $0$ and
$-1$. It is obvious that the function $\mathcal{A}$ is well defined and
analytic for all $w_{i}^{(1)},w_{i}^{(2)}\in\mathcal{D}\setminus\\{\\-1,0\\}$,
$1\leq i\leq N$, and for all
$(z_{1},z_{0})\in\mathbb{D}(\mathrm{r_{max}})\times\mathbb{D}$, here we choose

$\mathrm{r_{max}}=N^{N}(L-N)^{L-N}/L^{L}.$ (2.57)

We remark that we have a different ordering of indices compared to the
original formulas (2.49) and (2.50). This is because we want to make the
indices of $\hat{f}_{1}$ and $\hat{f}_{2}$ more natural by using $1$ to label
the parameters appearing in the first part of the last passage time and using
$2$ to label the parameters appearing in the second part of the last passage
time. On the other hand, we also want to make our indices in Propositions 2.10
and 2.11 consistent with [Liu19] so the readers can compare the results
easily. These different orderings might be confusing but they only appear in
this technical proof. We will keep reminding readers if needed.

The sum we are considering is

$\mathcal{G}(z_{1},z_{0})=\sum_{W^{(2)}\in\mathcal{R}_{\hat{z}_{2}}^{N}}\sum_{W^{(1)}\in\mathcal{R}_{\hat{z}_{1}}^{N}}J\left(W^{(1)}\right)J\left(W^{(2)}\right)\mathcal{H}\left(W^{(2)},W^{(1)};z_{1},z_{0}\right),$
(2.58)

where $\hat{z}_{2}=z_{1}$ and $\hat{z}_{1}=z_{1}z_{0}$. We assume that
$z_{1}\in\mathbb{D}_{0}(\mathrm{r_{max}})$ and $z_{0}\in\mathbb{D}_{0}$ hence
$0<|\hat{z}_{2}|<|\hat{z}_{1}|<\mathrm{r_{max}}$.

We need to verify that Proposition 2.10 is applicable for this function
(2.58). All other assumptions are trivial, except for the one that $q(w)$
dominates $\mathcal{H}\left(W^{(2)},W^{(1)};z_{1},z_{0}\right)$. We verify it
below.

There are only three types of Cauchy chains. The chains of single element
$w_{i_{1}}^{(1)}$ or $w_{i_{2}}^{(2)}$, and the chain of two elements
$w_{i_{2}}^{(2)},w_{i_{1}}^{(1)}$. For the first type of chains, we need to
verify $q(w_{i_{1}}^{(1)})\mathcal{A}\left(W^{(2)},W^{(1)};z_{1},z_{0}\right)$
is analytic at $0$ and $-1$. This follows from the fact that
$\hat{f}_{1}(w)q(w)w^{-1}=(w+1)^{L-N-m}w^{n-1}e^{s_{1}w}$ is an entire
function. Similarly we can verify it for the second type of Cauchy chains.
Finally, for the chain of two elements $w_{i_{2}}^{(2)},w_{i_{1}}^{(1)}$, we
need to show
$\left.q(w)\mathcal{A}\left(W^{(2)},W^{(1)};z_{1},z_{0}\right)\right|_{w_{i_{2}}^{(2)}=w_{i_{1}}^{(1)}=w}$
is analytic at $-1$ and $0$. It follows from the fact that
$\hat{f}_{1}(w)\hat{f}_{2}(w)q(w)=(w+1)^{L-N-M}e^{(s_{1}+s_{2})w}$ is entire.

So we can apply Proposition 2.10, and obtain

$\begin{split}&\mathcal{G}(0,z_{0}=\mathrm{z})=\prod_{i_{1}=1}^{N}\left(\frac{-\mathrm{z}}{1-\mathrm{z}}\int_{\Sigma_{\mathrm{out}}}\frac{{\mathrm{d}}w_{i_{1}}^{(1)}}{2\pi{\mathrm{i}}}+\frac{1}{1-\mathrm{z}}\int_{\Sigma_{\mathrm{in}}}\frac{{\mathrm{d}}w_{i_{1}}^{(1)}}{2\pi{\mathrm{i}}}\right)\prod_{i_{2}=1}^{N}\int_{\Sigma}\frac{{\mathrm{d}}w_{i_{2}}^{(2)}}{2\pi{\mathrm{i}}}\\\
&\quad\hat{f}_{1}\left(W^{(1)}\right)\hat{f}_{2}\left(W^{(2)}\right)\frac{\left(\Delta\left(W^{(1)}\right)\right)^{2}\left(\Delta\left(W^{(2)}\right)\right)^{2}}{\Delta\left(W^{(2)};W^{(1)}\right)}\cdot\left(\hat{H}\left(W^{(1)};W^{(2)}\right)+\mathrm{z}\frac{\prod_{i_{2}=1}^{N}w_{i_{2}}^{(2)}}{\prod_{i_{1}=1}^{N}w_{i_{1}}^{(1)}}\hat{H}\left(W^{(2)};W^{(1)}\right)\right).\end{split}$

