# Multi-point distribution of discrete time periodic TASEP

Yuchen Liao

###### Abstract.

We study the one-dimensional discrete time totally asymmetric simple exclusion
process with parallel update rules on a spatially periodic domain. A multi-
point space-time joint distribution formula is obtained for general initial
conditions. The formula involves contour integrals of Fredholm determinants
with kernels acting on certain discrete spaces. The large-time, large-period
limit is obtained under the relaxation time scale $t=O(L^{3/2})$ when the
height fluctuations are critically affected by the finite geometry for a class
of initial conditions satisfying certain technical assumptions. The
assumptions are verified for step and flat initial conditions. As a corollary
we obtain the multi-point distribution for discrete time TASEP on the whole
integer lattice $\mathbb{Z}$ by taking the period $L$ large enough so that the
finite-time distribution is not affected by the boundary. The large time limit
for multi-time distribution for discrete time TASEP on $\mathbb{Z}$ is also
obtained for step initial condition.

##  1\. Introduction

Models in the one-dimensional Kardar-Parisi-Zhang (KPZ) universality class are
expected to have the same limiting height fluctuations under the
$t:t^{2/3}:t^{1/3}$ KPZ scaling for temporal correlations, spatial
correlations and height fluctuations. Over the last two decades, the limiting
height fluctuations have been obtained for a large class of models, either for
the whole space models [2, 11, 8, 9, 20, 21, 16, 10, 13, 15], or for half
space models [1, 6, 7].

Recently similar limit theorems were obtained for models on periodic domains.
Such models are defined on a finite ring of size $L$ instead of on the
infinite or semi-infinite lattices and it is of crucial interest to understand
the effect of this finite geometry on the height fluctuations. For periodic
models the one-point marginals of limiting height fluctuations were obtained
for classical step, flat and stationary initial conditions in [3, 14], and
independently in [19] under the so-called relaxation time scale $t=O(L^{3/2})$
when the height fluctuations are critically affected by the finite geometry.
Further extensions including formulas for multi-point space-time joint
distributions were subsequently obtained in [4] for step initial condition and
[5] for more general initial conditions. These limiting formulas are expected
to be universal for all related models on periodic domains. So far all
limiting distribution formulas were obtained as scaling limits of finite-time
joint formulas for a single model in the KPZ universality class, namely the
continuous time periodic totally asymmetric simple exclusion process (TASEP on
a ring).

The goal of this paper is to study another classical model in the KPZ
universality class, the discrete time totally asymmetric exclusion process
with parallel updates, on periodic domains. On the infinite lattice
$\mathbb{Z}$, this model has been well studied. The one-point marginal
distribution for height function was obtained in [11] for the equivalent
geometric last passage percolation model and joint distributions of several
locations at equal time was obtained in [9]. Recently the joint distributions
along the time direction have also been studied, in [12] for two-time case and
[13] for general multi-time joint distributions. However on the periodic
domain there are less results concerning height fluctuations (see [18] for
some results on transition probability and stationary distributions), which
are the main focuses of this paper. The main results of this paper are
summarized as follows:

  1. (1)

For general initial conditions we obtain a finite-time multi-point (in both
space and time) joint distribution formula for discrete time parallel periodic
TASEP. The formula consists of an $m$-fold contour integral with integrand
involving a Fredholm determinant, where $m$ is the number of space-time points
we are considering and the Fredholm determinant has kernels acting on certain
discrete sets related to roots of some polynomials.

  2. (2)

Under the relaxation time scale $t=O(L^{3/2})$ and the $1:2:3$ KPZ scaling, we
obtain large-time, large-period limits for the multi-point joint distributions
under certain assumptions on the initial condition, which are verified for
step and flat cases. These limiting formulas agree with those obtained in [4,
5], thus providing an evidence that the height fluctuations for periodic
models in the KPZ class are in fact universal.

  3. (3)

We also obtain the same type of multi-point joint distribution formula for
discrete time TASEP on the infinite lattice $\mathbb{Z}$, by relating it with
the corresponding periodic model with large enough period so that the finite-
time distribution is not affected by the boundary. The large time limit for
the multi-time distribution formula under KPZ scaling is also obtained for
step initial condition, which agrees with the one in [15].

Comparing to the previous work [4, 5] and [15] on the multi-point
distributions of continuous time TASEP, on either periodic domain or
$\mathbb{Z}$, our work consists of formulas with similar structures but
involves an extra parameter $p$ describing the hopping probability, which
makes the algebraic properties a bit more complicated. In particular the
polynomial whose roots are related to the kernels in the periodic formulas now
depends on the extra parameter $p$. We point out that all the formulas for
continuous time TASEP can be obtained from our formulas by rescaling the time
and taking $p\to 0$.

On the infinite lattice $\mathbb{Z}$, the multi-time distribution of discrete
time TASEP (more precisely the equivalent geometric last passage percolation
model) has already been considered in [13], where a formula involving contour
integrals of Fredholm determinants was obtained. We expect our formulas to
agree with the formulas in [13], both in finite time and for the large time
limit. Currently we are not able to prove the equivalence of the two types of
formulas though they share a lot of similarities.

### Outline of the paper

In Section 2 we describe the discrete time parallel TASEP models and state the
main results involving several multi-point joint distribution formulas, for
both periodic domain and infinite lattice $\mathbb{Z}$, finite time and large
time limit. From Sections 3 to Section 5 we derive the main finite-time
algebraic formulas for multi-point distribution of discrete time parallel
periodic TASEP and we regard them as the main technical novelties in this
paper. In particular in Section 3 we prove a novel transition probability
formula for discrete time parallel TASEP on the periodic domain involving
integral of determinants using coordinate Bethe ansatz. In Section 4 we derive
the finite-time multi-point distribution formula by performing a multiple sum
over the transition probabilities. The key ingredients are certain Cauchy-type
summation identities over the eigenfunctions of the generator, which might be
of independent interests so we discuss the proof in Section 5. In Section 6
and Section 7 we discuss the large time, large period asymptotics for the
multi-point distribution under relaxation time scale $t=O(L^{3/2})$. Finally
in Section 8 we derive the multi-time distribution for infinite discrete time
parallel TASEP and perform large time asymptotics under KPZ scaling for step
initial condition.

### Acknowledgements

I would like to thank Jinho Baik for suggesting this problem and for useful
discussions. I would also like to thank Zhipeng Liu for helpful discussions.
This work is supported in part through Jinho Baik’s NSF grants DMS-1664531.

##  2\. Models and main results

Let $N<L$ be positive integers. We consider discrete time TASEP with parallel
updates with $N$ particles on a spatially periodic domain of size $L$. It is
convenient to view the dynamics as particles moving to the right on the
integer lattice $\mathbb{Z}$ while periodicity forces particle configurations
to be identical copies of each other every per $L$ sites, meaning that the
occupation functions $\eta_{j}(t)$ which equals $1$ if there is a particle at
site $j\in\mathbb{Z}$ at time $t$ and equalts $0$ otherwise should satisfy

$\eta_{j}(t)=\eta_{j+kL}(t),\quad\text{for all }j,k\in\mathbb{Z}\text{ and
}t\in\mathbb{N}.$ (2.1)

We fix a single period of size $L$ and denote the locations of totally $N$
particles in this period at time $t$ as

$x_{1}(t)>x_{2}(t)>\cdots>x_{N}(t).$

Here $x_{i}(t)\in\mathbb{Z}$ for $1\leq i\leq N$ and we index the particles
from right to left. The locations of all the particles then satisfy
$x_{i+kN}(t)=x_{i}(t)-kL$ for $1\leq i\leq N$ and $k\in\mathbb{Z}$ so that we
have

$\cdots>x_{N}(t)+L=x_{0}(t)>x_{1}(t)>x_{2}(t)>\cdots>x_{N}(t)>x_{N+1}(t)=x_{1}(t)-L>\cdots.$

Thus the natural configuration space for the particles should be

$\mathcal{X}_{N}^{(L)}:=\\{\vec{x}=(x_{1},\cdots,x_{N})\in\mathbb{Z}^{N}:x_{N}+L>x_{1}>x_{2}>\cdots>x_{N}\\}.$
(2.2)

The discrete time parallel periodic TASEP with $N$ particles, period $L$ and
hopping probability $0<p<1$, which we denote by $\mathrm{dpTASEP}(L,N,p)$, is
the following Markovian dynamics on particle configurations
$\vec{x}(t)\in\mathcal{X}_{N}^{(L)}$: at each time step, each particle in a
single period hops to its right neighbor site independently with probability
$p=1-q$ provided that the site is empty, otherwise it stays at its current
position. As a special case, the discrete time parallel TASEP on $\mathbb{Z}$
which we will denote by $\mathrm{dTASEP}(p)$ corresponds to particles
following the same evolution rules with the period $L\to\infty$ so that the
configuration space for the first $N$ particles(from right to left, we always
assume the existence of right-most particle) becomes

$\mathcal{X}_{N}^{(\infty)}:=\\{\vec{x}=(x_{1},\cdots,x_{N})\in\mathbb{Z}^{N}:x_{1}>x_{2}>\cdots>x_{N}\\}.$
(2.3)

######  Notation 2.1.

Throughout the paper there will be several very similar formulas and
quantities corresponding to either discrete time parallel TASEP on a periodic
domain with size $L$ or on the infinite lattice $\mathbb{Z}$, as well as their
scaling limits. To avoid confusion we will always follow the convention used
in (2.2) and (2.3) by putting superscripts $(L)$ to any quantities related to
periodic model in finite-time and $(\infty)$ to any quantities related to
model on the infinite lattice $\mathbb{Z}$ in finite-time. On the other hand
quantities describing the large-time limits will have superscripts
$(\mathrm{per})$ and $(\mathrm{kpz})$ for periodic and infinite models,
respectively.

The main results of the paper consist of several formulas for multi-point
joint distributions of discrete time parallel TASEP on periodic domain and on
$\mathbb{Z}$ as well as their large time scaling limits. All of them are
expressed as contour integrals of Fredholm determinants. The main difference
is the formulas for periodic models are expressed in terms of discrete sets
related to roots of certain algebraic equations while the Fredholm determinant
appearing in formulas for discrete time TASEP on $\mathbb{Z}$ has kernels
acting on continuous contours.

###  2.1. Multi-point distribution formulas for $\mathrm{dpTASEP}(L,N,p)$

The first theorem is a finite-time multi-point joint distribution formula for
discrete time TASEP on a periodic domain, which is the starting point of all
other results in this paper.

######  Theorem 2.2 (Finite-time multi-point joint distribution for
$\mathrm{dpTASEP}(L,N,p,\vec{y})$).

Let $N<L$ be integers. Consider discrete time periodic parallel TASEP with
hopping probability $0<p<1$, $N$ particles and $L$ sites per period (denoted
by $\mathrm{dpTASEP}(L,N,p,\vec{y})$) where $\vec{y}\in\mathcal{X}_{N}^{(L)}$
is the initial condition, i.e., $x_{i}(0)=y_{i}$ for $1\leq i\leq N$. We set
$\rho=N/L$ and

$\mathbbm{r}_{c}:=\left(\frac{-w_{c}}{1+pw_{c}}\right)^{\rho}(1+w_{c})^{1-\rho},$
(2.4)

where

$w_{c}=-\frac{2\rho}{1+\sqrt{1-4p\cdot\rho(1-\rho)}}:=-\frac{2\rho}{1+\nu}.$
(2.5)

Fix a positive integer $m$ and let $(k_{i},t_{i})$, $1\leq i\leq m$ be $m$
distinct points in $\mathbb{Z}\times\mathbb{Z}_{\geq 0}$ satisfying $0\leq
t_{1}\leq\cdots\leq t_{m}$. Then for any integers $a_{1},\cdots,a_{m}$,

$\mathbb{P}_{\vec{y}}^{(L)}\left(\bigcap_{i=1}^{m}\\{x_{k_{i}}(t_{i})\geq
a_{i}\\}\right)=\oint\cdots\oint\mathscr{C}_{\vec{y}}^{(L)}(\vec{z})\mathscr{D}_{\vec{y}}^{(L)}(\vec{z})\frac{{\mathrm{d}}z_{1}}{2\pi\mathrm{i}z_{1}}\cdots\frac{{\mathrm{d}}z_{m}}{2\pi\mathrm{i}z_{m}}.$
(2.6)

Where we set $\vec{z}=(z_{1},\cdots,z_{m})$ and the contours are over nested
circles centered at the origin: $0<|z_{m}|<\cdots<|z_{1}|<\mathbbm{r}_{c}$.
Here and in all the remaining results we suppress the dependence of the
integrand on the parameters $a_{i},k_{i},t_{i}$ as well as the hopping
probability $0<p<1$. The function $\mathscr{C}_{\vec{y}}^{(L)}(\vec{z})$ is
defined in Definition 2.14. The function
$\mathscr{D}_{\vec{y}}^{(L)}(\vec{z})$ is a Fredholm determinant

$\mathscr{D}_{\vec{y}}^{(L)}(\vec{z})=\det(1-\mathscr{K}_{1}^{(L)}\mathscr{K}_{\vec{y}}^{(L)}),$
(2.7)

where the operators $\mathscr{K}_{1}^{(L)}$ and $\mathscr{K}_{\vec{y}}^{(L)}$
are defined in Definition 2.15.

######  Remark 2.3.

Theorem 2.2 generalizes Theorem 3.1 of [5] (and also Theorem 4.6 of [4] for
the special step initial condition) on the finite-time multi-point
distribution of continuous time periodic TASEP. In fact their formulas can be
obtained from our formula (2.6) by taking $p=\epsilon$, $\hat{t}=\epsilon t$
and letting $\epsilon\to 0$.

Next we state the theorem on the large-time, large-period scaling limit of
(2.6) under the relaxation time scale. To emphasize the initial condition in
the limit theorem, we add the subscript “$\mathrm{ic}$” for the terms in the
limit which depend on the initial conditions. We make the following choice of
the labeling for convenience: we assume that $x_{1}(0)\leq 0<x_{0}(0)$. This
is equivalent to assume that the initial condition satisfies $y_{1}\leq
0<y_{N}+L$.

######  Theorem 2.4 (Relaxation time limit).

Consider a sequence $\mathrm{dpTASEP}(L,N,\vec{y}(L))$ where
$\rho=\rho_{L}=N/L$ stays in a compact subset of $(0,1)$ and $y_{1}\leq
0<y_{N}+L$. Suppose that the sequence of initial conditions $\vec{y}(L)$
satisfies certain assumptions (see Assumption 6.1). Fix a positive integer $m$
and let $\mathrm{p}_{j}=(\gamma_{j},\tau_{j})$ be $m$ points in the region

$\mathrm{R}:=[0,1]\times\mathbb{Z}_{>0}$

satisfying

$0<\tau_{1}<\tau_{2}<\cdots<\tau_{m}.$

Then for every fixed $\mathrm{x}_{1},\cdots,\mathrm{x}_{m}\in\mathbb{R}$ and
parameters $a_{i},k_{i},t_{i}$, $1\leq i\leq m$ given by

$t_{i}=c_{1}\tau_{i}L^{3/2}+O(1),\quad a_{i}=c_{2}t_{i}+\gamma_{i}L+O(1),\quad
k_{i}=c_{3}t_{i}+c_{4}\gamma_{i}L+c_{5}\mathrm{x}_{i}L^{1/2}+O(1),$ (2.8)

we have

$\displaystyle\lim_{L\to\infty}\mathbb{P}^{(L)}_{\vec{y}}\left(\bigcap_{j=1}^{m}\left\\{x_{k_{i}}(t_{i})\geq
a_{i}\right\\}\right)=\mathbb{F}_{\mathrm{ic}}^{\mathrm{per}}(\mathrm{x}_{1},\cdots,\mathrm{x}_{m};\mathrm{p}_{1},\cdots,\mathrm{p}_{m}).$
(2.9)

Here the constants $c_{i}$ depend explicitly on particle density $0<\rho<1$
and hopping probability $0<p<1$ and are given by

$\displaystyle
c_{1}=\frac{1}{p(1-p)}\frac{\nu^{5/2}}{\rho^{1/2}(1-\rho)^{1/2}},\quad
c_{2}=\frac{p(1-2\rho)}{\nu},$ (2.10) $\displaystyle
c_{3}=\frac{2\rho^{2}\cdot p(1-p)}{\nu(1+\nu-2p\rho)},\quad c_{4}=-\rho,\quad
c_{5}=-\rho^{1/2}(1-\rho)^{1/2}\nu^{1/2},$

where we set $\nu:=\sqrt{1-4p\cdot\rho(1-\rho)}$ for convenience. We recall
that in (2.9) $\mathbb{P}^{(L)}$ denotes the probability associated to
$\mathrm{dpTASEP}(L,N,\vec{y}(L))$. The function
$\mathbb{F}_{\mathrm{ic}}^{\mathrm{per}}$ agrees with the one defined in
Section 6.4 of [5] as the relaxation time limit of multi-point distribution of
continuous time periodic TASEP. We recall the definition of
$\mathbb{F}_{\mathrm{ic}}^{\mathrm{per}}$ in Section 6.3 for completeness. The
convergence is locally uniform in $\mathrm{x}_{j},\tau_{j}$, and $\gamma_{j}$.

###  2.2. Multi-point distribution formulas for $\mathrm{dTASEP}(p)$

The finite-time multi-point joint distribution for discrete time parallel
TASEP on $\mathbb{Z}$ can be obtained from Theorem 2.2 as a corollary. The
rough idea is to take the period $L$ sufficiently large so that the finite-
time formula is unaffected by the boundary. The precise procedure is more
delicate and will be explained in Section 8. For simplicity we will only state
result on step initial condition but we point out that in [15] similar multi-
time formulas for continuous time TASEP was obtained for general initial
conditions and we believe their arguments can also be adapted to our model.
The main result is the following:

######  Theorem 2.5 (Finite-time multi-point joint distribution for
$\mathrm{dTASEP}(p)$).

Assume $\vec{y}=(-1,-2,\cdots,-N)$. Consider discrete time parallel TASEP on
$\mathbb{Z}$ with initial particle locations $x_{i}(0)=y_{i}$ for $1\leq i\leq
N$. Let $m\geq 1$ be a positive integer and
$(k_{1},t_{1}),\cdots,(k_{m},t_{m})$ be $m$ distinct points in
$\\{1,\cdots,N\\}\times\mathbb{N}$. Assume that $0\leq t_{1}\leq\cdots\leq
t_{m}$. Then, for any integers $a_{1},\cdots,a_{m}$,

$\mathbb{P}_{\mathrm{step}}^{(\infty)}\left(\bigcap_{\ell=1}^{m}\left\\{x_{k_{\ell}}(t_{\ell})\geq
a_{\ell}\right\\}\right)=\oint\cdots\oint\left[\prod_{\ell=1}^{m-1}\frac{1}{1-\theta_{\ell}}\right]\mathcal{D}_{\mathrm{step}}^{(\infty)}(\theta_{1},\cdots,\theta_{m-1})\frac{{\mathrm{d}}\theta_{1}}{2\pi\mathrm{i}\theta_{1}}\cdots\frac{{\mathrm{d}}\theta_{m-1}}{2\pi\mathrm{i}\theta_{m-1}}$
(2.11)

where the integral contours are circles centered at the origin of radii less
than $1$. The function
$\mathcal{D}_{\mathrm{step}}^{(\infty)}(\theta_{1},\cdots,\theta_{m-1})$ is
defined in terms of a Fredholm determinant in Definition 2.16.