Hence we have an alternate expression for (2.5)

$p(s_{1},s_{2};m,n,M,N)=\frac{1}{(N!)^{2}}\oint_{0}\mathcal{G}(0,z_{0}=\mathrm{z})\frac{(1-\mathrm{z})^{N-2}\mathrm{d}\mathrm{z}}{2\pi\mathrm{i}\mathrm{z}^{n}}.$
(2.59)

####  2.4.3 Reformulation to a Fredholm-determinant-like expansion

In this subsubsection, we want to evaluate the summation (2.58) in a different
way. Recall $q(w)=w^{N}(w+1)^{L-N}$ and $\mathcal{R}_{z}$ are the roots of
$q(w)=z$. This equation is called the Bethe equation, and its roots are called
the Bethe roots. It is known [BL18] that when
$|z|<\mathrm{r_{max}}=N^{N}(L-N)^{L-N}/L^{L}$, the set $\mathcal{R}_{z}$ can
be split into two different subsets $\mathcal{R}_{z,\mathrm{L}}$ and
$\mathcal{R}_{z,\mathrm{R}}$ satisfying $|\mathcal{R}_{z,\mathrm{L}}|=L-N$ and
$|\mathcal{R}_{z,\mathrm{R}}|=N$. Intuitively, each root in
$\mathcal{R}_{z,\mathrm{L}}$ ($\mathcal{R}_{z,\mathrm{R}}$, respectively) can
be viewed as an continuous function of $z$ starting from $-1$ ($0$,
respectively) when $z=0$. We denote

$\mathcal{D}_{\mathrm{L}}=\cup_{|z|<\mathrm{r_{max}}}\mathcal{R}_{z,\mathrm{L}},\quad\text{and}\quad\mathcal{D}_{\mathrm{R}}=\cup_{|z|<\mathrm{r_{max}}}\mathcal{R}_{z,\mathrm{R}},$
(2.60)

and

$q_{z,\mathrm{L}}(w)=\prod_{u\in\mathcal{R}_{z,\mathrm{L}}}(w-u),\quad\text{and}\quad
q_{z,\mathrm{R}}(w)=\prod_{v\in\mathcal{R}_{z,\mathrm{L}}}(w-v)$ (2.61)

which will be used in later computations. Note that $\mathcal{D}_{\mathrm{L}}$
and $\mathcal{D}_{\mathrm{R}}$ are two disjoint bounded regions, and
$q_{z,\mathrm{L}}(w)q_{z,\mathrm{R}}(w)=q(w)-z$.

We will rewrite the summation (2.58) by treating
$w_{i_{k}}^{(k)}\in\mathcal{R}_{\hat{z}_{k},\mathrm{L}}$ and
$w_{i_{k}}^{(k)}\in\mathcal{R}_{\hat{z}_{k},\mathrm{R}}$ separately. We first
observe that, by checking the formulas (2.55) and (2.56), the summand is
invariant when we permute the coordinates of $W^{(k)}$, $k=1,2$. We also
observe that the summand is zero if any two coordinates of $W^{(k)}$ are equal
due to the Cauchy-type factor. Therefore we only need to consider the
summation for $W^{(k)}$ with different coordinates.

Assume that $n_{k}$ coordinates in $W^{(k)}$ are chosen from
$\mathcal{R}_{\hat{z}_{k},\mathrm{L}}$. Then the other $N-n_{k}$ coordinates
are chosen from $\mathcal{R}_{\hat{z}_{k},\mathrm{R}}$. Note that
$\mathcal{R}_{\hat{z}_{k},\mathrm{R}}$ has exactly $N$ elements, hence there
are $n_{k}$ elements which do not appear in $W^{(k)}$. We denote
$V^{(k)}=(v_{1}^{(k)},\cdots,v_{n_{k}}^{(k)})$ the vector formed by these
elements with any given order. We also denote
$U^{(k)}=(u_{1}^{(k)},\cdots,u_{n_{k}}^{(k)})$ the vector formed by the
coordinates of $W^{(k)}$ in $\mathcal{R}_{\hat{z}_{k},\mathrm{L}}$. Note the
invariance property we observed above. We write

$\sum_{W^{(2)}\in\mathcal{R}_{\hat{z}_{2}}^{N}}\sum_{W^{(1)}\in\mathcal{R}_{\hat{z}_{1}}^{N}}=(N!)^{2}\sum_{n_{1},n_{2}=0}^{N}\frac{1}{(n_{1}!)^{2}(n_{2}!)^{2}}\sum_{\begin{subarray}{c}U^{(2)}\in\mathcal{R}_{\hat{z}_{2},\mathrm{L}}^{n_{2}}\\\
V^{(2)}\in\mathcal{R}_{\hat{z}_{2},\mathrm{R}}^{n_{2}}\end{subarray}}\sum_{\begin{subarray}{c}U^{(1)}\in\mathcal{R}_{\hat{z}_{1},\mathrm{L}}^{n_{1}}\\\
V^{(1)}\in\mathcal{R}_{\hat{z}_{1},\mathrm{R}}^{n_{1}}\end{subarray}},$ (2.62)

where the factors $N!$, $n_{k}!$ come from the number of ways to permute the
coordinates of $W^{(k)}$, $U^{(k)}$ (and $V^{(k)}$) respectively. Now we need
to rewrite the summand in terms of $U^{(k)}$ and $V^{(k)}$, $k=1,2$. Such a
rewriting was mostly done in [BL18, BL19] except for one extra factor. We will
write down the formulas without proofs except for the one involving the extra
factor.

Recall the notation conventions (1.4), (1.5) and (1.6). We write, by simply
inserting the coordinates,

$\hat{f}_{k}\left(W^{(k)}\right)=\frac{\hat{f}_{k}\left(U^{(k)}\right)}{\hat{f}_{k}\left(V^{(k)}\right)}\cdot\hat{f}_{k}\left(\mathcal{R}_{\hat{z}_{k},\mathrm{R}}\right),\quad
J\left(W^{(k)}\right)=\frac{J\left(U^{(k)}\right)}{J\left(V^{(k)}\right)}\cdot
J\left(\mathcal{R}_{\hat{z}_{k},\mathrm{R}}\right),\quad k=1,2.$

We also have (see equation (4.43) of [BL19])

$\Delta\left(W^{(k)}\right)^{2}=(-1)^{N(N-1)/2}\frac{\Delta\left(U^{(k)}\right)^{2}\Delta\left(V^{(k)}\right)^{2}}{\Delta\left(U^{(k)};V^{(k)}\right)^{2}}\frac{q_{\hat{z}_{k},\mathrm{R}}^{2}\left(U^{(k)}\right)}{\left(q^{\prime}_{\hat{z}_{k},\mathrm{R}}\left(V^{(k)}\right)\right)^{2}}\cdot
q^{\prime}_{\hat{z}_{k},\mathrm{R}}\left(\mathcal{R}_{\hat{z}_{k},\mathrm{R}}\right)$

and (see equation (4.44) of [BL19])

$\Delta\left(W^{(2)};W^{(1)}\right)=\Delta\left(\mathcal{R}_{\hat{z}_{2},\mathrm{R}};\mathcal{R}_{\hat{z}_{1},\mathrm{R}}\right)\frac{\Delta\left(U^{(2)};U^{(1)}\right)\Delta\left(V^{(2)};V^{(1)}\right)}{\Delta\left(U^{(2)};V^{(1)}\right)\Delta\left(V^{(2)};U^{(1)}\right)}\cdot\frac{q_{\hat{z}_{1},\mathrm{R}}\left(U^{(2)}\right)q_{\hat{z}_{2},\mathrm{R}}\left(U^{(1)}\right)}{q_{\hat{z}_{1},\mathrm{R}}\left(V^{(2)}\right)q_{\hat{z}_{2},\mathrm{R}}\left(V^{(1)}\right)}.$

We need to further rewrite the above expressions so that we can apply
Proposition 2.11 later. Denote

$\mathfrak{h}(w;z):=\begin{dcases}q_{z,\mathrm{R}}(w)/w^{N},&w\in\mathcal{D}_{\mathrm{L}},\\\
q_{z,\mathrm{L}}(w)/(w+1)^{L-N},&w\in\mathcal{D}_{\mathrm{R}}.\end{dcases}$

It is easy to check that $\mathfrak{h}(w;z)$ is analytic and nonzero for
$w\in\mathcal{D}_{\mathrm{L}}\cup\mathcal{D}_{\mathrm{R}}$ and for
$z\in\mathbb{D}(\mathrm{r_{max}})$. Especially we have $\mathfrak{h}(w;0)=1$
for all $w\in\mathcal{D}_{\mathrm{L}}\cup\mathcal{D}_{\mathrm{R}}$. See
equation (5.5) in [Liu19] and the discussions below.