######  Remark 2.6.

Under the standard correspondence between discrete time parallel TASEP and
geometric last passage percolation, we have the following equality in
distributions:

$\mathbb{P}_{\mathrm{step}}^{(\infty)}\left(\bigcap_{\ell=1}^{m}\left\\{x_{k_{\ell}}(t_{\ell})\geq
a_{\ell}\right\\}\right)=\mathbb{P}_{\mathrm{GLPP}}\left(\bigcap_{\ell=1}^{m}\left\\{G(k_{\ell},a_{\ell}+k_{\ell})\leq
t_{\ell}\right\\}\right),$ (2.12)

where $G(m,n)$ is the point-to-point last passage time from $(0,0)$ to $(m,n)$
in the usual geometric last passage percolation model with parameter $q=1-p$.
For the right-hand side of (2.12) a formula of the similar type as (2.11) was
obtained in Theorem 2 of [13]. We expect the two formulas to be equivalent but
currently we are not able to prove it for $m\geq 2$.

Starting from the finite-time formula, we can then take large-time limit under
the $1:2:3$ KPZ scaling.

######  Theorem 2.7.

Consider discrete time parallel TASEP on $\mathbb{Z}$ with step initial
condition $x_{i}(0)=-i$. We rescale the parameters such that

$t_{i}=\frac{2q^{\frac{1}{4}}}{1-q}\tau_{i}T+O(1),\quad
a=\gamma_{i}T^{2/3}+O(1),\quad
k_{i}=\frac{q^{\frac{1}{4}}}{1+\sqrt{q}}\tau_{i}T-\frac{1}{2}\gamma_{i}T^{2/3}-\frac{q^{\frac{1}{4}}}{2}\mathrm{x}_{i}T^{1/3}+O(1),$
(2.13)

where $q=1-p$. Then

$\displaystyle\lim_{T\to\infty}\mathbb{P}_{\mathrm{step}}^{(\infty)}\left(\bigcap_{\ell=1}^{m}\left\\{x_{k_{\ell}}(t_{\ell})\geq
a_{\ell}\right\\}\right)$
$\displaystyle=\mathbb{F}_{\mathrm{step}}^{\mathrm{kpz}}(\mathrm{x}_{1},\cdots,\mathrm{x}_{m};(\tau_{1},\gamma_{1}),\cdots,(\tau_{m},\gamma_{m}))$
(2.14)
$\displaystyle=\oint\cdots\oint\left[\prod_{\ell=1}^{m-1}\frac{1}{1-\theta_{\ell}}\right]\mathrm{D}_{\mathrm{step}}^{\mathrm{kpz}}(\theta_{1},\cdots,\theta_{m-1})\frac{{\mathrm{d}}\theta_{1}}{2\pi\mathrm{i}\theta_{1}}\cdots\frac{{\mathrm{d}}\theta_{m}}{2\pi\mathrm{i}\theta_{m}}.$

Here
$\mathrm{D}_{\mathrm{step}}^{\mathrm{kpz}}(\theta_{1},\cdots,\theta_{m-1})$ is
a Fredholm determinant defined in Definition 8.4. We remark that the right-
hand side of (2.14) agrees with the right-hand side of equation (2.19) in
[15], with the re-scaled parameters changed from
$(\tau_{i},\gamma_{i},\mathrm{x}_{i})$ to $(\tau_{i},2x_{i},h_{i})$. Our
parameters are chosen to be consistent with the periodic cases.

###  2.3. Bethe equations and Bethe roots

For our analysis on the discrete parallel periodic TASEP the following
polynomial and its roots play essential role:

######  Definition 2.8 (Bethe roots).

Given $z\in\mathbb{C}$ and $0<p<1$. Define the degree $L$ polynomial
$q_{z}(w)$ by

$q_{z}(w):=w^{N}(1+w)^{L-N}-z^{L}(1+pw)^{N}.$ (2.15)

We call this polynomial the Bethe polynomial associated to $z$ and its roots
Bethe roots. We denote the set of all roots of the Bethe polynomial $q_{z}(w)$
by $\mathcal{S}_{z}$:

$\mathcal{S}_{z}=\\{w\in\mathbb{C}:q_{z}(w)=0\\}.$ (2.16)

The Bethe roots $\mathcal{S}_{z}$ is contained in the level set
$\\{w\in\mathbb{C}:|w|^{\rho}|1+w|^{1-\rho}=|z|\cdot|1+pw|^{\rho}\\}$, which
is sometimes called a deformed Cassini oval. It is not hard to check that for
$|z|<\mathbbm{r}_{c}$, the level set consists of two disjoint contours while
for $|z|>\mathbbm{r}_{c}$ the two contours merge to a single contour. For
$z=\mathbbm{r}_{c}$ there is a self-intersecting point for the contour at
$w=w_{c}$. Here $\mathbbm{r}_{c}$ and $w_{c}$ are defined in (2.4) and (2.5).
See figure 1. We remark that the Bethe polynomials (and their roots) we are
considering here is a one-parameter generalization of the one considered in
[3, 4, 5] for continuous time periodic TASEP which corresponds to $p=0$
degeneration of (2.15),

Figure 1. The solid dots are roots for $q_{z}(w)$ with $L=16$, $N=4$, $p=5/6$
and $|z|=\frac{11}{10}\mathbbm{r}_{c},\mathbbm{r}_{c}\text{ and
}\frac{9}{10}\mathbbm{r}_{c}$ from outside to inside. The dashed lines are the
corresponding level sets.

######  Definition 2.9 (Left and right Bethe roots).

For $|z|<\mathbbm{r}_{c}$, we define the sets

$\mathcal{L}_{z}:=\\{w\in\mathcal{S}_{z}:\text{Re}(w)<w_{c}\\},\quad\mathcal{R}_{z}:=\\{w\in\mathcal{S}_{z}:\text{Re}(w)>w_{c}\\},$
(2.17)

where $w_{c}$ and $\mathbbm{r}_{c}$ are defined in (2.5) and (2.4). Then it is
straightforward to check that $|\mathcal{L}_{z}|=L-N$ and
$|\mathcal{R}_{z}|=N$. Roots in $\mathcal{L}_{z}$ and $\mathcal{R}_{z}$ are
called left and right Bethe roots, respectively. We also define the left and
right Bethe polynomials $q_{z,\mathrm{L}}(w)$ and $q_{z,\mathrm{R}}(w)$ as the
monic polynomials with roots in $\mathcal{L}_{z}$ and $\mathcal{R}_{z}$:

$q_{z,\mathrm{L}}(w):=\prod_{u\in\mathcal{L}_{z}}(w-u),\quad
q_{z,\mathrm{R}}(w):=\prod_{v\in\mathcal{R}_{z}}(w-v).$ (2.18)

Then by definition we have

$\mathcal{S}_{z}=\mathcal{L}_{z}\cup\mathcal{R}_{z},\quad
q_{z}(w)=q_{z,\mathrm{L}}(w)q_{z,\mathrm{R}}(w).$

###  2.4. A symmetric function related to initial conditions

In our finite-time multi-point distribution formula (2.6), the quantities
encoding information in the initial condition are all related to the following
symmetric function:

######  Definition 2.10 (Symmetric function).

Given $p\in\mathbb{C}$ and
$\lambda=(\lambda_{1},\cdots,\lambda_{N})\in\mathbb{Z}^{N}$ with
$\lambda_{1}\geq\cdots\geq\lambda_{N}$. We define

$\mathcal{F}_{\lambda}(w_{1},\cdots,w_{N};p):=\frac{\det\left[w_{j}^{N-i}(pw_{j}+1)^{i-1}(w_{j}+1)^{\lambda_{i}}\right]_{i,j=1}^{N}}{\det[w_{j}^{N-i}]_{i,j=1}^{N}}.$
(2.19)

######  Remark 2.11.

The symmetric function $\mathcal{F}_{\lambda}$ defined here is a one-parameter
generalization of the Grothendieck-like symmetric function defined in equation
(3.6) of [5] which corresponds to the $p=0$ degeneration in our situation.

The following two quantities related to $\mathcal{F}_{\lambda}$ encode the
initial condition:

######  Definition 2.12 (Global energy and characteristic function).

For $\vec{y}\in\mathcal{X}_{N}^{(L)}$, we set

$\lambda(\vec{y})=(y_{1}+1,y_{2}+2,\cdots,y_{N}+N).$ (2.20)

For $|z|<\mathbbm{r}_{c}$, we define the global energy
$\mathcal{E}_{\vec{y}}(z)$ by

$\mathcal{E}_{\vec{y}}(z):=\mathcal{F}_{\lambda(\vec{y})}(\mathcal{R}_{z};p).$
(2.21)

When $\mathcal{E}_{\vec{y}}(z)\neq 0$, we define the characteristic function
$\chi_{\vec{y}}(v,u;z)$ for a left Bethe root $u$ and a right Bethe root $v$
by

$\chi_{\vec{y}}(v,u;z):=\frac{\mathcal{F}_{\lambda(\vec{y})}(\mathcal{R}_{z}\cup\\{u\\}\backslash\\{v\\};p)}{\mathcal{F}_{\lambda(\vec{y})}(\mathcal{R}_{z};p)},\quad\text{for
}u\in\mathcal{L}_{z}\text{ and }v\in\mathcal{R}_{z}.$ (2.22)

######  Remark 2.13.

A straightforward calculation shows that for step initial condition
$\vec{y}=(-1,\cdots,-N)$, we have $\mathcal{F}_{\lambda(\vec{y})}\equiv 1$.
Hence the global energy function and characteristic function are both constant
$1$ for step initial condition. In general since the roots of the Bethe
polynomial $q_{z}(w)$ depends analytically on $z$, the function
$\mathcal{E}_{\vec{y}}(z)$ is analytic for $|z|<\mathbbm{r}_{c}$. Furthermore
$\mathcal{E}_{\vec{y}}(z)$ can not be constant zero. In fact
$\mathcal{E}_{\vec{y}}(0)=1$ since when $|z|\to 0$ all the right Bethe roots
converge to $0$. As a consequence
$\mathcal{F}_{\lambda(\vec{y})}(\mathcal{R}_{z};p)$ is nonzero for all but
finitely many $z$ in any compact subset of $\\{|z|<\mathbbm{r}_{c}\\}$, which
means $\chi_{\vec{y}}(v,u;z)$ is a well-defined meromorphic function in $z$ on
$\\{|z|<\mathbbm{r}_{c}\\}$ for fixed $u,v$.

###  2.5. Definition of $\mathscr{C}_{\vec{y}}^{(L)}(\vec{z})$ and
$\mathscr{D}_{\vec{y}}^{(L)}(\vec{z})$

######  Definition 2.14 (Definition of
$\mathscr{C}_{\vec{y}}^{(L)}(\vec{z})$).

With the global energy function $\mathcal{E}_{\vec{y}}(z)$ defined in (2.21),
we define

$\mathscr{C}_{\vec{y}}^{(L)}(\vec{z}):=\mathcal{E}_{\vec{y}}(z_{1})\mathscr{C}_{\text{step}}^{(L)}(\vec{z}),$
(2.23)

where

$\displaystyle\mathscr{C}_{\text{step}}^{(L)}(\vec{z})$
$\displaystyle:=\left[\prod_{\ell=1}^{m}\frac{E_{\ell}(z_{\ell})}{E_{\ell-1}(z_{\ell})}\right]\left[\prod_{\ell=1}^{m}\frac{\prod_{u\in\mathcal{L}_{z_{\ell}}}(-u)^{N}\prod_{v\in\mathcal{R}_{z_{\ell}}}(v+1)^{L-N}}{\Delta(\mathcal{R}_{z_{\ell}};\mathcal{L}_{z_{\ell}})}\right]$
(2.24)
$\displaystyle\cdot\left[\prod_{\ell=2}^{m}\frac{z_{\ell-1}^{L}}{z_{\ell-1}^{L}-z_{\ell}^{L}}\right]\left[\prod_{\ell=2}^{m}\frac{\Delta(\mathcal{R}_{z_{\ell}};\mathcal{L}_{z_{\ell-1}})}{\prod_{u\in\mathcal{L}_{z_{\ell-1}}}(-u)^{N}\prod_{v\in\mathcal{R}_{z_{\ell}}}(v+1)^{L-N}}\right].$

Here

$E_{\ell}(z):=\prod_{u\in\mathcal{L}_{z}}(-u)^{-k_{\ell}}\prod_{v\in\mathcal{R}_{z}}(v+1)^{-a_{\ell}-k_{\ell}}(pv+1)^{t_{\ell}-k_{\ell}},$
(2.25)

for $1\leq i\leq m$ and $E_{0}(z):=1$.

$\mathscr{D}_{\vec{y}}^{(L)}(\vec{z})$ is a Fredholm determinant with kernel
acting on certain $\ell^{2}$ space over discrete sets related to the Bethe
roots. More precisely for $m$ distinct complex numbers $z_{i}$ satisfying
$|z_{i}|<\mathbbm{r}_{c}$, we define the discrete sets

$\mathscr{S}_{1}:=\mathcal{L}_{z_{1}}\cup\mathcal{R}_{z_{2}}\cup\mathcal{L}_{z_{3}}\cup\cdots\cup\begin{cases}\mathcal{L}_{z_{m}},&\text{if
$m$ is odd},\\\ \mathcal{R}_{z_{m}},&\text{if $m$ is even},\end{cases}$ (2.26)

and

$\mathscr{S}_{2}:=\mathcal{R}_{z_{1}}\cup\mathcal{L}_{z_{2}}\cup\mathcal{R}_{z_{3}}\cup\cdots\cup\begin{cases}\mathcal{R}_{z_{m}},&\text{if
$m$ is odd},\\\ \mathcal{L}_{z_{m}},&\text{if $m$ is even}.\end{cases}$ (2.27)

######  Definition 2.15 (Definition of
$\mathscr{D}_{\vec{y}}^{(L)}(\vec{z})$).

Let $0<|z_{m}|<\cdots<|z_{1}|<\mathbbm{r}_{c}$. Assume
$\mathcal{E}_{\vec{y}}(z_{1})\neq 0$ so that $\chi_{\vec{y}}(v,u;z_{1})$ is
well defined. Define

$\mathscr{D}_{\vec{y}}^{(L)}(\vec{z})=\det(I-\mathscr{K}_{1}^{(L)}\mathscr{K}_{\vec{y}}^{(L)}),$
(2.28)

where
$\mathscr{K}_{1}^{(L)}:\ell^{2}(\mathscr{S}_{2})\to\ell^{2}(\mathscr{S}_{1})$
and
$\mathscr{K}_{\vec{y}}^{(L)}:\ell^{2}(\mathscr{S}_{1})\to\ell^{2}(\mathscr{S}_{2})$
have kernels given by

$\mathscr{K}_{1}^{(L)}(w,w^{\prime}):=\left(\delta_{i}(j)+\delta_{i}(j+(-1)^{i})\right)\frac{J(w)f_{i}(w)(H_{z_{i}}(w))^{2}}{H_{z_{i-(-1)^{i}}}(w)H_{z_{j-(-1)^{j}}}(w^{\prime})(w-w^{\prime})}Q_{1}(j)$
(2.29)

and

$\mathscr{K}_{\vec{y}}^{(L)}(w^{\prime},w):=\left(\delta_{j}(i)+\delta_{j}(i-(-1)^{j})\right)\frac{J(w^{\prime})f_{j}(w^{\prime})(H_{z_{j}}(w^{\prime}))^{2}}{H_{z_{j+(-1)^{j}}}(w^{\prime})H_{z_{i+(-1)^{i}}}(w)(w^{\prime}-w)}Q_{2}(i)\Lambda(i,w,w^{\prime}),$
(2.30)

for

$w\in(\mathcal{L}_{z_{i}}\cup\mathcal{R}_{z_{i}})\cap\mathscr{S}_{1}\quad\text{and}\quad
w^{\prime}\in(\mathcal{L}_{z_{j}}\cup\mathcal{R}_{z_{j}})\cap\mathscr{S}_{2}$

with $1\leq i,j\leq m$. Here

$\Lambda(i,w,w^{\prime})=\begin{cases}\chi_{\vec{y}}(w,w^{\prime};z_{1})\quad&\text{for
}i=1,\\\ 1\quad&\text{for }2\leq i\leq m.\end{cases}$ (2.31)

The functions $J(w)$ and $Q_{i}(j)$ are defined as follows:

$J(w):=\frac{w(w+1)(pw+1)}{N+Lw+p(L-N)w^{2}},$ (2.32)

and

$Q_{1}(j):=1-\left(\frac{z_{j-(-1)^{j}}}{z_{j}}\right)^{L},\quad
Q_{2}(j):=1-\left(\frac{z_{j+(-1)^{j}}}{z_{j}}\right)^{L},$ (2.33)

for $j=1,\cdots,m$. Here we set $z_{m+1}=0$ for convenience.

To define $H_{z}(w)$ and $f_{i}(w)$ we recall the definition of left and right
Bethe polynomials and Bethe roots discussed in Section 2.3. With the notation
there we set

$H_{z}(w):=\begin{cases}\frac{q_{z,\mathrm{R}}(w)}{w^{N}}\quad&\text{for
}\text{Re}(w)<w_{c},\\\ \frac{q_{z,\mathrm{L}}(w)}{(w+1)^{L-N}}\quad&\text{for
}\text{Re}(w)>w_{c}.\end{cases}$ (2.34)

Finally the functions $f_{i}(w)$ encodes the information of the parameters
$a_{i},k_{i},t_{i}$’s:

$f_{\ell}(w):=\begin{cases}\frac{F_{\ell}(w)}{F_{\ell-1}(w)}\quad\text{for
}\text{Re}(w)<w_{c},\\\ \frac{F_{\ell-1}(w)}{F_{\ell}(w)}\quad\text{for
}\text{Re}(w)>w_{c},\end{cases}$ (2.35)

where

$F_{\ell}(w):=w^{k_{\ell}}(1+w)^{-a_{\ell}-k_{\ell}}(1+pw)^{t_{\ell}-k_{\ell}},$
(2.36)

for $1\leq\ell\leq m$ and $F_{0}(w):=1$.

###  2.6. Definition of
$\mathcal{D}_{\mathrm{step}}^{(\infty)}(\theta_{1},\cdots,\theta_{m-1})$

The function
$\mathcal{D}_{\mathrm{step}}^{(\infty)}(\theta_{1},\cdots,\theta_{m-1})$ has a
similar structure as $\mathscr{D}_{\vec{y}}^{(L)}(\vec{z})$ defined in
Definition 2.15 and is also a Fredholm determinant of the form
$\det(I-\mathcal{K}_{1}^{(\infty)}\mathcal{K}_{\mathrm{step}}^{(\infty)})$.
The kernels also share certain similarities and we will illustrate the
relationship between the two in Section 8.

####  2.6.1. Spaces of the operators

Instead of discrete sets, the operators appearing in
$\mathcal{D}_{\mathrm{step}}^{(\infty)}(\theta_{1},\cdots,\theta_{m-1})$ are
defined on specific spaces of nested contours defined as follows: Let
$\Omega_{\mathrm{L}}$ and $\Omega_{\mathrm{R}}$ be two simply connected
regions on the complex plane such that

  1. (1)

$\Omega_{\mathrm{L}}\subset\\{w\in\mathbb{C}:\mathrm{Re}(w)<-\frac{1}{1+\sqrt{1-p}}\\}$
and $\Omega_{\mathrm{L}}$ contains the point $-1$ but not $-1/p$.

  2. (2)

$\Omega_{\mathrm{R}}\subset\\{w\in\mathbb{C}:\mathrm{Re}(w)>-\frac{1}{1+\sqrt{1-p}}\\}$
and $\Omega_{\mathrm{R}}$ contains the point $0$.