One can write (see equation (4.51) of [BL19])

$q^{\prime}_{z,\mathrm{R}}(v)=\frac{v^{N}}{J(v)\mathfrak{h}(v;z)},\quad
v\in\mathcal{R}_{z,\mathrm{R}}.$

and (see (4.49) of [BL19])

$q_{z,\mathrm{R}}(v^{\prime})=\frac{z^{\prime}-z}{q_{z,\mathrm{L}}(v^{\prime})}=\frac{z^{\prime}-z}{(v^{\prime}+1)^{L-N}\mathfrak{h}(v^{\prime};z)},\quad
v^{\prime}\in\mathcal{R}_{z^{\prime},\mathrm{R}}.$

Note that
$\Delta\left(\mathcal{R}_{\hat{z}_{2},\mathrm{R}};\mathcal{R}_{\hat{z}_{1},\mathrm{R}}\right)=q_{\hat{z}_{1},\mathrm{R}}\left(\mathcal{R}_{\hat{z}_{2},\mathrm{R}}\right)$.
After inserting all these formulas and simplifying the expression, we end up
with

$\begin{split}&J\left(W^{(1)}\right)J\left(W^{(2)}\right)\hat{f}_{1}\left(W^{(1)}\right)\hat{f}_{2}\left(W^{(2)}\right)\frac{\left(\Delta\left(W^{(1)}\right)\right)^{2}\left(\Delta\left(W^{(2)}\right)\right)^{2}}{\Delta\left(W^{(2)};W^{(1)}\right)}=\mathcal{K}(\hat{z}_{2},\hat{z}_{1})\cdot\frac{\hat{z}_{1}^{n}\hat{z}_{2}^{N-n}}{(\hat{z}_{2}-\hat{z}_{1})^{N}}\\\
&\quad\cdot\left[\prod_{k=1}^{2}\frac{(\Delta(U^{(k)}))^{2}(\Delta(V^{(k)}))^{2}}{(\Delta(U^{(k)};V^{(k)}))^{2}}\cdot\frac{f_{k}(U^{(k)};s_{k})}{f_{k}(V^{(k)};s_{k})}\cdot\left(\mathfrak{h}(U^{(k)};\hat{z}_{k})\right)^{2}\cdot\left(\mathfrak{h}(V^{(k)};\hat{z}_{k})\right)^{2}\cdot
J(U^{(k)})J(V^{(k)})\right]\\\
&\quad\cdot\left[\frac{\Delta(U^{(2)};V^{(1)})\Delta(V^{(2)};U^{(1)})}{\Delta(U^{(2)};U^{(1)})\Delta(V^{(2)};V^{(1)})}\cdot\frac{(1-\hat{z}_{2}/\hat{z}_{1})^{n_{1}}(1-\hat{z}_{1}/\hat{z}_{2})^{n_{2}}}{\mathfrak{h}(U^{(2)};\hat{z}_{1})\mathfrak{h}(V^{(2)};\hat{z}_{1})\mathfrak{h}(U^{(1)};\hat{z}_{2})\mathfrak{h}(V^{(1)};\hat{z}_{2})}\right],\end{split}$
(2.63)

where the functions $f_{k}(w;s_{k})=\hat{f}_{k}(w)w^{N}$, $k=1,2$, are defined
in (1.9), and

$\begin{split}\mathcal{K}(\hat{z}_{2},\hat{z}_{1})&=\frac{1}{\hat{z}_{1}^{n}}\prod_{v\in\mathcal{R}_{\hat{z}_{1},\mathrm{R}}}\frac{(v+1)^{-m}v^{n}e^{s_{1}v}}{\mathfrak{h}(v;\hat{z}_{1})}\cdot\frac{1}{\hat{z}_{2}^{N-n}}\prod_{v\in\mathcal{R}_{\hat{z}_{2},\mathrm{R}}}\frac{(v+1)^{-M+m+L-N}v^{N-n}e^{s_{2}v}}{\mathfrak{h}(v;\hat{z}_{2})/\mathfrak{h}(v;\hat{z}_{1})}\\\
&=(-1)^{N(L-1)}\prod_{v\in\mathcal{R}_{\hat{z}_{1},\mathrm{R}}}\frac{(v+1)^{-m}e^{s_{1}v}}{\mathfrak{h}(v;\hat{z}_{1})}\prod_{u\in\mathcal{R}_{\hat{z}_{1},\mathrm{L}}}\frac{1}{u^{n}}\prod_{v\in\mathcal{R}_{\hat{z}_{2},\mathrm{R}}}\frac{(v+1)^{-M+m+L-N}e^{s_{2}v}}{\mathfrak{h}(v;\hat{z}_{2})/\mathfrak{h}(v;\hat{z}_{1})}\prod_{u\in\mathcal{R}_{\hat{z}_{2},\mathrm{L}}}\frac{1}{u^{N-n}}.\end{split}$
(2.64)