Let $\Sigma_{m,\mathrm{L}}^{+},\cdots,\Sigma_{2,\mathrm{L}}^{+}$,
$\Sigma_{1,\mathrm{L}}$,
$\Sigma_{2,\mathrm{L}}^{-},\cdots,\Sigma_{m,\mathrm{L}}^{-}$ be $2m-1$ nested
simple closed contours, from outside to inside, in $\Omega_{\mathrm{L}}$
enclosing the point $-1$ but not $-1/p$. Similarly,
$\Sigma_{m,\mathrm{R}}^{+},\cdots,\Sigma_{2,\mathrm{R}}^{+}$,
$\Sigma_{1,\mathrm{R}}$,
$\Sigma_{2,\mathrm{R}}^{-},\cdots,\Sigma_{m,\mathrm{R}}^{-}$ be $2m-1$ nested
simple closed contours, from outside to inside, in $\Omega_{\mathrm{R}}$
enclosing the point $0$. We define

$\Sigma_{\ell,\mathrm{L}}:=\Sigma_{\ell,\mathrm{L}}^{+}\cup\Sigma_{\ell,\mathrm{L}}^{-},\qquad\Sigma_{\ell,\mathrm{R}}:=\Sigma_{\ell,\mathrm{R}}^{+}\cup\Sigma_{\ell,\mathrm{R}}^{-},\qquad\Sigma_{\ell}=\Sigma_{\ell,\mathrm{L}}\cup\Sigma_{\ell,\mathrm{R}},\qquad\ell=2,\cdots,m,$

and

$\mathcal{S}_{1}:=\Sigma_{1,\mathrm{L}}\cup\Sigma_{2,\mathrm{R}}\cup\cdots\cup\begin{dcases}\Sigma_{m,\mathrm{L}},&\text{
if $m$ is odd},\\\ \Sigma_{m,\mathrm{R}},&\text{ if $m$ is even},\end{dcases}$

and

$\mathcal{S}_{2}:=\Sigma_{1,\mathrm{R}}\cup\Sigma_{2,\mathrm{L}}\cup\cdots\cup\begin{dcases}\Sigma_{m,\mathrm{R}},&\text{
if $m$ is odd},\\\ \Sigma_{m,\mathrm{L}},&\text{ if $m$ is even}.\end{dcases}$

####  2.6.2. Operators $\mathcal{K}_{1}^{(\infty)}$ and
$\mathcal{K}_{\mathrm{step}}^{\infty}$

Now we introduce the operators $\mathcal{K}_{1}^{(\infty)}$ and
$\mathcal{K}_{\mathrm{step}}^{(\infty)}$ to define
$\mathcal{D}_{\mathrm{step}}^{(\infty)}(\theta_{1},\cdots,\theta_{m-1})$ in
Theorem 2.5.

######  Definition 2.16.

We define

$\mathcal{D}_{\mathrm{step}}^{(\infty)}(\theta_{1},\cdots,\theta_{m-1})=\det\left(I-\mathcal{K}_{1}^{(\infty)}\mathcal{K}_{\mathrm{step}}^{(\infty)}\right),$

where two operators

$\mathcal{K}_{1}^{(\infty)}:L^{2}(\mathcal{S}_{2})\to
L^{2}(\mathcal{S}_{1}),\qquad\mathcal{K}_{\mathrm{step}}^{(\infty)}:L^{2}(\mathcal{S}_{1})\to
L^{2}(\mathcal{S}_{2})$

are defined by the kernels

$\mathcal{K}_{1}^{(\infty)}(w,w^{\prime}):=\left(\delta_{i}(j)+\delta_{i}(j+(-1)^{i})\right)\frac{f_{i}(w)}{w-w^{\prime}}Q_{1}^{(\infty)}(j)P_{j}(w^{\prime})\quad\text{for
}w^{\prime}\in\Sigma_{j}\cap\mathcal{S}_{2},\
w\in\Sigma_{i}\cap\mathcal{S}_{1},$ (2.37)

and

$\mathcal{K}_{\mathrm{step}}^{(\infty)}(w^{\prime},w):=\left(\delta_{j}(i)+\delta_{j}(i-(-1)^{j})\right)\frac{f_{j}(w^{\prime})}{w^{\prime}-w}Q_{2}^{(\infty)}(i)P_{i}(w)\quad\text{for
}w\in\Sigma_{i}\cap\mathcal{S}_{1},\
w^{\prime}\in\Sigma_{j}\cap\mathcal{S}_{2},$ (2.38)

where $1\leq i,j\leq m$. Here the function $f_{i}(w)$ is the same as the one
appearing in the kernels for periodic case defined in (2.35) and we define

$Q_{1}^{(\infty)}(j):=\begin{cases}1-\theta_{j},\quad&\text{if $j$ is odd and
$j<m$},\\\ 1-\theta_{j-1}^{-1},\quad&\text{if $j$ is even},\\\ 1,&\text{if
$j=m$ is even or $j=1$}.\end{cases}$ (2.39)

and

$Q_{2}^{(\infty)}(j):=\begin{cases}1-\theta_{i},\quad&\text{if $j$ is even and
$j<m$},\\\ 1-\theta_{j-1}^{-1},\quad&\text{if $j$ is odd and $j>1$},\\\
1,&\text{if $j=m$ is odd}.\end{cases}$ (2.40)

We also define

$P_{j}(w):=\begin{cases}\frac{1}{1-\theta_{j-1}},\quad&w\in\Sigma_{j,\mathrm{L}}^{+}\cup\Sigma_{j,\mathrm{R}}^{+},\quad
j=2,\cdots,m\\\
\frac{-\theta_{j-1}}{1-\theta_{j-1}},\quad&w\in\Sigma_{j,\mathrm{L}}^{-}\cup\Sigma_{j,\mathrm{R}}^{-},\quad
j=2,\cdots,m\\\
1,\quad&w\in\Sigma_{1,\mathrm{L}}\cup\Sigma_{1,\mathrm{R}}.\end{cases}$ (2.41)

##  3\. Transition probability

In this section we give an explicit integral formula for the transition
probability of discrete time parallel TASEP in the configuration space
$\mathcal{X}_{N}^{(L)}$. This is the starting point for deriving the finite-
time joint distribution formulas.

######  Proposition 3.1.

Given two particle configurations $\vec{x}=(x_{1},\cdots,x_{N}),\
\vec{y}=(y_{1},\cdots,y_{N})\in\mathcal{X}_{N}^{(L)}$. Let
$P_{t}(\vec{y}\to\vec{x})$ be the transition probability of observing
configuration $\vec{x}$ at time $t$ under the discrete time periodic TASEP
dynamics with initial configuration $\vec{y}$. With the convention
$x_{0}:=x_{N}+L$ we have

$P_{t}(\vec{y}\to\vec{x})=\prod_{i=1}^{N}(1-p\mathbf{1}_{x_{i-1}-x_{i}=1})\oint_{\Gamma}\frac{{\mathrm{d}}z}{2\pi\mathrm{i}z}\det\left[\sum_{w\in\mathcal{S}_{z}}F_{i,j}(w;\vec{x},\vec{y},t)J(w)\right]_{i,j=1}^{N}.$
(3.1)

Here $\Gamma$ is any simple closed contour with $0$ inside and
$\mathcal{S}_{z}$ consists of all the roots of the degree $L$ polynomial
$q_{z}(w):=w^{N}(w+1)^{L-N}-z^{L}(1+pw)^{N}$, i.e.,

$\mathcal{S}_{z}:=\\{w\in\mathbb{C}:w^{N}(w+1)^{L-N}-z^{L}(1+pw)^{N}=0\\}.$
(3.2)

The functions $F_{i,j}(w;t)$ and $J(w)$ are given by

$F_{i,j}(w;\vec{x},\vec{y},t)=w^{j-i}(w+1)^{-x_{N-i+1}+y_{N-j+1}+i-j-1}(1+pw)^{t+i-j},\quad
1\leq i,j\leq N,$ (3.3)

and

$J(w):=\frac{w(w+1)(1+pw)}{N+Lw+p(L-N)w^{2}}.$ (3.4)

######  Remark 3.2.

We remark that a different formula for the transition probability of discrete
time parallel TASEP on a ring was obtained in [18]. The key difference is the
formula in [18] is expressed as an infinite sum of determinants while our
formula is a single contour integral of determinants. The main reason for this
is in [18] the authors do not distinguish particle configurations differ by a
translation of an integer multiple of the period, so in our language their
transition probability really is

$P_{t}([\vec{y}]\to[\vec{x}])=\sum_{k\in\mathbb{Z}}P_{t}(\vec{y}\to\vec{x}+(kL,kL,\cdots,kL)).$

We believe our formula is simpler and more suitable for deriving finite-time
joint distributions.

######  Remark 3.3.

If we take the continuous time limit by setting $p=\epsilon$, $t=T/\epsilon$
and send $\epsilon\to 0$, the dynamics then becomes continuous time periodic
TASEP considered in [3] and our formula (3.1) reduces to equation (5.4) in [3]
for the transition probability of continuous time periodic TASEP up to an
index reversing.

We first list a few elementary properties of the transition probability
formula (3.1) before discussing the proof of proposition 3.1.

######  Proposition 3.4 (Properties of the transition probability formula).

The right hand side of (3.1) satisfies the following properties:

  1. (i)

The right hand side of (3.1) can also be written as:

$\displaystyle\prod_{i=1}^{N}(1-p\mathbf{1}_{x_{i-1}-x_{i}=1})\oint_{\Gamma}\frac{{\mathrm{d}}z}{2\pi\mathrm{i}z}\det\left[\oint_{\Gamma_{\mathcal{S}_{z}}}\frac{{\mathrm{d}}w}{2\pi\mathrm{i}}F_{i,j}(w;\vec{x},\vec{y},t)\left(1+\frac{z^{L}}{\hat{q}_{z}(w)}\right)\right]_{i,j=1}^{N},$
(3.5)

where $\hat{q}_{z}(w)=q_{z}(w)(1+pw)^{-N}=w^{N}(1+pw)^{-N}(1+w)^{L-N}-z^{L}$
has the same roots as $q_{z}(w)$ for any $|z|>0$ and
$\Gamma_{\mathcal{S}_{z}}$ is any simple closed contour with all the roots in
$\mathcal{S}_{z}$ inside and $-1$ and $-1/p$ outside.

  2. (ii)

The outer integral with respect to $z$ in (3.1) (and also (3.5)) does not
depend on the contour $\Gamma$.

  3. (iii)

Assume further that $L\geq x_{1}-y_{N}+2$. Then the right-hand side of (3.1)
can be further written as

$\prod_{i=1}^{N}(1-p\mathbf{1}_{x_{i-1}-x_{i}=1})\det\left[\oint_{\Gamma_{0,-1}}\frac{{\mathrm{d}}w}{2\pi\mathrm{i}}F_{i,j}(w;\vec{x},\vec{y},t)\right]_{i,j=1}^{N},$
(3.6)

where $\Gamma_{0,-1}$ is any simple closed contour enclosing $0$ and $-1$ as
the only possible poles for the integrand. Note that (3.6) agrees with the
transition probability for discrete time parallel TASEP on $\mathbb{Z}$, see
for example equation (3.21) of [9].

  4. (iv)

The right-hand side of (3.1) is invariant under cyclic translation and re-
indexing. Namely, for any fixed $1\leq k\leq N$, set
$\vec{x}^{\prime}:=(x_{k},x_{k+1},\cdots,x_{N},x_{1}-L,x_{2}-L,\cdots,x_{k-1}-L)$
and
$\vec{y}^{\prime}:=(y_{k},y_{k+1},\cdots,y_{N},y_{1}-L,y_{2}-L,\cdots,y_{k-1}-L)$.
Then the right-hand side of (3.1) is invariant if we replace $\vec{x}$ and
$\vec{y}$ by $\vec{x}^{\prime}$ and $\vec{y}^{\prime}$.

###### Proof.

  1. (i)

It is easy to check that
$J(w)=\frac{\hat{q}_{z}(w)+z^{L}}{\frac{{\mathrm{d}}}{{\mathrm{d}}w}\hat{q}_{z}(w)}$.
Hence by residue theorem we have

$\sum_{w\in\mathcal{S}_{z}}F_{i,j}(w;\vec{x},\vec{y},t)J(w)=\oint_{\Gamma_{\mathcal{R}_{z}}}\frac{{\mathrm{d}}w}{2\pi\mathrm{i}}F_{i,j}(w;\vec{x},\vec{y},t)\frac{\hat{q}_{z}(w)+z^{L}}{\hat{q}_{z}(w)}.$

Note that
$F_{i,j}(w,\vec{x},\vec{y},t)\frac{\hat{q}_{z}(w)+z^{L}}{\hat{q}_{z}(w)}$ is
analytic at $w=0$ for any $1\leq i,j\leq N$ and the only possible poles
besides $\mathcal{R}_{z}$ are $w=-1$ and $w=-1/p$.

  2. (ii)

Choose $R>0$ large enough and $\epsilon>0$ small enough so that all the roots
in $\mathcal{S}_{z}$ are inside the region
$\\{\epsilon<|w+1|<R\\}\backslash\\{|1+pw|\leq\epsilon\\}$. Then

$\sum_{w\in\mathcal{S}_{z}}F_{i,j}(w;\vec{x},\vec{y},t)J(w)=\oint_{|w+1|=R}-\oint_{|w+1|=\epsilon}-\oint_{|pw+1|=\epsilon}\frac{{\mathrm{d}}w}{2\pi\mathrm{i}}F_{i,j}(w;\vec{x},\vec{y},t)\frac{\hat{q}_{z}(w)+z^{L}}{\hat{q}_{z}(w)}.$
(3.7)

Since $R$ and $\epsilon$ can be arbitrarily large or small, the right hand
side of (3.7) as a function in $z$ is analytic for any $|z|>0$. Hence the
integral with respect to $z$ in (3.1) is independent of $\Gamma$ since the
integrand has an analytic continuation to $\\{|z|>0\\}$.

  3. (iii)

For $L>x_{1}-y_{N}+2$, we have $-x_{N-i+1}+y_{N-j+1}+i-j+L-N-1\geq 0$ for all
$1\leq i,j\leq N$. Hence for any $|z|>0$, the integrand
$F_{i,j}(w;\vec{x},\vec{y},t)\frac{\hat{q}_{z}(w)+z^{L}}{\hat{q}_{z}(w)}$ in
(3.7) is analytic at $w=-1$, which implies

$\sum_{w\in\mathcal{S}_{z}}F_{i,j}(w;\vec{x},\vec{y},t)J(w)=\oint_{|w+1|=R}-\oint_{|pw+1|=\epsilon}\frac{{\mathrm{d}}w}{2\pi\mathrm{i}}F_{i,j}(w;\vec{x},\vec{y},t)\frac{\hat{q}_{z}(w)+z^{L}}{\hat{q}_{z}(w)}.$
(3.8)

Now for fixed $R$ large enough and $\epsilon$ small enough, the right-hand
side of (3.8) is an analytic function in $z$ for $|z|$ sufficiently small such
that all the roots of $\hat{q}_{z}$ are in the region $\\{|w+1|\leq
R\\}\backslash\\{|pw+1|\leq\epsilon\\}$. Now by residue theorem the outer
contour integral with respect to $z$ in (3.5) equals the integrand evaluated
at $z=0$, which equals

$\det\left[\oint_{|w+1|=R}-\oint_{|pw+1|=\epsilon}\frac{{\mathrm{d}}w}{2\pi\mathrm{i}}F_{i,j}(w;\vec{x},\vec{y},t)\right]_{i,j=1}^{N}.$

  4. (iv)

We remark that this property can be easily understood if we use the
probabilistic interpretation since $\vec{x}$ and $\vec{x}^{\prime}$ (and also
$\vec{y}$ and $\vec{y}^{\prime}$) actually represent the same particle
configuration on $\mathbb{Z}$ (we just use particles in different period as
representatives), hence the transition probability between $\vec{y}$ and
$\vec{x}$ and $\vec{y}^{\prime}$ and $\vec{x}^{\prime}$ should be the same.
Here however we can not directly use this since we have not proven (3.1). In
fact, we will need this fact in our proof of (3.1) hence we give an
independent algebraic proof here. It suffices to assume $k=2$ and hence
$\vec{x}^{\prime}=(x_{2},\cdots,x_{N},x_{1}-L)$,
$\vec{y}^{\prime}=(y_{2},\cdots,y_{N},y_{1}-L)$. Clearly we have
$\prod_{i=1}^{N}(1-p\mathbf{1}_{x_{i-1}-x_{i}}=1)=\prod_{i=1}^{N}(1-p\mathbf{1}_{x^{\prime}_{i-1}-x^{\prime}_{i}}=1)$.
So it suffices to show

$\det\left[\sum_{w\in\mathcal{S}_{z}}F_{i,j}(w;\vec{x},\vec{y},t)J(w)\right]_{i,j=1}^{N}=\det\left[\sum_{w\in\mathcal{S}_{z}}\tilde{F}_{i,j}(w;\vec{x},\vec{y},t)J(w)\right]_{i,j=1}^{N},$

where
$\tilde{F}_{i,j}(w;\vec{x},\vec{y},t)=w^{j-i}(w+1)^{-x^{\prime}_{N-i+1}+y^{\prime}_{N-j+1}+i-j-1}(1+pw)^{t+i-j}$.
By multi-linearity we have

$\displaystyle\det\left[\sum_{w\in\mathcal{S}_{z}}F_{i,j}(w;\vec{x},\vec{y},t)J(w)\right]_{i,j=1}^{N}=\sum_{w_{1},\cdots,w_{N}\in\mathcal{S}_{z}}\det\left[F_{i,j}(w_{i};\vec{x},\vec{y},t)J(w_{i})\right]_{i,j=1}^{N}$
$\displaystyle=\sum_{w_{1},\cdots,w_{N}\in\mathcal{S}_{z}}\sum_{\sigma\in
S_{N}}\text{sgn}(\sigma)\prod_{i=1}^{N}w_{i}^{\sigma(i)-i}(w_{i}+1)^{-x_{N-i+1}+y_{N-\sigma(i)+1}+i-\sigma(i)-1}(1+pw_{i})^{t+i-\sigma(i)}$
$\displaystyle=\sum_{w_{1},\cdots,w_{N}\in\mathcal{S}_{z}}\sum_{\sigma\in
S_{N}}\text{sgn}(\sigma)\prod_{i=1}^{N}w_{i}^{\sigma(i)-i}(w_{i}+1)^{-x^{\prime}_{N-i}+y^{\prime}_{N-\sigma(i)}+i-\sigma(i)-1}(1+pw_{i})^{t+i-\sigma(i)}.$

Here again $x^{\prime}_{0}=x^{\prime}_{N}+L=x_{1}-L+L=x_{1}$. Now we fix
$\tau=(N\cdots 21)\in S_{N}$ and set $\tilde{w}_{i}:=w_{\tau(i)}$ for $1\leq
i\leq N$ and $\tilde{\sigma}:=\tau^{-1}\sigma\tau$. Then the last line in the
above equation equals

$\sum_{\tilde{w}_{1},\cdots,\tilde{w}_{N}\in\mathcal{R}_{z}}\sum_{\tilde{\sigma}\in
S_{N}}\text{sgn}(\tilde{\sigma})\prod_{i=1}^{N}\tilde{w}_{i}^{\tau\tilde{\sigma}(i)-\tau(i)}(\tilde{w}_{i}+1)^{-x^{\prime}_{N-\tau(i)}+y^{\prime}_{N-\tau\tilde{\sigma}(i)}+\tau(i)-\tau\tilde{\sigma}(i)-1}(1+p\tilde{w}_{i})^{t+\tau(i)-\tau\tilde{\sigma}(i)}.$
(3.9)

We claim that for any fixed $\tilde{\sigma}\in S_{N}$,

$\displaystyle\prod_{i=1}^{N}\tilde{w}_{i}^{\tau\tilde{\sigma}(i)-\tau(i)}(\tilde{w}_{i}+1)^{-x^{\prime}_{N-\tau(i)}+y^{\prime}_{N-\tau\tilde{\sigma}(i)}+\tau(i)-\tau\tilde{\sigma}(i)-1}(1+p\tilde{w}_{i})^{t+\tau(i)-\tau\tilde{\sigma}(i)}$
$\displaystyle=\prod_{i=1}^{N}\tilde{w}_{i}^{\tilde{\sigma}(i)-i}(\tilde{w}_{i}+1)^{-x^{\prime}_{N-i+1}+y^{\prime}_{N-\tilde{\sigma}(i)+1}+i-\tilde{\sigma}(i)-1}(1+p\tilde{w}_{i})^{t+i-\tilde{\sigma}(i)}.$

This then implies

$\displaystyle\det\left[\sum_{w\in\mathcal{S}_{z}}F_{i,j}(w;\vec{x},\vec{y},t)J(w)\right]_{i,j=1}^{N}$
$\displaystyle=\sum_{\tilde{w}_{1},\cdots,\tilde{w}_{N}\in\mathcal{S}_{z}}\sum_{\tilde{\sigma}\in
S_{N}}\text{sgn}(\tilde{\sigma})\prod_{i=1}^{N}\tilde{w}_{i}^{\tilde{\sigma}(i)-i}(\tilde{w}_{i}+1)^{-x^{\prime}_{N-i+1}+y^{\prime}_{N-\tilde{\sigma}(i)+1}+i-\tilde{\sigma}(i)-1}(1+p\tilde{w}_{i})^{t+i-\tilde{\sigma}(i)}$
$\displaystyle=\det\left[\sum_{w\in\mathcal{S}_{z}}\tilde{F}_{i,j}(w;\vec{x},\vec{y},t)J(w)\right]_{i,j=1}^{N}.$

To see the claim note that for $i\neq 1$ we have $\tau(i)=i-1$ and for
$\tilde{\sigma}(i)\neq 1$ we have $\tau\tilde{\sigma}(i)=\tilde{\sigma}(i)-1$.
Hence

$\displaystyle\tilde{w}_{i}^{\tau\tilde{\sigma}(i)-\tau(i)}(\tilde{w}_{i}+1)^{-x^{\prime}_{N-\tau(i)}+y^{\prime}_{N-\tau\tilde{\sigma}(i)}+\tau(i)-\tau\tilde{\sigma}(i)-1}(1+p\tilde{w}_{i})^{t+\tau(i)-\tau\tilde{\sigma}(i)}$
$\displaystyle=\tilde{w}_{i}^{\tilde{\sigma}(i)-i}(\tilde{w}_{i}+1)^{-x^{\prime}_{N-i+1}+y^{\prime}_{N-\tilde{\sigma}(i)+1}+i-\tilde{\sigma}(i)-1}(1+p\tilde{w}_{i})^{t+i-\tilde{\sigma}(i)},$

for $i\neq 1$ and $\tilde{\sigma}(i)\neq 1$. For the other situations we split
into two cases:

Case 1: $\tilde{\sigma}(1)\neq 1$. Then we have

$\displaystyle\tilde{w}_{1}^{\tau\tilde{\sigma}(1)-\tau(1)}(\tilde{w}_{1}+1)^{-x^{\prime}_{N-\tau(1)}+y^{\prime}_{N-\tau\tilde{\sigma}(1)}+\tau(1)-\tau\tilde{\sigma}(1)-1}(1+p\tilde{w}_{1})^{t+\tau(1)-\tau\tilde{\sigma}(1)}$
$\displaystyle=\tilde{w}_{1}^{\tilde{\sigma}(1)-1}(\tilde{w}_{1}+1)^{-x^{\prime}_{N}+y^{\prime}_{N-\tilde{\sigma}(1)+1}-\tilde{\sigma}(1)}(1+p\tilde{w}_{i})^{t+1-\tilde{\sigma}(1)}\cdot\tilde{w}_{1}^{-N}(\tilde{w}_{1}+1)^{N-L}(1+p\tilde{w}_{1})^{N}$
$\displaystyle=\tilde{w}_{1}^{\tilde{\sigma}(1)-1}(\tilde{w}_{1}+1)^{-x^{\prime}_{N}+y^{\prime}_{N-\tilde{\sigma}(1)+1}-\tilde{\sigma}(1)}(1+p\tilde{w}_{i})^{t+1-\tilde{\sigma}(1)}\cdot
z^{-L}.$

Similarly if we set $\tilde{\sigma}^{-1}(1)=j$, then the term involving
$\tilde{w}_{j}$ in (3.9) equals

$\displaystyle\tilde{w}_{j}^{\tau(1)-\tau(j)}(\tilde{w}_{j}+1)^{-x^{\prime}_{N-\tau(j)}+y^{\prime}_{N-\tau(1)}+\tau(j)-\tau(1)-1}(1+p\tilde{w}_{j})^{t+\tau(j)-\tau(1)}$
$\displaystyle=\tilde{w}_{j}^{\tilde{\sigma}(j)-j}(\tilde{w}_{1}+1)^{-x^{\prime}_{N-j+1}+y^{\prime}_{N-\tilde{\sigma}(j)+1}-\tilde{\sigma}(j)+j-1}(1+p\tilde{w}_{j})^{t+j-\tilde{\sigma}(j)}\cdot\tilde{w}_{j}^{N}(\tilde{w}_{j}+1)^{L-N}(1+p\tilde{w}_{j})^{-N}$
$\displaystyle=\tilde{w}_{j}^{\tilde{\sigma}(j)-j}(\tilde{w}_{1}+1)^{-x^{\prime}_{N-j+1}+y^{\prime}_{N-\tilde{\sigma}(j)+1}-\tilde{\sigma}(j)+j-1}(1+p\tilde{w}_{j})^{t+j-\tilde{\sigma}(j)}\cdot
z^{L}.$

Hence

$\displaystyle\prod_{i=1}^{N}\tilde{w}_{i}^{\tau\tilde{\sigma}(i)-\tau(i)}(\tilde{w}_{i}+1)^{-x^{\prime}_{N-\tau(i)}+y^{\prime}_{N-\tau\tilde{\sigma}(i)}+\tau(i)-\tau\tilde{\sigma}(i)-1}(1+p\tilde{w}_{i})^{t+\tau(i)-\tau\tilde{\sigma}(i)}$
$\displaystyle=z^{L}\cdot
z^{-L}\cdot\prod_{i=1}^{N}\tilde{w}_{i}^{\tilde{\sigma}(i)-i}(\tilde{w}_{i}+1)^{-x^{\prime}_{N-i+1}+y^{\prime}_{N-\tilde{\sigma}(i)+1}+i-\tilde{\sigma}(i)-1}(1+p\tilde{w}_{i})^{t+i-\tilde{\sigma}(i)}$
$\displaystyle=\prod_{i=1}^{N}\tilde{w}_{i}^{\tilde{\sigma}(i)-i}(\tilde{w}_{i}+1)^{-x^{\prime}_{N-i+1}+y^{\prime}_{N-\tilde{\sigma}(i)+1}+i-\tilde{\sigma}(i)-1}(1+p\tilde{w}_{i})^{t+i-\tilde{\sigma}(i)}.$

Case 2: $\tilde{\sigma}(1)=1$. Then we have

$\displaystyle\tilde{w}_{1}^{\tau\tilde{\sigma}(1)-\tau(1)}(\tilde{w}_{1}+1)^{-x^{\prime}_{N-\tau(1)}+y^{\prime}_{N-\tau\tilde{\sigma}(1)}+\tau(1)-\tau\tilde{\sigma}(1)-1}(1+p\tilde{w}_{1})^{t+\tau(1)-\tau\tilde{\sigma}(1)}$
$\displaystyle=\tilde{w}_{1}^{\tilde{\sigma}(1)-1}(\tilde{w}_{1}+1)^{-x^{\prime}_{N}+y^{\prime}_{N-\tilde{\sigma}(1)+1}-\tilde{\sigma}(1)}(1+p\tilde{w}_{i})^{t+1-\tilde{\sigma}(1)},$

so the claim follows.

∎

Now we turn to the proof of formula (3.1). The proof basically follows the
idea of [9, 3] where we replace the Kolmogorov forward equation by a free
evolution equation with additional boundary conditions. The main difficulty
here is due to parallel update rule, the stationary distribution for the
dynamics is non-uniform. In fact one can check
$\mu(\vec{x})\propto\prod_{i=1}^{N}(1-p\mathbf{1}_{x_{i-1}-x_{i}=1})$ is the
stationary distribution. As a result it turns out that
$\frac{P_{t}(\vec{y}\to\vec{x})}{\prod_{i=1}^{N}(1-p\mathbf{1}_{x_{i-1}-x_{i}=1})}$
satisfies a relatively simpler dynamics than $P_{t}(\vec{y}\to\vec{x})$. In
fact we have

######  Lemma 3.5.

Given $\vec{x},\vec{y}\in\mathbb{Z}^{N}$. Let $G(\vec{x},t;\vec{y},0)$ be the
(unique) solution of the following free evolution equation

$G(\vec{x},t+1;\vec{y},0)=\sum_{b_{1},\cdots,b_{N}\in\\{0,1\\}}\prod_{i=1}^{N}p^{b_{i}}(1-p)^{1-b_{i}}G(\vec{x}-\vec{b},t;\vec{y},0),$
(3.10)

together with the boundary conditions:

$\displaystyle(1-p)\left[G(\cdots,x_{i-1}=x_{i}+1,x_{i},x_{i+1},\cdots,t;\vec{y},0)-G(\cdots,x_{i-1}=x_{i},x_{i},x_{i+1},\cdots,t;\vec{y},0)\right]$
(3.11)
$\displaystyle=p\left[G(\cdots,x_{i},x_{i}-1,x_{i+1},\cdots,t;\vec{y},0)-G(\cdots,x_{i}+1,x_{i}-1,x_{i+1},\cdots,t;\vec{y},0)\right],\text{
for }1\leq i\leq N,$

and initial condition:

$G(\vec{x},0;\vec{y},0)=\frac{\mathbf{1}_{\vec{x}=\vec{y}}}{\prod_{i=1}^{N}(1-p\mathbf{1}_{x_{i-1}-x_{1}=1})}.$
(3.12)

Then

$G(\vec{x},t;\vec{y},0)=\frac{P_{t}(\vec{y}\to\vec{x})}{\prod_{i=1}^{N}(1-p\mathbf{1}_{x_{i-1}-x_{1}=1})},\quad\text{for
all }\vec{x},\vec{y}\in\mathcal{X}_{N}^{(L)}.$ (3.13)

Note that we used the convention $x_{0}=x_{N}+L$ so when $i=1$, (3.11) should
be interpreted as

$\displaystyle(1-p)\left[G(x_{1},\cdots,x_{1}-L+1,t;\vec{y},0)-G(x_{1},\cdots,x_{1}-L,t;\vec{y},0)\right]$
(3.14)
$\displaystyle=p\left[G(x_{1}-1,\cdots,x_{1}-L,t;\vec{y},0)-G(x_{1}-1,\cdots,x_{1}-L+1,t;\vec{y},0)\right].$

###### Proof.

Set

$H(\vec{x},t;\vec{y},0):=\frac{P_{t}(\vec{y}\to\vec{x})}{\prod_{i=1}^{N}(1-p\mathbf{1}_{x_{i-1}-x_{i}=1})}.$
(3.15)

It suffices to show that for $\vec{x},\vec{y}\in\mathcal{X}_{N}^{(L)}$,
$H(\vec{x},t;\vec{y},0)$ and $G(\vec{x},t;\vec{y},0)$ satisfy the same
evolution equation to conclude that
$H(\vec{x},t;\vec{y},0)=G(\vec{x},t;\vec{y},0)$ for all
$\vec{x},\vec{y}\in\mathcal{X}_{N}^{(L)}$ since they have the same initial
condition. To better describe the evolution equation for
$H(\vec{x},t;\vec{y},0)$ it is convenient to introduce the notion of clusters
of a particle configuration $\vec{x}$. Given
$\vec{x}=(x_{1},\cdots,x_{N})\in\mathcal{X}_{N}^{(L)}$, for $1\leq i\leq N$
and $1\leq k\leq N$ we say $[x_{i},x_{i-1},\cdots,x_{i-k+1}]$ is a cluster of
size $k$ of $\vec{x}$ if

$x_{i+1}+1<x_{i}=x_{i-1}-1=\cdots=x_{i-k+1}-k+1<x_{i-k}-k,$

namely particle $i$ through $i-k+1$ are right next to each other while there
are at least one empty site to the left of $x_{i}$ and right of $x_{i-k+1}$.
Here we abuse notation by allowing the index to exceed $\\{1,\cdots,N\\}$ and
this should be understood with the convention $x_{i+kN}=x_{i}-kL$ for $1\leq
i\leq N$ and $k\in\mathbb{Z}$. For convenience when
$[x_{i},x_{i-1},\cdots,x_{1},x_{0},\cdots x_{-j}]$ is a cluster for some
$0\leq j<N$, we will also say $[x_{i},\cdots,x_{1},x_{N},\cdots,x_{N-j}]$
forms a cluster so that all the indices appearing will be between $1$ and $N$.

Let $\mathcal{N}_{c}(\vec{x})$ be the number of clusters in configuration
$\vec{x}$ and let $x_{c_{j}}$, $1\leq j\leq\mathcal{N}_{c}(\vec{x})$ be the
locations of the left-most particles in each cluster. Then it is
straightforward to check that $H(\vec{x},t;\vec{y},0)$ satisfies

$H(\vec{x},t+1;\vec{y},0)=\sum_{\begin{subarray}{c}b_{c_{j}}\in\\{0,1\\},\\\
1\leq
j\leq\mathcal{N}_{c}(\vec{x})\end{subarray}}\prod_{j=1}^{\mathcal{N}_{c}(\vec{x})}p^{b_{c_{j}}}(1-p)^{1-b_{c_{j}}}H(\vec{x}-\sum_{j=1}^{\mathcal{N}_{c}(\vec{x})}b_{c_{j}}\vec{e}_{c_{j}},t;\vec{y},0),$
(3.16)

where $\vec{e}_{c_{j}}\in\mathbb{Z}^{N}$ has $1$ in the $c_{j}$-th coordinate
and $0$ in the other coordinates.

We claim that for $\vec{x},\vec{y}\in\mathcal{X}_{N}^{(L)}$, (3.16) and (3.10)
takes the same form (with $H$ replaced by $G$) provided
$G(\vec{x},t;\vec{y},0)$ satisfies boundary conditions (3.11). Due to the sum
of products form of (3.16) and (3.10) it suffices to check

$\sum_{b_{i}\in\\{0,1\\}}p^{b_{i}}(1-p)^{1-b_{i}}G(\vec{x}-b_{i}\vec{e}_{i},t;\vec{y},0)=\sum_{\begin{subarray}{c}b_{j}\in\\{0,1\\}\\\
j=i-m+1,\cdots,i\end{subarray}}\prod_{j}p^{b_{j}}(1-p)^{1-b_{j}}G(\vec{x}-\sum_{j}b_{j}\vec{e}_{j},t;\vec{y},0),$
(3.17)

for $\vec{x},\vec{y}\in\mathcal{X}_{N}^{(L)}$ and a single cluster
$[x_{i},x_{i-1},\cdots,x_{i-m+1}]$ of size $m$. We will show the stronger
statement: (3.17) actually holds for any
$\vec{x}=(x_{1},\cdots,x_{N})\in\mathbb{Z}^{N}$ with
$x_{i},x_{i-1},\cdots,x_{i-m+1}$ merely satisfying
$x_{i}=x_{i-1}-1=\cdots=x_{i-m+1}-m+1$. We do not require empty sites at the
left and right ends so they may not form a cluster.

We prove this by induction on $m$. For $m=1$ this is trivial. Assume the claim
is true for any clusters of size $\leq m$. Now let $\vec{x}\in\mathbb{Z}^{N}$
with $x_{i}=x_{i-1}-1=\cdots=x_{i-m}-m$ for some $1\leq i\leq N$. Here without
loss of generality we can assume $i-m\geq 1$, otherwise replace $j$ by $j+N$
for indices $j\leq 0$. Then

$\displaystyle\sum_{\begin{subarray}{c}b_{j}\in\\{0,1\\}\\\
j=i-m,\cdots,i\end{subarray}}\prod_{j}p^{b_{j}}(1-p)^{1-b_{j}}G(\vec{x}-\sum_{j}b_{j}\vec{e}_{j},t;\vec{y},0)$
(3.18)
$\displaystyle=\sum_{b_{i}\in\\{0,1\\}}p^{b_{i}}(1-p)^{1-b_{i}}\cdot\Big{(}\sum_{\begin{subarray}{c}b_{j}\in\\{0,1\\}\\\
j=i-m,\cdots,i-1\end{subarray}}\prod_{j=i-m}^{i-1}p^{b_{j}}(1-p)^{1-b_{j}}G(\vec{x}-\sum_{j=i-m}^{i}b_{j}\vec{e}_{j},t;\vec{y},0)\Big{)}$
$\displaystyle=\sum_{b_{i}\in\\{0,1\\}}p^{b_{i}}(1-p)^{1-b_{i}}\cdot\Big{(}\sum_{b_{i-1}\in\\{0,1\\}}p^{b_{i-1}}(1-p)^{1-b_{i-1}}G(\vec{x}-b_{i}\vec{e}_{i}-b_{i-1}\vec{e}_{i-1},t;\vec{y},0)\Big{)}$
$\displaystyle=(1-p)^{2}G(\vec{x},t)+p(1-p)G(\vec{x}-\vec{e}_{i},t)+p(1-p)G(\vec{x}-\vec{e}_{i-1},t)+p^{2}G(\vec{x}-\vec{e}_{i}-\vec{e}_{i-1},t),$

where we used induction hypothesis in the second equality of (3.18) for the
sum inside the brackets and we suppress the dependence on $\vec{y}$ in the
last line to save space. Now by the boundary conditions (3.11)(possibly
(3.14)) we have

$(1-p)[G(\vec{x},t;\vec{y},0)-G(\vec{x}-\vec{e}_{i-1},t;\vec{y},0)]=p[G(\vec{x}-\vec{e}_{i}-\vec{e}_{i-1},t;\vec{y},0)-G(\vec{x}-\vec{e}_{i},t;\vec{y},0)].$
(3.19)

Inserting (3.19) into (3.18) we see the last line of (3.18) simplifies to

$(1-p)G(\vec{x},t;\vec{y},0)+pG(\vec{x}-\vec{e}_{i},t;\vec{y},0),$ (3.20)

which is precisely the left hand side of (3.17) and this completes the proof
of Lemma 3.5. ∎

###### Proof of Proposition 3.1.

By Lemma 3.5 it suffices to prove

$G(\vec{x},t;\vec{y},0)=\oint_{\Gamma}\frac{{\mathrm{d}}z}{2\pi\mathrm{i}z}\det\left[\sum_{w\in\mathcal{S}_{z}}F_{i,j}(w;\vec{x},\vec{y},t)J(w)\right]_{i,j=1}^{N}.$
(3.21)

To see this, we check that the right-hand side of (3.21) satisfies free
evolution equation (3.10), boundary conditions (3.11) and initial condition
(3.12).