We observe that $\mathcal{K}(\hat{z}_{2},\hat{z}_{1})$ is analytic for both
$\hat{z}_{2}\in\mathbb{D}_{\mathrm{r_{max}}}$ and
$\hat{z}_{1}\in\mathbb{D}_{\mathrm{r_{max}}}$ since $\mathfrak{h}$ is analytic
and nonzero, and
$z^{-1}\prod_{v\in\mathcal{R}_{z,\mathrm{R}}}v=(-1)^{L-1}\prod_{u\in\mathcal{R}_{z,\mathrm{L}}}u^{-1}$
is analytic for $z\in\mathbb{D}_{\mathrm{r_{max}}}$. Moreover, we have
$\mathcal{K}(0,0)=1$.

As we mentioned before, there is an extra factor in the summand of (2.62)
which comes from (2.56),

$\hat{H}\left(W^{(1)};W^{(2)}\right)+z_{0}\prod_{i=1}^{N}\frac{w_{i}^{(2)}}{w_{i}^{(1)}}\hat{H}\left(W^{(2)};W^{(1)}\right).$

Here $\hat{H}$ is defined in (2.7). Recall that $\\{w_{i}^{(k)}:1\leq i\leq
N\\}=\mathcal{R}_{\hat{z}_{k},\mathrm{R}}\cup\\{u_{i}^{(k)}:1\leq i\leq
n_{k}\\}\setminus\\{v_{i}^{(k)}:1\leq i\leq n_{k}\\}$. We write, for each
$k,k^{\prime}\in\\{1,2\\}$,

$\sum_{i=1}^{N}\left(w_{i}^{(k)}-w_{i}^{(k^{\prime})}\right)=\sum_{i_{k}=1}^{n_{k}}\left(u_{i_{k}}^{(k)}-v_{i_{k}}^{(k)}\right)-\sum_{i^{\prime}_{k^{\prime}}=1}^{n_{k^{\prime}}}\left(u_{i^{\prime}_{k^{\prime}}}^{(k^{\prime})}-v_{i^{\prime}_{k^{\prime}}}^{(k^{\prime})}\right)+\mathcal{S}_{1}(\hat{z}_{k})-\mathcal{S}_{1}(\hat{z}_{k^{\prime}})$

and

$\sum_{i=1}^{N}\left((w_{i}^{(k)})^{2}-(w_{i}^{(k^{\prime})})^{2}\right)=\sum_{i_{k}=1}^{n_{k}}\left((u_{i_{k}}^{(k)})^{2}-(v_{i_{k}}^{(k)})^{2}\right)-\sum_{i^{\prime}_{k^{\prime}}=1}^{n_{k^{\prime}}}\left((u_{i^{\prime}_{k^{\prime}}}^{(k^{\prime})})^{2}-(v_{i^{\prime}_{k^{\prime}}}^{(k^{\prime})})^{2}\right)+\mathcal{S}_{2}(\hat{z}_{k})-\mathcal{S}_{2}(\hat{z}_{k^{\prime}}),$

where

$\mathcal{S}_{k}(\hat{z}):=\sum_{v\in\mathcal{R}_{\hat{z},\mathrm{R}}}v^{k},\quad
k=1,2$ (2.65)

is analytic in $\hat{z}\in\mathbb{D}_{\mathrm{r_{max}}}$. Moreover, it is easy
to see that $\mathcal{S}_{k}(0)=0$ for both $k=1,2$. We also write

$z_{0}\prod_{i=1}^{N}\frac{w_{i}^{(2)}}{w_{i}^{(1)}}=\frac{\hat{z}_{1}}{\hat{z}_{2}}\prod_{i=1}^{N}\frac{w_{i}^{(2)}}{w_{i}^{(1)}}=\prod_{i_{1}=1}^{n_{1}}\frac{v_{i_{1}}^{(1)}}{u_{i_{1}}^{(1)}}\prod_{i_{2}=1}^{n_{2}}\frac{u_{i_{2}}^{(2)}}{v_{i_{2}}^{(2)}}\cdot\frac{\pi(\hat{z}_{2})}{\pi(\hat{z}_{1})},$
(2.66)

where

$\pi(\hat{z}):=\frac{1}{\hat{z}}\prod_{v\in\mathcal{R}_{\hat{z},\mathrm{R}}}v=\frac{(-1)^{L-1}}{\prod_{u\in\mathcal{R}_{\hat{z},\mathrm{L}}}u}$

is analytic in $\mathbb{D}_{\mathrm{r_{max}}}$. Moreover, it is easy to see
that $\pi(0)=(-1)^{N-1}$.