For the free evolution equation (3.10) note first that it is straightforward
to check

$\sum_{b_{N-i+1}=0}^{1}p^{b_{N-i+1}}(1-p)^{1-b_{N-i+1}}F_{i,j}(w;\vec{x}-\vec{b},\vec{y};t)=F_{i,j}(w;\vec{x}-\vec{b}+b_{N-i+1}\vec{e}_{N-i+1},\vec{y};t+1).$
(3.22)

Hence by multi-linearity of determinants we have

$\displaystyle\sum_{\begin{subarray}{c}b_{k}\in\\{0,1\\}\\\
k=1,\cdots,N\end{subarray}}\prod_{k=1}^{N}p^{b_{k}}(1-p)^{1-b_{k}}\det\left[\sum_{w\in\mathcal{R}_{z}}F_{i,j}(w;\vec{x}-\vec{b},\vec{y},t)J(w)\right]_{i,j=1}^{N}$
$\displaystyle=\det\left[\sum_{w\in\mathcal{S}_{z}}\sum_{b_{N-i+1}=0}^{1}p^{b_{N-i+1}}(1-p)^{1-b_{N-i+1}}F_{i,j}(w;\vec{x}-\vec{b},\vec{y},t)J(w)\right]_{i,j=1}^{N}$
$\displaystyle=\det\left[\sum_{w\in\mathcal{S}_{z}}F_{i,j}(w;\vec{x}-\vec{b}+b_{N-i+1}\vec{e}_{N-i+1},\vec{y},t+1)J(w)\right]_{i,j=1}^{N}$
$\displaystyle=\det\left[\sum_{w\in\mathcal{S}_{z}}F_{i,j}(w;\vec{x},\vec{y},t+1)J(w)\right]_{i,j=1}^{N}.$

Here $\vec{e}_{i}\in\mathbb{Z}^{N}$ has $1$ in the $i$-th entry and $0$ for
the other entries. In the last equality we used the fact that
$F_{i,j}(w;\vec{x},\vec{y},t)$ only depends on the $N-i+1$-th entry of
$\vec{x}$. Now (3.10) follows from linearity of integral.

Next we check the boundary conditions (3.11). Given $1\leq k\leq N$, let
$\vec{x}=(x_{1},\cdots,x_{N})\in\mathbb{Z}^{N}$ be satisfying
$x_{k-1}=x_{k}+1$. Note that when $k=1$ this means $x_{1}=x_{N}+L-1$. Then the
boundary conditions (3.11) can be expressed as

$(1-p)[G(\vec{x},t;\vec{y},0)-G(\vec{x}-\vec{e}_{k-1},t;\vec{y},0)]=p[G(\vec{x}-\vec{e}_{k-1}-\vec{e}_{k},t;\vec{y},0)-G(\vec{x}-\vec{e}_{i},t;\vec{y},0)],$
(3.23)

for $2\leq k\leq N$ and

$(1-p)[G(\vec{x},t;\vec{y},0)-G(\vec{x}-\vec{e}_{N},t;\vec{y},0)]=p[G(\vec{x}-\vec{e}_{N}-\vec{e}_{1},t;\vec{y},0)-G(\vec{x}-\vec{e}_{1},t;\vec{y},0)],$
(3.24)

for $k=1$. We prove (3.23) first. Note that for $2\leq k\leq N$ with
$x_{k-1}=x_{k}+1$ we have

$F_{i,j}(w;\vec{x},\vec{y},t)-F_{i,j}(w;\vec{x}-\vec{e}_{k-1},\vec{y},t)=\begin{cases}0,\quad&\text{if
}i\neq N-k+2,\\\ -w\cdot F_{i,j}(w;\vec{x},\vec{y},t),\quad&\text{if
}i=N-k+2.\end{cases}$

Hence by multi-linearity we have

$\displaystyle\det\left[\sum_{w\in\mathcal{S}_{z}}F_{i,j}(w;\vec{x},\vec{y},t)J(w)\right]_{i,j=1}^{N}-\det\left[\sum_{w\in\mathcal{S}_{z}}F_{i,j}(w;\vec{x}-\vec{e}_{k-1},\vec{y},t)J(w)\right]_{i,j=1}^{N}$
$\displaystyle=\det\left[\sum_{w\in\mathcal{S}_{z}}F_{i,j}(w;\vec{x},\vec{y},t)J(w)\cdot(1-(1+w)\mathbf{1}_{i=N-k+2})\right]_{i,j=1}^{N}.$

Similarly

$\displaystyle\det\left[\sum_{w\in\mathcal{S}_{z}}F_{i,j}(w;\vec{x}-\vec{e}_{k},\vec{y},t)J(w)\right]_{i,j=1}^{N}-\det\left[\sum_{w\in\mathcal{S}_{z}}F_{i,j}(w;\vec{x}-\vec{e}_{k}-\vec{e}_{k-1},\vec{y},t)J(w)\right]_{i,j=1}^{N}$
$\displaystyle=\det\left[\sum_{w\in\mathcal{S}_{z}}F_{i,j}(w;\vec{x}-\vec{e}_{k},\vec{y},t)J(w)\cdot(1-(1+w)\mathbf{1}_{i=N-k+2})\right]_{i,j=1}^{N}.$

Now since
$F_{i,j}(w;\vec{x}-\vec{e}_{k},\vec{y},t)=F_{i,j}(w;\vec{x},\vec{y},t)$ for
$i\neq N-k+1$ and

$(1-p)F_{N-k+1,j}(w;\vec{x},\vec{y},t)+pF_{N-k+1,j}(w;\vec{x}-\vec{e}_{k},\vec{y},t)=w\cdot
F_{N-k+2,j}(w;\vec{x},\vec{y},t),$

we have

$\displaystyle(1-p)\cdot\det\left[\sum_{w\in\mathcal{S}_{z}}F_{i,j}(w;\vec{x},\vec{y},t)J(w)\cdot(1-(1+w)\mathbf{1}_{i=N-k+2})\right]_{i,j=1}^{N}$
(3.25)
$\displaystyle+p\cdot\det\left[\sum_{w\in\mathcal{S}_{z}}F_{i,j}(w;\vec{x}-\vec{e}_{k},\vec{y},t)J(w)\cdot(1-(1+w)\mathbf{1}_{i=N-k+2})\right]_{i,j=1}^{N}$
$\displaystyle=\det\left[M_{i,j}^{(k)}(\vec{x},\vec{y},t)\right]_{i,j=1}^{N}=0,$

where
$M_{i,j}^{(k)}(\vec{x},\vec{y},t)=\sum_{w\in\mathcal{S}_{z}}F_{i,j}(w;\vec{x},\vec{y},t)J(w)$
for $i\neq N-k+1,N-k+2$ and
$M_{N-k+1,j}^{(k)}(\vec{x},\vec{y},t)=-M_{N-k+2,j}^{(k)}(\vec{x},\vec{y},t)=\sum_{w\in\mathcal{S}_{z}}F_{N-k+2,j}(w;\vec{x},\vec{y},t)J(w)\cdot
w$. These two rows are proportional so the determinant is $0$. Now (3.23)
follows from linearity of the integral.

The proof for (3.24) is similar. The only thing changes is when $k=1$, we have
$M_{i,j}^{(1)}(\vec{x},\vec{y},t)=\sum_{w\in\mathcal{S}_{z}}F_{i,j}(w;\vec{x},\vec{y},t)J(w)$
for $i\neq 1,N$ while

$M_{N,j}^{(1)}(\vec{x},\vec{y},t)=\sum_{w\in\mathcal{S}_{z}}w^{-N}(w+1)^{N-L}(1+pw)^{N}F_{1,j}(w;\vec{x},\vec{y},t)J(w)\cdot
w=-z^{-L}M_{1,j}^{(1)}(\vec{x},\vec{y},t).$ (3.26)

Hence $\det[M^{(1)}_{i,j}(\vec{x},\vec{y},t)]=0$ since row $1$ and $N$ are
proportional. Note that in the last equality of (3.26) we used the fact that
$w\in\mathcal{S}_{z}$.

Finally we check the initial condition (3.12). We need to show

$\oint_{\Gamma}\frac{{\mathrm{d}}z}{2\pi\mathrm{i}z}\det\left[\sum_{w\in\mathcal{S}_{z}}F_{i,j}(w;\vec{x},\vec{y},0)J(w)\right]_{i,j=1}^{N}=\frac{\mathbf{1}_{\vec{x}=\vec{y}}}{\prod_{i=1}^{N}(1-p\mathbf{1}_{x_{i-1}-x_{1}=1})}$
(3.27)

Thanks to the cyclic-shift invariance of both sides of (3.27) (see (iv) of
Proposition 3.4) we can assume without loss of generality that
$\vec{x}=(x_{1},\cdots,x_{N})\in\mathcal{X}_{N}^{(L)}$ satisfies
$x_{1}<x_{N}+L-1$. In fact since $N<L$ there is at least one $1\leq i\leq N$
such that $x_{i}<x_{i-1}-1$ and we can replace $\vec{x}$ and $\vec{y}$ by
$\vec{x}^{\prime}:=(x_{i},x_{i+1},\cdots,x_{N},x_{1}-L,\cdots,x_{i-1}-L)$ and
$\vec{y}^{\prime}:=(y_{i},y_{i+1},\cdots,y_{N},y_{1}-L,\cdots,y_{i-1}-L)$ if
necessary since the two sides of (3.27) remain the same. By (3.7) we have

$\displaystyle\sum_{w\in\mathcal{S}_{z}}F_{i,j}(w;\vec{x},\vec{y},0)J(w)=\oint_{|w+1|=R}-\oint_{|w+1|=\epsilon}-\oint_{|pw+1|=\epsilon}\frac{{\mathrm{d}}w}{2\pi\mathrm{i}}F_{i,j}(w;\vec{x},\vec{y},0)\frac{\hat{q}_{z}(w)+z^{L}}{\hat{q}_{z}(w)}$
$\displaystyle=\oint_{|w+1|=R}-\oint_{|w+1|=\epsilon}-\oint_{|pw+1|=\epsilon}\frac{{\mathrm{d}}w}{2\pi\mathrm{i}}\frac{w^{j-i+N}(w+1)^{-x_{N-i+1}+y_{N-j+1}+i-j-1+L-N}(1+pw)^{i-j}}{w^{N}(w+1)^{L-N}-z^{L}(1+pw)^{N}}$
$\displaystyle:=I_{1}(i,j)-I_{2}(i,j)-I_{3}(i,j),$

where $I_{1}(i,j),I_{2}(i,j),I_{3}(i,j)$ are the integrals over the three
contours, respectively. Here we recall that $R$ and $\epsilon$ are
large(small) enough so that $\mathcal{S}_{z}$ is contained in the region
$\\{\epsilon<|w+1|<R\\}\backslash\\{|1+pw|\leq\epsilon\\}$. For $I_{3}(i,j)$
note that

$\displaystyle I_{3}(i,j)$
$\displaystyle=\oint_{|pw+1|=\epsilon}\frac{{\mathrm{d}}w}{2\pi\mathrm{i}}F_{i,j}(w;\vec{x},\vec{y},0)+z^{L}\oint_{|pw+1|=\epsilon}\frac{{\mathrm{d}}w}{2\pi\mathrm{i}}\frac{F_{i,j}(w;\vec{x},\vec{y},0)(1+pw)^{N}}{q_{z}(w)}$
(3.28)
$\displaystyle=\oint_{|pw+1|=\epsilon}\frac{{\mathrm{d}}w}{2\pi\mathrm{i}}F_{i,j}(w;\vec{x},\vec{y},0),$

since $F_{i,j}(w;\vec{x},\vec{y},0)(1+pw)^{N}$ is analytic at $w=-1/p$ for all
$1\leq i,j\leq N$ and
$q_{z}(w)=w^{N}(w+1)^{L-N}-z^{L}(1+pw)^{N}=(1+pw)^{N}\hat{q}_{z}(w)$ is
nonzero at $w=-1/p$. For the other parts we write

$\displaystyle I_{1}(i,j)$
$\displaystyle=\oint_{|w+1|=R}\frac{{\mathrm{d}}w}{2\pi\mathrm{i}}F_{i,j}(w;\vec{x},\vec{y},0)+z^{L}\oint_{|w+1|=R}\frac{{\mathrm{d}}w}{2\pi\mathrm{i}}\frac{F_{i,j}(w;\vec{x},\vec{y},0)(1+pw)^{N}}{q_{z}(w)}$
(3.29)
$\displaystyle:=\oint_{|w+1|=R}\frac{{\mathrm{d}}w}{2\pi\mathrm{i}}F_{i,j}(w;\vec{x},\vec{y},0)+z^{L}I_{1}^{\prime}(i,j),$

and

$\displaystyle I_{2}(i,j)$
$\displaystyle=z^{-L}\oint_{|w+1|=\epsilon}\frac{{\mathrm{d}}w}{2\pi\mathrm{i}}\frac{w^{j-i+N}(w+1)^{-x_{N-i+1}+y_{N-j+1}+i-j-1+L-N}(1+pw)^{i-j-N}}{z^{-L}w^{N}(w+1)^{L-N}(1+pw)^{-N}-1}$
(3.30) $\displaystyle:=z^{-L}I_{2}^{\prime}(i,j).$

Depending on the analytical properties of integrands in $I_{1}^{\prime}$ and
$I_{2}^{\prime}$ we split into two cases:

Case 1: $x_{1}<y_{1}$. First note the basic fact that for any
$\vec{x},\vec{y}\in\mathcal{X}_{N}^{(L)}$ we have

$x_{1}-L+i\leq x_{N-i+1}\leq x_{1}-N+i,\quad\text{and}\quad y_{1}-L+j\leq
y_{N-j+1}\leq y_{1}-N+j,\quad 1\leq i,j\leq N.$ (3.31)

Now if $x_{1}<y_{1}$, then for any $1\leq i,j\leq N$ we have

$-x_{N-i+1}+y_{N-j+1}+i-j-1+L-N\geq y_{1}-x_{1}-1\geq 0.$

Hence the integrand of $I_{2}^{\prime}(i,j)$ is analytic at $w=-1$ so
$I_{2}^{\prime}(i,j)=0$. Therefore in this case we have

$\displaystyle\oint_{\Gamma}\frac{{\mathrm{d}}z}{2\pi\mathrm{i}z}\det\left[\sum_{w\in\mathcal{S}_{z}}F_{i,j}(w;\vec{x},\vec{y},0)J(w)\right]_{i,j=1}^{N}$
(3.32)
$\displaystyle=\oint_{\Gamma}\frac{{\mathrm{d}}z}{2\pi\mathrm{i}z}\det\left[\oint_{\Gamma_{0,-1}}\frac{{\mathrm{d}}w}{2\pi\mathrm{i}}F_{i,j}(w;\vec{x},\vec{y},0)+z^{L}I_{1}^{\prime}(i,j)\right]_{i,j=1}^{N}$
$\displaystyle=\det\left[\oint_{\Gamma_{0,-1}}\frac{{\mathrm{d}}w}{2\pi\mathrm{i}}F_{i,j}(w;\vec{x},\vec{y},0)\right]_{i,j=1}^{N}.$

Here in the last equality of (3.32) we take the outer integral contour
$\Gamma$ to be $|z|=r$ and let $r\to 0$. $\Gamma_{0,-1}$ is any simple closed
contour with $0$ and $-1$ inside and $-1/p$ outside.

Case 2: $x_{1}\geq y_{1}$. Write

$I_{1}^{\prime}(i,j)=\oint_{|w+1|=R}\frac{{\mathrm{d}}w}{2\pi\mathrm{i}}\frac{w^{j-i-N}(w+1)^{-x_{N-i+1}+y_{N-j+1}+i-j-1-L+N}(1+pw)^{i-j+N}}{1-z^{L}w^{-N}(w+1)^{-L+N}(1+pw)^{N}}.$
(3.33)

Again by (3.31) we have

$-x_{N-i+1}+y_{N-j+1}+i-j-1-L+N\leq-x_{1}+L-i+y_{1}-N+j+i-j-1-L+N\leq-1.$
(3.34)

We claim that the first inequality in (3.34) is strict and hence
$-x_{N-i+1}+y_{N-j+1}+i-j-1-L+N\leq-2$. This is due to our original assumption
that $x_{N}+L-1>x_{1}$ and hence $x_{N-i+1}>x_{1}-L+i$ for all $1\leq i\leq
N$. Owing to this fact, the integrand in (3.33) is $O(R^{-2})$ since
$w^{j-i-N}(1+pw)^{i-j+N}$ remains bounded. Hence $I_{1}^{\prime}(i,j)\to 0$ as
$R\to\infty$ but since it is independent of large enough $R$ , we have
$I_{1}^{\prime}(i,j)=0$ for all $R$ large enough. Now a similar argument as in
(3.32) with $\Gamma$ instead be large circle $|z|=r$ with $r\to\infty$
implies:

$\displaystyle\oint_{\Gamma}\frac{{\mathrm{d}}z}{2\pi\mathrm{i}z}\det\left[\sum_{w\in\mathcal{S}_{z}}F_{i,j}(w;\vec{x},\vec{y},0)J(w)\right]_{i,j=1}^{N}$
(3.35)
$\displaystyle=\oint_{\Gamma}\frac{{\mathrm{d}}z}{2\pi\mathrm{i}z}\det\left[\oint_{\Gamma_{0,-1}}\frac{{\mathrm{d}}w}{2\pi\mathrm{i}}F_{i,j}(w;\vec{x},\vec{y},0)+z^{-L}I_{2}^{\prime}(i,j)\right]_{i,j=1}^{N}$
$\displaystyle=\det\left[\oint_{\Gamma_{0,-1}}\frac{{\mathrm{d}}w}{2\pi\mathrm{i}}F_{i,j}(w;\vec{x},\vec{y},0)\right]_{i,j=1}^{N}.$

In conclusion we have reduced checking (3.27) to check the following:

$\det\left[\oint_{\Gamma_{0,-1}}\frac{{\mathrm{d}}w}{2\pi\mathrm{i}}F_{i,j}(w;\vec{x},\vec{y},0)\right]_{i,j=1}^{N}=\frac{\mathbf{1}_{\vec{x}=\vec{y}}}{\prod_{i=1}^{N}(1-p\mathbf{1}_{x_{i-1}-x_{1}=1})},$
(3.36)

for any $\vec{x},\vec{y}\in\mathcal{X}_{N}^{L}$ with $x_{1}<x_{N}+L-1$. But
this is precisely equation $(3.30)$ of [9] which appears in checking the
determinantal formula for the transition probability of discrete parallel
TASEP on $\mathbb{Z}$ satisfies the proper initial condition. We will not
repeat the proof here but just point out that due to assumption
$x_{1}<x_{N}+L-1$ we have $1-p\mathbf{1}_{x_{0}-x_{1}=1}=1$ so (3.36) is
really identical to equation $(3.30)$ of [9]. ∎

##  4\. Finite-time Multi-point joint distribution under general initial
conditions

###  4.1. A Toeplitz-like determinant formula

In this section we derive a formula for the finite-time multi-point joint
distributions for discrete time parallel periodic TASEP under arbitrary
initial condition. The proof basically follows the strategy of [4] by
performing a multiple sum of transition probabilities over suitable particle
configurations. The main technical part is a Cauchy-type identity for
summation of left and right eigenfunctions (see Proposition 5.4) which
generalizes Proposition 3.4 of [4] and some new difficulties appear.