Combing the above calculations we have

$\hat{H}\left(W^{(1)};W^{(2)}\right)+z_{0}\prod_{i=1}^{N}\frac{w_{i}^{(2)}}{w_{i}^{(1)}}\hat{H}\left(W^{(1)};W^{(2)}\right)=\tilde{H}(U^{(1)},U^{(2)};V^{(1)},V^{(2)};\hat{z}_{1},\hat{z}_{2})$
(2.67)

for some function $\tilde{H}$ which is analytic for all
$u_{i_{1}}^{(1)},u_{i_{2}}^{(2)}\in\mathcal{D}_{\mathrm{L}}$,
$v_{i_{1}}^{(1)},v_{i_{2}}^{(2)}\in\mathcal{D}_{\mathrm{R}}\setminus\\{0\\}$,
$1\leq i_{1}\leq n_{1},1\leq i_{2}\leq n_{2}$, and for
$\hat{z}_{1},\hat{z}_{2}\in\mathbb{D}_{\mathrm{r_{max}}}$. Moreover, we have

$\tilde{H}(U^{(1)},U^{(2)};V^{(1)},V^{(2)};0,0)=H(U^{(1)},U^{(2)};V^{(1)},V^{(2)}),$
(2.68)

where $H$ is defined in (1.10).

Now we combine (2.63) and (2.67), and note (2.62). Note
$\hat{z}_{1}/\hat{z}_{2}=z_{0}$. We have

$\begin{split}&\frac{1}{(N!)^{2}}\mathcal{G}(z_{1},z_{0})\frac{(1-z_{0})^{N}}{z_{0}^{n}}\\\
&=\mathcal{K}(\hat{z}_{2},\hat{z}_{1})\sum_{n_{1},n_{2}=0}^{N}\frac{(1-z_{0}^{-1})^{n_{1}}(1-z_{0})^{n_{2}}}{(n_{1}!)^{2}(n_{2}!)^{2}}\sum_{\begin{subarray}{c}U^{(2)}\in\mathcal{R}_{\hat{z}_{2},\mathrm{L}}^{n_{2}}\\\
V^{(2)}\in\mathcal{R}_{\hat{z}_{2},\mathrm{R}}^{n_{2}}\end{subarray}}\sum_{\begin{subarray}{c}U^{(1)}\in\mathcal{R}_{\hat{z}_{1},\mathrm{L}}^{n_{1}}\\\
V^{(1)}\in\mathcal{R}_{\hat{z}_{1},\mathrm{R}}^{n_{1}}\end{subarray}}\mathcal{C}(U^{(2)};U^{(1)})\mathcal{C}(V^{(2)};V^{(1)})\\\
&\quad\cdot\left[\prod_{k=1}^{2}\frac{(\Delta(U^{(k)}))(\Delta(V^{(k)}))}{(\Delta(U^{(k)};V^{(k)}))^{2}}\cdot\frac{f_{k}(U^{(k)};s_{k})}{f_{k}(V^{(k)};s_{k})}\cdot\left(\mathfrak{h}(U^{(k)};\hat{z}_{k})\right)^{2}\cdot\left(\mathfrak{h}(V^{(k)};\hat{z}_{k})\right)^{2}\cdot
J(U^{(k)})J(V^{(k)})\right]\\\
&\quad\cdot\left[{\Delta(U^{(2)};V^{(1)})\Delta(V^{(2)};U^{(1)})}\cdot\frac{\tilde{H}(U^{(1)},U^{(2)};V^{(1)},V^{(2)};\hat{z}_{1},\hat{z}_{2})}{\mathfrak{h}(U^{(2)};\hat{z}_{1})\mathfrak{h}(V^{(2)};\hat{z}_{1})\mathfrak{h}(U^{(1)};\hat{z}_{2})\mathfrak{h}(V^{(1)};\hat{z}_{2})}\right].\end{split}$
(2.69)

####  2.4.4 Completing the proof

Now we are ready to complete the proof. We will take $z_{1}\to 0$ on both
sides of (2.69). Recall that we have already proven that
$\mathcal{G}(z_{1},z_{0})$ is analytic for
$(z_{1},z_{0})\in\mathbb{D}_{\mathrm{r_{max}}}\times\mathbb{D}$ and
$\mathcal{G}(0,z_{0})$ is well defined. For the right hand side, recall
$\hat{z}_{2}=z_{1}$ and $\hat{z}_{1}=z_{1}z_{0}$. When $z_{1}\to 0$, both
$\hat{z}_{1}$ and $\hat{z}_{2}$ go to $0$. We also recall
$\mathcal{K}(0,0)=1$.

For the summand over $U^{(2)},V^{(2)},U^{(1)},V^{(1)}$, it is a Cauchy type
summation as we discussed in Proposition 2.11. Our previous discussions on the
functions $\mathfrak{h}$ and $\tilde{H}$ implies that this summand satisfies
the analyticity assumption. The proof that $q(w)$ dominates the corresponding
factor in this summand is also similar to the previous case discussed in
Section 2.4.2. The only minor difference is that we have a factor
$\prod_{i_{1}}v_{i_{1}^{(1)}}\prod_{i_{2}}(v_{i_{2}}^{(2)})^{-1}$ in
$\tilde{H}$ but the proof does not change even with this factor. Hence we know
that this summation is also analytic for
$(z_{1},z_{0})\in\mathbb{D}_{\mathrm{r_{max}}}\times\mathbb{D}$. Moreover, by
inserting $z_{1}=0$ in the equation, we obtain

$\begin{split}&\frac{1}{(N!)^{2}}\mathcal{G}(0,z_{0})\frac{(1-z_{0})^{N}}{z_{0}^{n}}=\sum_{n_{1},n_{2}=0}^{N}\frac{(1-z_{0}^{-1})^{n_{1}}(1-z_{0})^{n_{2}}}{(n_{1}!)^{2}(n_{2}!)^{2}}\\\
&\quad\prod_{i_{1}=1}^{n_{1}}\left(\frac{1}{1-{z_{0}}}\int_{\Sigma_{\mathrm{L},\mathrm{in}}}\frac{{\mathrm{d}}u^{(1)}_{i_{1}}}{2\pi{\mathrm{i}}}-\frac{{z_{0}}}{1-{z_{0}}}\int_{\Sigma_{\mathrm{L},\mathrm{out}}}\frac{{\mathrm{d}}u^{(1)}_{i_{1}}}{2\pi{\mathrm{i}}}\right)\left(\frac{1}{1-{z_{0}}}\int_{\Sigma_{\mathrm{R},\mathrm{in}}}\frac{{\mathrm{d}}v^{(1)}_{i_{1}}}{2\pi{\mathrm{i}}}-\frac{{z_{0}}}{1-{z_{0}}}\int_{\Sigma_{\mathrm{R},\mathrm{out}}}\frac{{\mathrm{d}}v^{(1)}_{i_{1}}}{2\pi{\mathrm{i}}}\right)\\\
&\quad\cdot\prod_{i_{2}=1}^{n_{2}}\int_{\Sigma_{\mathrm{L}}}\frac{{\mathrm{d}}u^{(2)}_{i_{2}}}{2\pi{\mathrm{i}}}\int_{\Sigma_{\mathrm{L}}}\frac{{\mathrm{d}}v^{(2)}_{i_{2}}}{2\pi{\mathrm{i}}}\mathcal{C}(U^{(2)};U^{(1)})\mathcal{C}(V^{(2)};V^{(1)})\prod_{k=1}^{2}\frac{(\Delta(U^{(k)}))(\Delta(V^{(k)}))}{(\Delta(U^{(k)};V^{(k)}))^{2}}\cdot\frac{f_{k}(U^{(k)};s_{k})}{f_{k}(V^{(k)};s_{k})}\\\
&\quad\cdot{\Delta(U^{(2)};V^{(1)})\Delta(V^{(2)};U^{(1)})}\cdot{H(U^{(1)},U^{(2)};V^{(1)},V^{(2)}).}\end{split}$
(2.70)