######  Theorem 4.1 (Multi-point joint distribution for discrete time parallel
TASEP in $\mathcal{X}_{N}^{(L)}$).

Let $\vec{y}\in\mathcal{X}_{N}^{(L)}$ and
$\vec{x}(t)=(x_{1}(t),\cdots,x_{N}(t))\in\mathcal{X}_{N}^{(L)}$ be particle
configurations evolving according to the discrete time parallel TASEP in
$\mathcal{X}_{N}^{(L)}$ at time $t$ with initial configuration
$\vec{x}(0)=\vec{y}$. Fix a positive integer $m$. Let
$(k_{1},t_{1}),\cdots,(k_{m},t_{m})\in\\{1,\cdots,N\\}\times\mathbb{N}$ be
distinct with $0\leq t_{1}\leq\cdots\leq t_{m}$. Let $a_{i}\in\mathbb{Z}$ for
$1\leq i\leq m$. Then

$\mathbb{P}_{\vec{y}}^{(L)}\left(\bigcap_{i=1}^{m}\\{x_{k_{i}}(t_{i})\geq
a_{i}\\}\right)=\oint\cdots\oint\frac{{\mathrm{d}}z_{m}}{2\pi\mathrm{i}z_{m}}\cdots\frac{{\mathrm{d}}z_{1}}{2\pi\mathrm{i}z_{1}}\mathcal{C}^{(L)}(\vec{z})\mathcal{D}_{\vec{y}}^{(L)}(\vec{z}),$
(4.1)

where the contours for the integrals are nested circles
$0<|z_{m}|<\cdots<|z_{1}|$. Here $\vec{z}=(z_{1},\cdots,z_{m})$. The functions
$\mathcal{C}^{(L)}(\vec{z})$ and $\mathcal{D}^{(L)}_{\vec{y}}(\vec{z})$ are
defined by

$\mathcal{C}^{(L)}(\vec{z})=(-1)^{(N-k_{m})(N-1)}z^{(N-k_{1})L}\prod_{\ell=2}^{m}\left[z_{\ell}^{(k_{\ell-1}-k_{\ell})L}\left(\left(\frac{z_{\ell}}{z_{\ell-1}}\right)^{L}-1\right)^{N-1}\right],$
(4.2)

and

$\mathcal{D}_{\vec{y}}^{(L)}(\vec{z})=\det\left[\sum_{\begin{subarray}{c}w_{\ell}\in\mathcal{S}_{z_{\ell}}\\\
\ell=1,\cdots,m\end{subarray}}\frac{w_{1}^{i-1-N}(1+pw_{1})^{N-i+1}(1+w_{1})^{y_{N-i+1}+N-i+1}\cdot
w_{m}^{-j}}{\prod_{\ell=2}^{m}(w_{\ell}-w_{\ell-1})}\prod_{\ell=1}^{N}G_{\ell}(w_{\ell})\right]_{i,j=1}^{N},$
(4.3)

where for $1\leq\ell\leq m$

$G_{\ell}(w):=\frac{w(w+1)(1+pw)}{N+Lw+p(L-N)w^{2}}\cdot\frac{w^{k_{\ell}}(1+w)^{-a_{\ell}-k_{\ell}}(1+pw)^{t_{\ell}-k_{\ell}}}{w^{k_{\ell-1}}(1+w)^{-a_{\ell-1}-k_{\ell-1}}(1+pw)^{t_{\ell-1}-k_{\ell-1}}}.$
(4.4)

Here $k_{0}=t_{0}=a_{0}:=0$ and we suppress the dependence on $a_{i}$, $k_{i}$
and $t_{i}$’s in $\mathcal{C}^{(L)}(\vec{z})$ and
$\mathcal{D}_{\vec{y}}^{(L)}(\vec{z})$.

###### Proof.

We start with the case $m=1$ for a warm-up. First by Cauchy-Binet formula we
can rewrite the transition probability (3.1) as

$P_{t}(\vec{x}\to\vec{x}^{\prime})=\oint_{\Gamma}\frac{{\mathrm{d}}z}{2\pi\mathrm{i}z}\sum_{\vec{w}\in(\mathcal{R}_{z})^{N}}\Psi_{\vec{x}}^{\ell}(\vec{w})\Psi_{\vec{x}^{\prime}}^{r}(\vec{w})\mathcal{Q}(\vec{w};t),$

where

$\displaystyle\Psi_{\vec{x}}^{\ell}(\vec{w})=\det\left[\left(\frac{w_{i}}{1+pw_{i}}\right)^{j-1}(1+w_{i})^{x_{N-j+1}-j+1}\right]_{i,j=1}^{N},$
(4.5)
$\displaystyle\Psi_{\vec{x}}^{r}(\vec{w})=\prod_{i=1}^{N}(1-p\mathbf{1}_{x_{i-1}-x_{i}=1})\cdot\det\left[\left(\frac{w_{i}}{1+pw_{i}}\right)^{1-j}(1+w_{i})^{-x_{N-j+1}+j-1}\right]_{i,j=1}^{N},$
(4.6)

and

$\mathcal{Q}(\vec{w};t)=\frac{1}{N!}\prod_{j=1}^{N}\frac{(1+pw_{j})^{t}J(w_{j})}{w_{j}+1}=\frac{1}{N!}\prod_{j=1}^{N}\frac{w_{j}(1+pw_{j})^{t+1}}{N+Lw_{j}+p(L-N)w_{j}^{2}}.$
(4.7)

Now to get the one-point distribution $\mathbb{P}_{\vec{y}}^{(L)}(x_{k}(t)\geq
a)$ we perform a summation over all configurations
$\vec{x}\in\mathcal{X}_{N}^{(L)}$ with $x_{k}\geq a$ of the transition
probability and interchange the order of integration and summation:

$\displaystyle\mathbb{P}_{\vec{y}}^{(L)}(x_{k}(t)\geq a)$
$\displaystyle=\sum_{\vec{x}\in\mathcal{X}_{N}^{(L)}\cap\\{x_{k}\geq
a\\}}P_{t}(\vec{y}\to\vec{x})$ (4.8)
$\displaystyle=\oint_{\Gamma}\frac{{\mathrm{d}}z}{2\pi\mathrm{i}}\sum_{\vec{w}\in(\mathcal{S}_{z})^{N}}\Psi_{\vec{y}}^{\ell}(\vec{w})Q(\vec{w};t)\left(\sum_{\vec{x}\in\mathcal{X}_{N}^{(L)}\cap\\{x_{k}\geq
a\\}}\Psi_{\vec{x}}^{r}(\vec{w})\right).$

Now by Corollary 5.3 in Section 5 we have
$\sum_{\vec{x}\in\mathcal{X}_{N}^{(L)}\cap\\{x_{k}\geq
a\\}}\Psi_{\vec{x}}^{r}(\vec{w})$ equals

$(-1)^{(N-k)(N-1)}z^{(N-k)L}\prod_{i=1}^{N}\left[\left(\frac{1+pw_{i}}{w_{i}}\right)^{N-k}\cdot(1+w_{i})^{-a+N-k+1}\right]\cdot\det[w_{i}^{-j}]_{i,j=1}^{N}.$
(4.9)

Inserting (4.9) back to (4.8) and use Cauchy-Binet formula backwards we
conclude that

$\displaystyle\mathbb{P}_{\vec{y}}^{(L)}(x_{k}(t)\geq
a)=\frac{(-1)^{(N-k)(N-1)}}{2\pi\mathrm{i}}$ (4.10) $\displaystyle\cdot$
$\displaystyle\oint_{\Gamma}\frac{{\mathrm{d}}z}{z^{1-(N-k)L}}\det\left[\sum_{w\in\mathcal{S}_{z}}\frac{w^{j-i+k-N}(1+pw)^{t-j+N-k+2}(1+w)^{y_{N-j+1}-j-a+N-k+2}}{N+Lw+p(L-N)w^{2}}\right]_{i,j=1}^{N}.$

Here in order to interchange the order of summation and integration as in
(4.8) we need to make sure that the summation over
$\vec{x}\in\mathcal{X}_{N}^{(L)}\cap\\{x_{k}\geq a\\}$ converges absolutely,
which is guaranteed if we assume $|w+1|>1$ for all $w\in\mathcal{S}_{z}$(see
the discussion in Proposition 5.2). This can then be achieved by choosing the
contour $\Gamma$ to be circle with large radius $R$ since for $|z|=R$ large we
have $|w+1|=O(|z|^{\frac{L}{L-N}})>1$ for all $w$ satisfying
$w^{N}(1+pw)^{-N}(w+1)^{L-N}=z^{L}$. Finally a similar argument as in (ii) of
Proposition 3.4 shows that the right-hand side of (4.10) does not depend on
the choice of $\Gamma$ so we can deform $\Gamma$ to be any simple closed
contour containing $0$.

Now assume $m\geq 2$. Then

$\mathbb{P}_{\vec{y}}^{(L)}\left(\cap_{i=1}^{m}\\{x_{k_{i}}(t_{i})\geq
a_{i}\\}\right)=\sum_{\begin{subarray}{c}\vec{x}^{(\ell)}\in\mathcal{X}_{N}^{(L)}\cap\\{x^{(\ell)}_{k_{\ell}}\geq
a_{\ell}\\}\\\
\ell=1,\cdots,m\end{subarray}}P_{t_{1}-t_{0}}(\vec{y}\to\vec{x}^{(1)})\cdots
P_{t_{m}-t_{m-1}}(\vec{x}^{(m-1)}\to\vec{x}^{(m)}),$ (4.11)

where $t_{0}:=0$. As in the $m=1$ case we rewrite the transition probability
using Cauchy-Binet formula and interchange the order of summation and
integration (will be justified later) so that
$\mathbb{P}_{\vec{y}}(\cap_{i=1}^{m}\\{x_{k_{i}}(t_{i})\geq a_{i}\\})$ equals

$\displaystyle\oint\frac{{\mathrm{d}}z_{1}}{2\pi\mathrm{i}z_{1}}\cdots\oint\frac{{\mathrm{d}}z_{m}}{2\pi\mathrm{i}z_{m}}\sum_{\begin{subarray}{c}\vec{w}^{(\ell)}\in(\mathcal{S}_{z_{\ell}})^{N}\\\
\ell=1,\cdots,m\end{subarray}}\mathcal{P}(\vec{w}^{(1)},\cdots,\vec{w}^{(m)})\prod_{\ell=1}^{m}\mathcal{Q}(\vec{w}^{(\ell)};t_{\ell}-t_{\ell-1}).$
(4.12)

Here $\vec{w}^{(\ell)}=(w^{(\ell)}_{1},\cdots,w^{(\ell)}_{m})$ and

$\mathcal{P}(\vec{w}^{(1)},\cdots,\vec{w}^{(m)})=\Psi_{\vec{y}}^{(\ell)}(\vec{w}^{(1)})\cdot\left[\prod_{\ell=1}^{m-1}\mathcal{H}_{k_{\ell},a_{\ell}}(\vec{w}^{(\ell)};\vec{w}^{(\ell+1)})\right]\cdot\left[\sum_{\vec{x}\in\mathcal{X}_{N}^{(L)}\cap\\{x_{k_{m}}\geq
a_{m}\\}}\Psi_{\vec{x}}^{r}(\vec{w}^{(m)})\right].$ (4.13)

Where

$\mathcal{H}_{k,a}(\vec{w};\vec{w}^{\prime}):=\sum_{\vec{x}\in\mathcal{X}_{N}^{(L)}\cap\\{x_{k}\geq
a\\}}\Psi_{\vec{x}}^{r}(\vec{w})\Psi_{\vec{x}}^{\ell}(\vec{w}^{\prime}).$
(4.14)

Now similar as in the $m=1$ case we evaluate the sums appearing in (4.13) and
(4.14) using Corollary 5.3 and Corollary 5.5 in Section 5 and apply Cauchy-
Binet formula backwards to conclude that

$\mathbb{P}_{\vec{y}}^{(L)}\left(\bigcap_{i=1}^{m}\\{x_{k_{i}}(t_{i})\geq
a_{i}\\}\right)=\oint\cdots\oint\frac{{\mathrm{d}}z_{m}}{2\pi\mathrm{i}z_{m}}\cdots\frac{{\mathrm{d}}z_{1}}{2\pi\mathrm{i}z_{1}}\mathcal{C}^{(L)}(\vec{z})\mathcal{D}_{\vec{y}}^{(L)}(\vec{z}),$
(4.15)

for $\mathcal{C}^{(L)}(\vec{z})$ and $\mathcal{D}_{\vec{y}}^{(L)}(\vec{z})$
defined in (4.2) and (4.3). Similar as the discussion before, in order to
interchange summation and integration we need the absolute convergence of all
the infinite sums which is guaranteed if we assume

$\prod_{j=1}^{N}|w_{j}^{(1)}+1|>\prod_{j=1}^{N}|w_{j}^{(2)}+1|>\cdots>\prod_{j=1}^{N}|w_{j}^{(m)}+1|>1.$
(4.16)

By the same reasoning as in $m=1$ case this can be achieved assuming the
integral contours for $z_{i}$’s are large nested contours
$|z_{\ell}|=r_{\ell}$ with $r_{\ell}-r_{\ell+1}$ also large enough for all
$1\leq\ell\leq m$. Finally we can deform the integral contours in (4.1) into
arbitrary nested contours with $0$ inside, not necessarily with large radius
thanks to the analyticity of $\mathcal{C}^{(L)}(\vec{z})$ and
$\mathcal{D}_{\vec{y}}^{(L)}(\vec{z})$ in $\vec{z}$ for any $z_{i}$’s nonzero
and distinct which can be shown in a similar way as (ii) of Proposition 3.4. ∎

###  4.2. Toeplitz-like determinant to Fredholm determinant

The finite time formula obtained in Theorem 4.1 contains a factor
$\mathcal{D}_{\vec{y}}^{(L)}(\vec{z})$ inside the integrals which is a
Toeplitz-like determinant and is hard to take large-time limits. In this
section we rewrite the formula based on a remarkable finite determinant
identity obtained in [5] between a Toeplitz-like determinant with symbol
supported on a finite set and a Fredholm determinant with kernel acting on the
$\ell^{2}$ space supported on the same finite set.

###### Proof of Theorem 4.1.

The proof is almost verbatim to the proof of Theorem 3.1 in [5] so we omit
most of the details. We apply Proposition 4.1 of [5] with the functions
$p_{i}(w)$, $q_{i}(w)$ and $h_{i}(w)$ given by

$p_{i}(w)=w^{i-1}(1+pw)^{N-i}(1+w)^{y_{N-i+1}+N-i+1},\quad\text{and }\
q_{i}(w)=w^{N-i},$ (4.17)

for $1\leq i\leq N$ and

$h_{i}(w):=\begin{cases}G_{i}(w)w^{-N}(1+pw)\quad&i=1,\\\ G_{i}(w)\quad&2\leq
i\leq m-1,\\\ G_{i}(w)w^{-N}\quad&i=m.\end{cases}$ (4.18)

Then the Toepliz-like determinant $\mathcal{D}_{\vec{y}}^{(L)}(\vec{z})$ can
be rewritten as the product of a function and a Fredholm determinant.
Combining with the definition of the function
$\mathcal{C}_{\vec{y}}^{(L)}(\vec{z})$ and using the algebraic relation
satisfied by the Bethe roots we obtain the identity

$\mathcal{C}_{\vec{y}}^{(L)}(\mathrm{z})\mathcal{D}_{\vec{y}}^{(L)}(\mathrm{z})=\mathscr{C}_{\vec{y}}^{(L)}(\vec{z})\mathscr{D}_{\vec{y}}^{(L)}(\vec{z}).$
(4.19)

∎

##  5\. Summation identities of eigenfunctions

###  5.1. Summation over single eigenfunction

In this section we state and prove the summation identities used in computing
multi-time joint distribution in Section 4. For convenience we recall the left
and right eigenfunctions defined in (4.5) and (4.6).

######  Definition 5.1 (Left and right eigenfunctions).

Given $\vec{x}=(x_{1},\cdots,x_{N})\in\mathcal{X}_{N}^{(\infty)}$ and
$p\in\mathbb{C}$, we define the functions $\Psi_{\vec{x}}^{\ell}(\vec{w})$ and
$\Psi_{\vec{x}}^{r}(\vec{w})$ for $\vec{w}\in\mathbb{C}^{N}$ as follows:

$\displaystyle\Psi_{\vec{x}}^{\ell}(\vec{w})=\det\left[\left(\frac{w_{i}}{1+pw_{i}}\right)^{j-1}(1+w_{i})^{x_{N-j+1}-j+1}\right]_{i,j=1}^{N},$
(5.1)
$\displaystyle\Psi_{\vec{x}}^{r}(\vec{w})=\prod_{i=1}^{N}(1-p\mathbf{1}_{x_{i-1}-x_{i}=1})\cdot\det\left[\left(\frac{w_{i}}{1+pw_{i}}\right)^{1-j}(1+w_{i})^{-x_{N-j+1}+j-1}\right]_{i,j=1}^{N}.$
(5.2)

We start with a summation identity of $\Psi_{\vec{x}}^{r}(\vec{w})$:

######  Proposition 5.2 (Summation over a single eigenfunction).

Let $z\in\mathbb{C}$ be nonzero. Let $\Psi_{\vec{x}}^{r}(\vec{w})$ be as in
(4.6) where $\vec{w}=(w_{1},\cdots,w_{N})\in(\mathcal{S}_{z})^{N}$ such that
$\prod_{j=1}^{N}|w_{j}+1|>1$. Then

$\sum_{\vec{x}\in\mathcal{X}_{N}^{(L)}\cap\\{x_{N}\geq
0\\}}\Psi_{\vec{x}}^{r}(\vec{w})=\prod_{i=1}^{N}(1+w_{i})\cdot\det[w_{i}^{-j}]_{i,j=1}^{N}.$
(5.3)

###### Proof.

First we write

$\displaystyle\sum_{\vec{x}\in\mathcal{X}_{N}^{(L)}\cap\\{x_{N}\geq
0\\}}\Psi_{\vec{x}}^{r}(\vec{w})$
$\displaystyle=\sum_{a=0}^{\infty}\sum_{\vec{x}\in\mathcal{X}_{N}^{(L)}\cap\\{x_{N}=a\\}}\Psi_{\vec{x}}^{r}(\vec{w})$
$\displaystyle=\sum_{a=0}^{\infty}\prod_{j=1}^{N}(1+w_{j})^{-a}\sum_{\vec{x}^{\prime}\in\mathcal{X}_{N}^{(L)}\cap\\{x_{N}^{\prime}=0\\}}\Psi_{\vec{x}^{\prime}}^{r}(\vec{w}),$

where the summation over $a$ converges absolutely for
$\prod_{j=1}^{N}|1+w_{j}|>1$. Here $\vec{x}^{\prime}=\vec{x}-(a,\cdots,a)$.
Now we start with computing the summation over
$\vec{x}\in\mathcal{X}_{N}^{(L)}$ with $x_{N}=0$ (this is a finite sum so
there is no convergence issue). For this we split the sum according the number
of particles to the right of the $N$-th particle in the cluster containing the
$N$-th particle for a given configuration $\vec{x}$:

$\displaystyle\sum_{\vec{x}\in\mathcal{X}_{N}^{(L)}\cap\\{x_{N}=0\\}}\Psi_{\vec{x}}^{r}(\vec{w})=\sum_{k=0}^{N-1}\sum_{\vec{x}\in\mathcal{X}_{N}^{(L)}(k)}\Psi_{\vec{x}}^{r}(\vec{w}):=\sum_{k=0}^{N-1}S_{k}.$
(5.4)

Where

$\mathcal{X}_{N}^{(L)}(k):=\\{\vec{x}\in\mathcal{X}_{N}^{(L)}:x_{N}=x_{N-1}-1=\cdots
x_{N-k}-k=0,x_{N-k-1}-k-1>0\\}.$ (5.5)

To compute $S_{k}$ we perform the sum in the order $k+2\leq
x_{N-k-1}<x_{N-k-2}<\cdots<x_{1}\leq L-1$. Note that configurations
$\vec{x}\in\mathcal{X}_{N}^{(L)}(k)$ takes the form
$(x_{1},\cdots,x_{N-k-1},k,\cdots,0)$ where $x_{N-k-1}>k+1$. Hence for
$\vec{x}\in\mathcal{X}_{N}^{(L)}(k)$ we have

$\Psi_{\vec{x}}^{r}(\vec{w})=(1-p)^{k}\prod_{i=1}^{N-k-1}(1-p\mathbf{1}_{x_{i-1}-x_{i}=1})\det[R_{i,j}^{(k,k+1)}(\vec{w})]_{i,j=1}^{N},$

where

$R_{i,j}^{(k,k+1)}(\vec{w})=\begin{cases}w_{i}^{1-j}(1+pw_{i})^{j-1},\quad&1\leq
j\leq k+1,\\\
w_{i}^{1-j}(1+pw_{i})^{j-1}(1+w_{i})^{-x_{N-j+1}+j-1},\quad&k+2\leq j\leq
N.\end{cases}$

Note that

$\displaystyle\sum_{x_{N-k-1}=k+2}^{x_{N-k-2}-1}(1-p\mathbf{1}_{x_{N-k-2}-x_{N-k-1}=1})R_{i,k+2}^{(k+1)}(\vec{w})$
$\displaystyle=w_{i}^{-2-k}(1+pw_{i})^{1+k}-w_{i}^{-2-k}(1+pw_{i})^{2+k}(1+w_{i})^{-x_{N-k-2}+k+2}$
$\displaystyle=w_{i}^{-2-k}(1+pw_{i})^{1+k}-R_{i,k+3}^{(k,k+1)}(\vec{w}).$

Adding the $k+3$-th column to the $k+2$-th column we get

$\sum_{x_{N-k-1}=k+2}^{x_{N-k-2}-1}\Psi_{\vec{x}}^{r}(\vec{w})=(1-p)^{k}\prod_{i=1}^{N-k-2}(1-p\mathbf{1}_{x_{i-1}-x_{i}=1})\det[R_{i,j}^{(k,k+2)}(\vec{w})]_{i,j=1}^{N},$
(5.6)

where

$R_{i,j}^{(k,k+2)}(\vec{w})=\begin{cases}w_{i}^{1-j}(1+pw_{i})^{j-1},\quad&1\leq
j\leq k+1,\\\ w_{i}^{-j}(1+pw_{i})^{j-1},\quad&j=k+2,\\\
w_{i}^{1-j}(1+pw_{i})^{j-1}(1+w_{i})^{-x_{N-j+1}+j-1},\quad&k+3\leq j\leq
N.\end{cases}$

Now repeat this procedure and perform the sum over $x_{N-k-2},\cdots,x_{2}$ we
get

$\sum_{x_{2}=N-1}^{x_{1}-1}\cdots\sum_{x_{N-k-1}=k+2}^{x_{N-k-2}-1}\Psi_{\vec{x}}^{r}(\vec{w})=(1-p)^{k}(1-p\mathbf{1}_{x_{0}-x_{1}=1})\det[R_{i,j}^{(k,N-1)}(\vec{w})]_{i,j=1}^{N},$

where

$R_{i,j}^{(k,N-1)}(\vec{w})=\begin{cases}w_{i}^{1-j}(1+pw_{i})^{j-1},\quad&1\leq
j\leq k+1,\\\ w_{i}^{-j}(1+pw_{i})^{j-1},\quad&k+2\leq j\leq N-1,\\\
w_{i}^{1-j}(1+pw_{i})^{j-1}(1+w_{i})^{-x_{N-j+1}+j-1},\quad&j=N.\end{cases}$

Finally note that

$\displaystyle\sum_{x_{1}=N}^{L-1}(1-p\mathbf{1}_{x_{0}-x_{1}=1})R_{i,N}^{(k,N-1)}(\vec{w})$
$\displaystyle=w_{i}^{-N}(1+pw_{i})^{N-1}-w_{i}^{-N}(1+pw_{i})^{N}(1+w_{i})^{-L-N}$
$\displaystyle=w_{i}^{-2-k}(1+pw_{i})^{1+k}-z^{-L}R_{i,1}^{(k,N-1)}(\vec{w}).$

Here we used the fact that $w_{i}\in\mathcal{S}_{z}$. Multiplying the first
column by $z^{-L}$ and adding to last column we get

$S_{k}:=\sum_{x_{1}=N}^{L-1}\cdots\sum_{x_{N-k-1}=k+2}^{x_{N-k-2}-1}\Psi_{\vec{x}}^{r}(\vec{w})=(1-p)^{k}\det[R_{i,j}^{(k,N)}(\vec{w})]_{i,j=1}^{N},$
(5.7)

where

$R_{i,j}^{(k,N)}(\vec{w})=\begin{cases}w_{i}^{1-j}(1+pw_{i})^{j-1},\quad&1\leq
j\leq k+1,\\\ w_{i}^{-j}(1+pw_{i})^{j-1},\quad&k+2\leq j\leq N.\\\
\end{cases}$

For the purpose of summing over $S_{k}$ we further rewrite
$R_{i,j}^{(k,N)}(\vec{w})$ slightly. Given $0\leq k\leq N-3$, we add the
$j$-th column of $R^{(k,N)}(\vec{w})$ to its $j+1$-th column, $j=N,\cdots,k+2$
so that

$\det[R_{i,j}^{(k,N)}(\vec{w})]_{i,j=1}^{N}=\det[\hat{R}_{i,j}^{(k,N)}(\vec{w})]_{i,j=1}^{N},$
(5.8)

where

$\hat{R}_{i,j}^{(k,N)}(\vec{w})=\begin{cases}w_{i}^{1-j}(1+pw_{i})^{j-1},\quad&1\leq
j\leq k+1,\\\ w_{i}^{-j}(1+pw_{i})^{j-1},\quad&j=k+2,\\\
w_{i}^{-j}(1+pw_{i})^{j-1}(1+w_{i}),\quad&k+3\leq j\leq N.\\\ \end{cases}$

For $k=N-2,N-1$ we just set $\hat{R}^{(k,N)}(\vec{w})=R^{(k,N)}(\vec{w})$. Now
we perform the sum over $S_{k}$ in the order $S_{N-1}+S_{N-2}+\cdots+S_{0}$.
Note that for each $0\leq k\leq N-1$,
$\hat{R}_{i,j}^{k,N}(\vec{w})=\hat{R}^{k+1,N}_{i,j}(\vec{w})$ except for
$j=k+1$. Hence by multi-linearity of the determinants we have

$\displaystyle S_{N-1}+S_{N-2}$
$\displaystyle=(1-p)^{N-2}\left((1-p)\det[\hat{R}_{i,j}^{(N-1,N)}(\vec{w})]+\det[\hat{R}_{i,j}^{(N-2,N)}(\vec{w})]\right)$
$\displaystyle=(1-p)^{N-2}\det[T_{i,j}^{(N-2)}(\vec{w})]_{i,j=1}^{N},$

where

$T_{i,j}^{(N-2)}(\vec{w})=\begin{cases}w_{i}^{1-j}(1+pw_{i})^{j-1},\quad&1\leq
j\leq N-1,\\\ w_{i}^{-j}(1+pw_{i})^{j-2}(1+w_{i}),&j=N.\\\ \end{cases}$

Here to simplify the expression we have multiplied the $(N-1)$-th column by
$-p(1-p)$ and added to the sum of the $N$-th column of
$\hat{R}^{(N-1,N)}(\vec{w})$ and $\hat{R}^{(N-2,N)}(\vec{w})$, using the
simple fact that

$\left[(1-p)w_{i}+1\right]\cdot
w_{i}^{-N}(1+pw_{i})^{N-1}=w_{i}^{-N}(1+pw_{i})^{N-2}(1+w_{i})+p(1-p)w_{i}^{2-N}(1+pw_{i})^{N-2}.$

Now repeat this procedure we get

$\sum_{k=0}^{N-1}S_{k}=\det[T^{(0)}_{i,j}(\vec{w})]_{i,j=1}^{N},$ (5.9)

where

$T_{i,j}^{(0)}(\vec{w})=\begin{cases}w_{i}^{1-j}(1+pw_{i})^{j-1},\quad&j=1,\\\
w_{i}^{-j}(1+pw_{i})^{j-2}(1+w_{i}),&2\leq j\leq N.\\\ \end{cases}$ (5.10)

Multiplying the $j$-th column by $-p$ and add to the $j+1$-th column, for
$j=N-1,\cdots,2$, we get

$\sum_{k=0}^{N-1}S_{k}=\det[\hat{T}^{(0)}_{i,j}(\vec{w})]_{i,j=1}^{N}=\prod_{i=1}^{N}(1+w_{i})\cdot\det[w_{i}^{-j}]_{i,j=1}^{N}-\det[w_{i}^{-j}]_{i,j=1}^{N},$
(5.11)

where

$\hat{T}_{i,j}^{(0)}(\vec{w})=\begin{cases}w_{i}^{1-j},\quad&j=1,\\\
w_{i}^{-j}(1+w_{j}),&2\leq j\leq N.\\\ \end{cases}$ (5.12)

Thus we conclude that

$\displaystyle\sum_{\vec{x}\in\mathcal{X}_{N}^{(L)}\cap\\{x_{N}\geq
0\\}}\Psi_{\vec{x}}^{r}(\vec{w})$
$\displaystyle=\sum_{a=0}^{\infty}\left[\prod_{i=1}^{N}(1+w_{i})^{-a+1}-\prod_{i=1}^{N}(1+w_{i})^{-a}\right]\det[w_{i}^{-j}]_{i,j=1}^{N}$
(5.13)
$\displaystyle=\prod_{i=1}^{N}(1+w_{i})\cdot\det[w_{i}^{-j}]_{i,j=1}^{N}.$

∎

The following corollary is a simple consequence of Proposition 5.2 and the
periodic nature of the form of $\Psi_{\vec{x}}^{r}(\vec{w})$.

######  Corollary 5.3.

Under the same assumption as in Proposition 5.2 we have

$\displaystyle\sum_{\vec{x}\in\mathcal{X}_{N}^{(L)}\cap\\{x_{N-k}\geq
a\\}}\Psi_{\vec{x}}^{r}(\vec{w})=(-1)^{k(N-1)}z^{kL}\prod_{i=1}^{N}\left[\left(\frac{1+pw_{i}}{w_{i}}\right)^{k}\cdot(1+w_{i})^{-a+k+1}\right]\cdot\det[w_{i}^{-j}]_{i,j=1}^{N},$
(5.14)

for all $0\leq k\leq N-1$ and $a\in\mathbb{Z}$.

###### Proof.

For given $\vec{x}=(x_{1},\cdots,x_{N})\in\mathcal{X}_{N}^{(L)}$ and $0\leq
k\leq N-1$, set
$\vec{x}^{\prime}:=(x_{1}^{\prime},\cdots,x_{N}^{\prime})=(x_{N-k+1}+L,\cdots,x_{N}+L,x_{1},\cdots,x_{N-k})$.
Then the condition $\vec{x}\in\mathcal{X}_{N}^{(L)}\cap\\{x_{N-k}\geq a\\}$ is
the same as $\vec{x}^{\prime}\in\mathcal{X}_{N}^{(L)}\cap\\{x_{N}^{\prime}\geq
a\\}$. Now consider

$\Psi_{\vec{x}}^{r}(\vec{w})=\det\left[\left(\frac{w_{i}}{1+pw_{i}}\right)^{1-j}(1+w_{i})^{-x_{N-j+1}+j-1}\right]_{i,j=1}^{N}.$

We move the first $k$ columns of the matrix to the end. The resulting
determinant equals $(-1)^{k(N-1)}$ times the determinant of the matrix whose
$(i,j)$-th entry has the form
$\left(\frac{w_{i}}{1+pw_{i}}\right)^{1-j-k}(1+w_{i})^{-x_{N-j+1}^{\prime}+j+k-1}$
for $1\leq j\leq N-k$ and the form
$\left(\frac{w_{i}}{1+pw_{i}}\right)^{1-j-k+N}(1+w_{i})^{-x_{N-j+1}^{\prime}+j+k-1+L-N}$
for $N-k+1\leq j\leq N$. But since
$\left(\frac{w_{i}}{1+pw_{i}}\right)^{N}(1+w_{i})^{L-N}=z^{L}$ for all $1\leq
i\leq N$, we have

$\left(\frac{w_{i}}{1+pw_{i}}\right)^{1-j-k+N}(1+w_{i})^{-x_{N-j+1}^{\prime}+j+k-1+L-N}=z^{L}\cdot\left(\frac{w_{i}}{1+pw_{i}}\right)^{1-j-k}(1+w_{i})^{-x_{N-j+1}^{\prime}+j+k-1}.$

After factoring out the common factor
$\left(\frac{w_{i}}{1+pw_{i}}\right)^{-k}(1+w_{i})^{k}$ from each row and
$z^{L}$ from the last $k$ columns we conclude that

$\Psi_{\vec{x}}^{r}(\vec{w})=(-1)^{k(N-1)}z^{kL}\prod_{i=1}^{N}\left[\left(\frac{1+pw_{i}}{w_{i}}\right)^{k}\cdot(1+w_{i})^{k}\right]\Psi_{\vec{x}^{\prime}}^{r}(\vec{w}).$
(5.15)

Hence

$\displaystyle\sum_{\vec{x}\in\mathcal{X}_{N}^{(L)}\cap\\{x_{N-k}\geq
a\\}}\Psi_{\vec{x}}^{r}(\vec{w})$
$\displaystyle=(-1)^{k(N-1)}z^{kL}\prod_{i=1}^{N}\left[\left(\frac{1+pw_{i}}{w_{i}}\right)^{k}\cdot(1+w_{i})^{k}\right]\cdot\sum_{\vec{x}^{\prime}\in\Omega_{L,N}\cap\\{x_{N}^{\prime}\geq
a\\}}\Psi_{\vec{x}^{\prime}}^{r}(\vec{w})$ (5.16)
$\displaystyle=(-1)^{k(N-1)}z^{kL}\prod_{i=1}^{N}\left[\left(\frac{1+pw_{i}}{w_{i}}\right)^{k}\cdot(1+w_{i})^{-a+k+1}\right]\cdot\det[w_{i}^{-j}]_{i,j=1}^{N}.$

∎

###  5.2. Cacuchy-type summation identity for left and right eigenfunctions

######  Proposition 5.4 (Cauchy-type summation identity over left and right
eigenfunctions).

Let $z,z^{\prime}\in\mathbb{C}$ be nonzero such that $(z^{\prime})^{L}\neq
z^{L}$. For $\vec{x}=(x_{1},\cdots,x_{N})\in\mathcal{X}_{N}^{(L)}$, let
$\Psi_{\vec{x}}^{r}(\vec{w})$ and $\Psi_{\vec{x}}^{\ell}(\vec{w}^{\prime})$ be
as in (4.5) and (4.6) where
$\vec{w}=(w_{1},\cdots,w_{N})\in(\mathcal{S}_{z})^{N}$ and
$\vec{w}^{\prime}=(w_{1}^{\prime},\cdots,w_{N}^{\prime})\in(\mathcal{S}_{z^{\prime}})^{N}$
satisfy $\prod_{j=1}^{N}|w_{j}+1|>\prod_{j=1}^{N}|w_{j}^{\prime}+1|$. Then

$\sum_{\vec{x}\in\mathcal{X}_{N}^{(L)}\cap\\{x_{N}\geq
0\\}}\Psi_{\vec{x}}^{r}(\vec{w})\Psi_{\vec{x}}^{\ell}(\vec{w}^{\prime})=\left(1-\left(\frac{z^{\prime}}{z}\right)^{L}\right)^{N-1}\prod_{j=1}^{N}(w_{j}+1)\cdot\det\left[\frac{1}{w_{i}-w_{i^{\prime}}^{\prime}}\right]_{i,i^{\prime}=1}^{N}.$
(5.17)

Similar as in Corollary 5.3, we can easily extend Proposition 5.4 using
periodicity to summation over
$\vec{x}\in\mathcal{X}_{N}^{(L)}\cap\\{x_{N-k}\geq a\\}$ for any $0\leq k\leq
N-1$ and $a\in\mathbb{Z}$:

######  Corollary 5.5.

Under the same assumption as in Proposition 5.4 we have

$\displaystyle\sum_{\vec{x}\in\mathcal{X}_{N}^{(L)}\cap\\{x_{N-k}\geq
a\\}}\Psi_{\vec{x}}^{r}(\vec{w})\Psi_{\vec{x}}^{\ell}(\vec{w}^{\prime})$
$\displaystyle=\left(\frac{z}{z^{\prime}}\right)^{kL}\left(1-\left(\frac{z^{\prime}}{z}\right)^{L}\right)^{N-1}\cdot\det\left[\frac{1}{w_{i}-w_{i^{\prime}}^{\prime}}\right]_{i,i^{\prime}=1}^{N}$
(5.18)
$\displaystyle\cdot\prod_{j=1}^{N}\frac{w_{j}^{-k}(1+pw_{j})^{k}(w_{j}+1)^{-a+k+1}}{(w_{j}^{\prime})^{-k}(1+pw_{j}^{\prime})^{k}(w_{j}^{\prime}+1)^{-a+k}},$

for all $0\leq k\leq N-1$ and $a\in\mathbb{Z}$.

###### Proof.

Similar as the proof of Corollary 5.3. ∎

######  Remark 5.6.

It is interesting to note that the right-hand side of (5.17) does not depend
on $p$ explicitly(of course the $w_{i}$’ should satisfy certain algebraic
equations which depend on $p$). Take $p\to 0$, Proposition 5.4 degenerates to
Proposition $3.4$ of [4], which can be understood as a (periodic version) of
Cauchy identity for the Grothendieck polynomial (and its dual), and can be
derived from the deformed Cauchy identity for Grothendieck polynomials
obtained in Theorem 5.3 of [17]. For $0<p<1$, to the best of our knowledge the
corresponding Cauchy-type identity (5.17) has not been discussed in the
existing literature, at least for the periodic case. The key point here is
instead of summing over all configuration $\vec{x}$ with $0\leq
x_{N}<x_{N-1}<\cdots<x_{1}$ as in the usual Cauchy identity, we are only
summing over those configurations satisfying the extra constraint
$x_{1}<x_{N}+L$. For general spectral parameters $\vec{w}$ and
$\vec{w}^{\prime}$ this sum only gives a deformed or generalized Cauchy
determinant. It further reduces to a genuine Cauchy determinant when we impose
the conditions as in Proposition 5.4 that the spectral parameters satisfy
suitable Bethe equations.

The proof of Proposition 5.4 is rather lengthy so we divide it into three
steps and discuss them one by one in the next few subsections. The proof
mainly follows the strategy of Proposition 3.4 of [4] but there are several
new technicalities. The non-uniform term
$\prod_{i=1}^{N}(1-p\mathbf{1}_{x_{i-1}-x_{i}=1})$ appearing in
$\Psi_{\vec{x}}^{r}(\vec{w})$ leads to extra difficulty and in Step 1 we
overcome this by introducing a different way (and slightly more convenient way
in our opinion) of expressing the sum in (5.17) compare to the proof in [4],
see Lemma 5.7 for details. In Step 2 we establish a key summation identity
(see Lemma 5.10) which generalizes Lemma 5.4 in [4] while the computation is
more delicate. Finally in step 3 we combine the formula obtained in Step 1 and
the summation identity obtained in Step 2 to conclude the final result.
Throughout the proof several rank-one perturbation formulas for Cauchy
determinants are used frequently so we collect all these elementary formulas
in a separate section for convenience, see Section 5.6 for details.