Inserting it in (2.59) and replacing $n_{0},n_{1}$ by $k_{1},k_{2}$, we obtain

$\begin{split}&p(s_{1},s_{2};m,n,M,N)=\oint_{0}\frac{{\mathrm{d}}\mathrm{z}}{2\pi{\mathrm{i}}\mathrm{(1-z)^{2}}}\sum_{k_{1},k_{2}\geq
0}\frac{1}{(k_{1}!k_{2}!)^{2}}\\\
&\quad\prod_{i_{1}=1}^{k_{1}}\left(\frac{1}{1-\mathrm{z}}\int_{\Sigma_{\mathrm{L},\mathrm{in}}}\frac{{\mathrm{d}}u^{(1)}_{i_{1}}}{2\pi{\mathrm{i}}}-\frac{\mathrm{z}}{1-\mathrm{z}}\int_{\Sigma_{\mathrm{L},\mathrm{out}}}\frac{{\mathrm{d}}u^{(1)}_{i_{1}}}{2\pi{\mathrm{i}}}\right)\left(\frac{1}{1-\mathrm{z}}\int_{\Sigma_{\mathrm{R},\mathrm{in}}}\frac{{\mathrm{d}}v^{(1)}_{i_{1}}}{2\pi{\mathrm{i}}}-\frac{\mathrm{z}}{1-\mathrm{z}}\int_{\Sigma_{\mathrm{R},\mathrm{out}}}\frac{{\mathrm{d}}v^{(1)}_{i_{1}}}{2\pi{\mathrm{i}}}\right)\\\
&\quad\cdot\prod_{i_{2}=1}^{k_{2}}\int_{\Sigma_{\mathrm{L}}}\frac{{\mathrm{d}}u^{(2)}_{i_{2}}}{2\pi{\mathrm{i}}}\int_{\Sigma_{\mathrm{R}}}\frac{{\mathrm{d}}v^{(2)}_{i_{2}}}{2\pi{\mathrm{i}}}\cdot\left(1-\mathrm{z}\right)^{k_{2}}\left(1-\frac{1}{\mathrm{z}}\right)^{k_{1}}\cdot\frac{f_{1}(U^{(1)};s_{1})f_{2}(U^{(2)};s_{2})}{f_{1}(V^{(1)};s_{1})f_{2}(V^{(2)};s_{2})}\cdot
H(U^{(1)},U^{(2)};V^{(1)},V^{(2)})\\\
&\quad\cdot\prod_{\ell=1}^{2}\frac{\left(\Delta(U^{(\ell)})\right)^{2}\left(\Delta(V^{(\ell)})\right)^{2}}{\left(\Delta(U^{(\ell)};V^{(\ell)})\right)^{2}}\cdot\frac{\Delta(U^{(1)};V^{(2)})\Delta(V^{(1)};U^{(2)})}{\Delta(U^{(1)};U^{(2)})\Delta(V^{(1)};V^{(2)})}.\end{split}$
(2.71)

Note that when $k_{1}=0$, the summand is analytic for $\mathrm{z}=0$ hence the
integral of $\mathrm{z}$ vanishes. When $k_{2}=0$, there is no
$u^{(2)}_{i_{2}}$ or $v_{i_{2}}^{(2)}$ variable, hence the $u_{i_{1}}^{(1)}$
and $v_{i_{1}}^{(1)}$ contours can be deformed to $\Sigma_{\mathrm{L}}$ and
$\Sigma_{\mathrm{R}}$ respectively. As a result, the $\mathrm{z}$ integral can
be separately written as

$\oint_{0}\frac{{\mathrm{d}}\mathrm{z}}{2\pi{\mathrm{i}}\mathrm{(1-z)^{2}}}\left(1-\frac{1}{\mathrm{z}}\right)^{k_{1}}=\begin{dcases}-1,&k_{1}=1,\\\
0,&k_{1}=0,\text{or }k_{1}\geq 2.\end{dcases}$

However, it is direct to check that $H(U^{(1)},U^{(2)};V^{(1)},V^{(2)})=0$
when $k_{1}=1$ and $k_{2}=0$. Therefore the summand when $k_{2}=0$ also
vanishes. Thus we can replace the sum $\sum_{k_{1},k_{2}\geq 0}$ by
$\sum_{k_{1},k_{2}\geq 1}$, and arrive at the formula (1.7).

##  3 Asymptotic analysis and proof of Theorem 1.3

In this section, we will perform asymptotic analysis for the formulas obtained
in Theorem 1.1 and prove Theorem 1.3. The main technical result of this
section is as follows.

######  Proposition 3.1.

Suppose $\alpha>0,\gamma\in(0,1)$ are fixed constants. Assume that

$\begin{split}M&=[\alpha N],\\\ m&=[\gamma\alpha
N+x_{1}\alpha^{2/3}(1+\sqrt{\alpha})^{2/3}N^{2/3}],\\\ n&=[\gamma
N+x_{2}\alpha^{-1/3}(1+\sqrt{\alpha})^{2/3}N^{2/3}],\\\
t_{1}&=d((1,1),(m,n))+\mathrm{t}_{1}\cdot\alpha^{-1/6}(1+\sqrt{\alpha})^{4/3}N^{1/3},\\\
t_{2}&=d((m+1,n),(M,N))+\mathrm{t}_{2}\cdot\alpha^{-1/6}(1+\sqrt{\alpha})^{4/3}N^{1/3},\\\
t_{2}^{\prime}&=d((m,n+1),(M,N))+\mathrm{t}_{2}\cdot\alpha^{-1/6}(1+\sqrt{\alpha})^{4/3}N^{1/3},\end{split}$
(3.1)

for some real numbers $x_{1},x_{2}$. Then

$\begin{split}&\mathbb{P}\left((m,n),(m+1,n)\in{\mathcal{G}}_{(1,1)}(M,N),L_{(1,1)}(m,n)\geq
t_{1},L_{(m+1,n)}(M,N)\geq t_{2}\right)\\\