###  5.3. Proof of Proposition 5.4: Step 1

Similar as in the proof of Proposition 5.2 we first write

$\displaystyle\sum_{\vec{x}\in\mathcal{X}_{N}^{(L)}\cap\\{x_{N}\geq
0\\}}\Psi_{\vec{x}}^{r}(\vec{w})\Psi_{\vec{x}}^{\ell}(\vec{w}^{\prime})$
$\displaystyle=\sum_{a=0}^{\infty}\prod_{j=1}^{N}\left(\frac{1+w_{j}^{\prime}}{1+w_{j}}\right)^{a}\sum_{\vec{x}\in\mathcal{X}_{N}^{(L)}\cap\\{x_{N}=0\\}}\Psi_{\vec{x}}^{r}(\vec{w})\Psi_{\vec{x}}^{\ell}(\vec{w}^{\prime})$
$\displaystyle:=\sum_{a=0}^{\infty}\prod_{j=1}^{N}\left(\frac{1+w_{j}^{\prime}}{1+w_{j}}\right)^{a}\cdot\mathcal{H}_{N}(\vec{w},\vec{w}^{\prime}),$

so that it suffices to compute the sum over
$\vec{x}\in\mathcal{X}_{N}^{(L)}\cap\\{x_{N}=0\\}$, which is a finite sum so
there is no convergence issue. The summation over $a$ convergences absolutely
again by our assumption that
$\prod_{j=1}^{N}|w_{j}+1|>\prod_{j=1}^{N}|w_{j}^{\prime}+1|$. Expanding the
determinants in $\Psi_{\vec{x}}^{\ell}(\vec{w})$ and
$\Psi_{\vec{x}}^{r}(\vec{w}^{\prime})$ we get

$\displaystyle\mathcal{H}_{N}(\vec{w},\vec{w}^{\prime})=\sum_{\sigma,\sigma^{\prime}\in
S_{N}}\text{sgn}(\sigma\sigma^{\prime})\prod_{j=1}^{N}\left(\frac{w^{\prime}_{\sigma^{\prime}(j)}(1+pw_{\sigma(j)})}{w_{\sigma(j)}(1+pw^{\prime}_{\sigma^{\prime}(j)})}\right)^{j-1}\mathcal{H}^{(N)}_{\sigma,\sigma^{\prime}}(\vec{w},\vec{w}^{\prime}),$

where

$\mathcal{H}^{(N)}_{\sigma,\sigma^{\prime}}(\vec{w},\vec{w}^{\prime})=\sum_{L=x_{0}>x_{1}>\cdots>x_{N}=0}\prod_{j=1}^{N}(1-p\mathbf{1}_{x_{j-1}-x_{j}=1})\left(\frac{w^{\prime}_{\sigma^{\prime}(j)}+1}{w_{\sigma(j)}+1}\right)^{x_{N-j+1}-j+1}.$
(5.19)

By Lemma 5.7 below we see
$\mathcal{H}^{(N)}_{\sigma,\sigma^{\prime}}(\vec{w},\vec{w}^{\prime})$ equals

$\displaystyle\sum_{k=1}^{N}\left[\prod_{i=2}^{k}\left(-p+\frac{1}{1-\prod_{j=i}^{k}\frac{w_{\sigma^{\prime}(j)}^{\prime}+1}{w_{\sigma(j)}+1}}\right)\cdot\prod_{i=k+1}^{N}\left(\frac{w_{\sigma^{\prime}(i)}^{\prime}+1}{w_{\sigma(i)}+1}\right)^{L-N}\cdot\left(-p+\frac{1}{1-\prod_{j=k+1}^{i}\frac{w_{\sigma(j)}+1}{w_{\sigma^{\prime}(j)}^{\prime}+1}}\right)\right].$

Hence

$\displaystyle\mathcal{H}_{N}(\vec{w},\vec{w}^{\prime})=\sum_{k=1}^{N}\sum_{\sigma,\sigma^{\prime}\in
S_{N}}\text{sgn}(\sigma\sigma^{\prime})\prod_{j=1}^{N}\left(\frac{w^{\prime}_{\sigma^{\prime}(j)}(1+pw_{\sigma(j)})}{w_{\sigma(j)}(1+pw^{\prime}_{\sigma^{\prime}(j)})}\right)^{j-1}$
(5.20)
$\displaystyle\cdot\left[\prod_{i=2}^{k}\left(-p+\frac{1}{1-\prod_{j=i}^{k}\frac{w_{\sigma^{\prime}(j)}^{\prime}+1}{w_{\sigma(j)}+1}}\right)\cdot\prod_{i=k+1}^{N}\left(\frac{w_{\sigma^{\prime}(i)}^{\prime}+1}{w_{\sigma(i)}+1}\right)^{L-N}\cdot\left(-p+\frac{1}{1-\prod_{j=k+1}^{i}\frac{w_{\sigma(j)}+1}{w_{\sigma^{\prime}(j)}^{\prime}+1}}\right)\right].$

######  Lemma 5.7.

Let $N\geq 1$ be an integer and $w_{1},\cdots,w_{N}$ and
$w_{1}^{\prime},\cdots,w_{N}^{\prime}$ be distinct complex numbers. Then for
any integer $L>N$ we have

$\displaystyle\sum_{L=x_{0}>x_{1}>\cdots>x_{N}=0}\prod_{i=1}^{N}(1-p\mathbf{1}_{x_{i-1}-x_{i}=1})\prod_{i=1}^{N}\left(\frac{w_{i}^{\prime}+1}{w_{i}+1}\right)^{x_{N-i+1}-i+1}$
(5.21)
$\displaystyle=\sum_{k=1}^{N}\left[\prod_{i=2}^{k}\left(-p+\frac{1}{1-\prod_{j=i}^{k}\frac{w_{j}^{\prime}+1}{w_{j}+1}}\right)\cdot\prod_{i=k+1}^{N}\left(\frac{w_{i}^{\prime}+1}{w_{i}+1}\right)^{L-N}\cdot\left(-p+\frac{1}{1-\prod_{j=k+1}^{i}\frac{w_{j}+1}{w_{j}^{\prime}+1}}\right)\right].$

Here any empty product is set to be $1$.

###### Proof.

We use an induction on $N$. For $N=1$ the identity is obvious. Assume now
$N\geq 2$ and the identity holds for all indices less than $N$. We split the
sum into two sums depending on whether $x_{1}=L-1$ or not:

$\sum_{L=x_{0}>x_{1}>\cdots>x_{N}=0}=\sum_{L-1=x_{1}>\cdots>x_{N}=0}+\sum_{L-1>x_{1}>\cdots>x_{N}=0}:=T_{1}+T_{2}.$

For $T_{1}$ we first relabel the indices so that
$(x_{0}^{\prime},x_{1}^{\prime},\cdots,x_{N-1}^{\prime}):=(x_{1},x_{2},\cdots,x_{N})$
and $L^{\prime}:=L-1$. Then by induction hypothesis we have

$\displaystyle T_{1}$
$\displaystyle=(1-p)\left(\frac{w_{N}^{\prime}+1}{w_{N}+1}\right)^{L-N}\sum_{L^{\prime}=x_{0}^{\prime}>x_{1}^{\prime}>\cdots>x_{N-1}^{\prime}=0}\prod_{i=1}^{N-1}(1-p\mathbf{1}_{x^{\prime}_{i-1}-x^{\prime}_{i}=1})\prod_{i=1}^{N-1}\left(\frac{w_{i}^{\prime}+1}{w_{i}+1}\right)^{x_{N-i}^{\prime}-i+1}$
(5.22) $\displaystyle=\
\sum_{k=1}^{N-1}\left[\prod_{i=2}^{k}\left(-p+\frac{1}{1-\prod_{j=i}^{k}\frac{w_{j}^{\prime}+1}{w_{j}+1}}\right)\cdot\prod_{i=k+1}^{N}\left(\frac{w_{i}^{\prime}+1}{w_{i}+1}\right)^{L-N}\cdot
T_{1}^{(k)}(\vec{w},\vec{w}^{\prime})\right].$

Where

$T_{1}^{(k)}(\vec{w},\vec{w}^{\prime}):=(1-p)\prod_{i=k+1}^{N-1}\left(-p+\frac{1}{1-\prod_{j=k+1}^{i}\frac{w_{j}+1}{w_{j}^{\prime}+1}}\right),$
(5.23)

for $1\leq k\leq N-1$ and we set $T_{1}^{(N)}(\vec{w},\vec{w}^{\prime}):=0$
for convenience.

For $T_{2}$ we calculate the sum directly in the order
$L-1>x_{1}>\cdots>x_{N}=0$ using Lemma 5.8 below which gives:

$\displaystyle T_{2}$
$\displaystyle=\sum_{x_{N-1}=1}^{L-N-1}\cdots\sum_{x_{1}=x_{2}+1}^{L-2}\prod_{i=2}^{N}(1-p\mathbf{1}_{x_{i-1}-x_{i}=1})\prod_{i=1}^{N}\left(\frac{w_{i}^{\prime}+1}{w_{i}+1}\right)^{x_{N-i+1}-i+1}$
(5.24)
$\displaystyle=\sum_{k=1}^{N}\left[\prod_{i=2}^{k}\left(-p+\frac{1}{1-\prod_{j=i}^{k}\frac{w_{j}^{\prime}+1}{w_{j}+1}}\right)\cdot\prod_{i=k+1}^{N}\left(\frac{w_{i}^{\prime}+1}{w_{i}+1}\right)^{L-N}\cdot
T_{2}^{(k)}(\vec{w},\vec{w}^{\prime})\right].$

Where

$\displaystyle\quad\ T_{2}^{(k)}(\vec{w},\vec{w}^{\prime})$ (5.25)
$\displaystyle:=\sum_{\ell=1}^{N-k}\sum_{k+1=s_{1}<\cdots<s_{\ell}\leq
N}\prod_{i=1}^{\ell}\left(\frac{1}{\left(\prod_{j=s_{i}}^{s_{i+1}-1}\frac{w_{j}^{\prime}+1}{w_{j}+1}\right)-1}\cdot\prod_{j=s_{i}+1}^{s_{i+1}-1}\left(-p+\frac{1}{1-\prod_{m=j}^{s_{i+1}-1}\frac{w_{m}^{\prime}+1}{w_{m}+1}}\right)\right),$

for $1\leq k\leq N-1$ and $T_{2}^{(N)}(\vec{w},\vec{w}^{\prime}):=1$. Here we
are summing over all possible partitions of $\\{k+1,\cdots,N\\}$ and
$s_{i+1}:=N+1$. Now compare (5.22) and (5.24) with (5.21) we see it suffices
to prove

$T_{1}^{(k)}(\vec{w},\vec{w}^{\prime})+T_{2}^{(k)}(\vec{w},\vec{w}^{\prime})=\prod_{i=k+1}^{N}\left(-p+\frac{1}{1-\prod_{j=k+1}^{i}\frac{w_{j}+1}{w_{j}^{\prime}+1}}\right),$
(5.26)

for all $1\leq k\leq N$. Using the simple identity

$\frac{1}{1-\prod_{j=m}^{n}\frac{w_{j}+1}{w_{j}^{\prime}+1}}+\frac{1}{1-\prod_{j=m}^{n}\frac{w_{j}^{\prime}+1}{w_{j}+1}}=1,$
(5.27)

(5.26) is further reduced to showing

$T_{k}^{(2)}(\vec{w},\vec{w}^{\prime})=\prod_{i=k+1}^{N-1}\left(-p+\frac{1}{1-\prod_{j=k+1}^{i}\frac{w_{j}+1}{w_{j}^{\prime}+1}}\right)\cdot\frac{1}{\prod_{j=k+1}^{N}\frac{w_{j}^{\prime}+1}{w_{j}+1}-1},$
(5.28)

which follows from Lemma 5.9 below by taking
$z_{j}=\frac{w_{j}^{\prime}+1}{w_{j}+1}$ and properly shifting the indices. ∎

######  Lemma 5.8.

For complex numbers $f_{j}$, set

$F_{m,n}=\prod_{j=m}^{n}f_{j}\quad\text{for }1\leq m\leq n.$ (5.29)

Then

$\displaystyle\sum_{x_{N-1}=1}^{L-N-1}\cdots\sum_{x_{1}=x_{2}+1}^{L-2}\prod_{j=2}^{N}(1-p\mathbf{1}_{x_{N-j+1}-x_{N-j+2}=1})(f_{j})^{x_{N-j+1}-j+1}$
(5.30)
$\displaystyle=\sum_{k=1}^{N-1}\left[\prod_{i=2}^{k}\left(-p+\frac{1}{1-F_{i,k}}\right)\right]\cdot(F_{k+1,N})^{L-N}$
$\displaystyle\cdot\left[\sum_{\ell=1}^{N-k}\sum_{k<s_{1}<\cdots<s_{\ell}\leq
N}\prod_{i=1}^{\ell}\left(\prod_{j=s_{i}+1}^{s_{i+1}-1}\left(-p+\frac{1}{1-F_{j,s_{i+1}-1}}\right)\cdot\frac{1}{(F_{s_{i},s_{i+1}-1})-1}\right)\right].$

Where we set $s_{\ell+1}=N+1$.

###### Proof.

This lemma is a slightly modified version of Lemma 5.3 of [4]. The proof is
elementary and almost identical to the proof in [4] so we omit it. ∎

######  Lemma 5.9.

Let $n\geq 1$ be an integer and $z_{1},\cdots,z_{n}$ be complex numbers such
that $\prod_{j=\ell}^{k}z_{j}\neq 1$ for all $1\leq\ell\leq k\leq n$. Then

$\displaystyle\sum_{\ell=1}^{n}\sum_{1=s_{1}<\cdots<s_{\ell}\leq
n}\prod_{i=1}^{\ell}\left(\frac{1}{\left(\prod_{j=s_{i}}^{s_{i+1}-1}z_{j}\right)-1}\cdot\prod_{j=s_{i}+1}^{s_{i+1}-1}\left(-p+\frac{1}{1-\prod_{m=j}^{s_{i+1}-1}z_{m}}\right)\right)$
(5.31)
$\displaystyle=\prod_{i=1}^{n-1}\left(-p+\frac{1}{1-\prod_{j=1}^{i}z_{j}^{-1}}\right)\cdot\frac{1}{\left(\prod_{j=1}^{n}z_{j}\right)-1}.$

###### Proof.

Fix integer $M>n$, consider the following sum:

$\sum_{M>x_{1}>\cdots>x_{n}\geq
0}\prod_{i=2}^{n}(1-p\mathbf{1}_{x_{i-1}-x_{i}}=1)\prod_{j=1}^{n}z_{j}^{x_{n-j+1}-j+1}.$

We calculate the sum in two different ways: from right to left or from left to
right. Namely we set

$S^{r\to\ell}_{M}:=\sum_{x_{n}=0}^{M-n}\cdots\sum_{x_{1}=x_{2}+1}^{M-1}\prod_{i=2}^{n}(1-p\mathbf{1}_{x_{i-1}-x_{i}}=1)\prod_{j=1}^{n}z_{j}^{x_{n-j+1}-j+1},$

and

$S^{\ell\to
r}_{M}:=\sum_{x_{1}=n-1}^{M-1}\cdots\sum_{x_{n}=0}^{x_{n-1}-1}\prod_{i=2}^{n}(1-p\mathbf{1}_{x_{i-1}-x_{i}}=1)\prod_{j=1}^{n}z_{j}^{x_{n-j+1}-j+1}.$

Then clearly $S^{r\to\ell}_{M}=S^{\ell\to r}_{M}$ since they represent the
same sum. Now the two sums are calculated by calculating the (almost)
geometric sums one by one either from left to right or vice versa. There are
$2^{n}$ terms in total for both sums since every single term produces two
terms after performing the geometric sum once. Each of the terms consists of a
factor of the form $\prod_{j=n-k+1}^{n}z_{j}^{M-n+1}$ for some $0\leq k\leq n$
where $k=0$ corresponds to the terms consisting no such factor. For each $k$
we combine all the terms with the same factor
$\prod_{j=n-k+1}^{n}z_{j}^{M-n+1}$ and write

$S^{r\to\ell}_{M}=\sum_{k=0}^{n}\left[C^{r\to\ell}_{k}(z_{1},\cdots,z_{n})\cdot\prod_{j=n-k+1}^{n}z_{j}^{M-n+1}\right],$

and

$S^{r\to\ell}_{M}=\sum_{k=0}^{n}\left[C^{\ell\to
r}_{k}(z_{1},\cdots,z_{n})\cdot\prod_{j=n-k+1}^{n}z_{j}^{M-n+1}\right].$

Where $C^{r\to\ell}_{k}(z_{1},\cdots,z_{n})$’s and $C^{\ell\to
r}_{k}(z_{1},\cdots,z_{n})$’s are some very explicit functions in
$z_{1},\cdots,z_{n}$ independent of $M$ which are analytic for all $z_{i}$’s
satisfying the assumption that $\prod_{j=\ell}^{k}z_{j}\neq 1$ for all $1\leq
k\leq\ell\leq n$. In particular it is straightforward to check(see Lemma 5.8
for example) that

$\text{LHS of }\eqref{eq: reduction of the sum over
partitions}=C^{r\to\ell}_{n}(z_{1},\cdots,z_{n}),\quad\text{RHS of }\eqref{eq:
reduction of the sum over partitions}=C^{\ell\to r}_{n}(z_{1},\cdots,z_{n}).$

We claim that $C^{r\to\ell}_{k}(z_{1},\cdots,z_{n})=C^{\ell\to
r}_{k}(z_{1},\cdots,z_{n})$ for all $0\leq k\leq n$. This in particular
implies (5.31). Due to analyticity it suffices to check this for the $z_{j}$’s
satisfying $|z_{j}|<1$ for all $1\leq j\leq n$. In this case by letting
$M\to\infty$ in the equality $S^{r\to\ell}_{M}=S^{\ell\to r}_{M}$ we see
$C_{0}^{r\to\ell}=C_{0}^{\ell\to r}$. Similarly

$C_{1}^{r\to\ell}=\lim_{M\to\infty}z_{n}^{-M+n-1}\cdot(S_{M}^{r\to\ell}-C_{0}^{r\to\ell})=\lim_{M\to\infty}z_{n}^{-M+n-1}\cdot(S_{M}^{\ell\to
r}-C_{0}^{\ell\to r})=C_{1}^{\ell\to r}.$

Repeat this procedure we see $C_{k}^{r\to\ell}=C_{k}^{\ell\to r}$ for all
$0\leq k\leq n$. ∎

###  5.4. Proof of Proposition 5.4: Step 2

In this section we simplify the sum (5.20). We rewrite the sum further by
first deciding two index sets $J$ and $J^{\prime}$ with $|J|=|J^{\prime}|=k$
and then express the sum in terms of summation over index sets $J,J^{\prime}$:

$\mathcal{H}_{N}(\vec{w},\vec{w}^{\prime})=\sum_{k=1}^{N}\sum_{\begin{subarray}{c}J,J^{\prime}\subset\\{1,\cdots,N\\}\\\
|J|=|J^{\prime}|=k\end{subarray}}(-1)^{\\#(J,J^{c})+\\#(J^{\prime},(J^{\prime})^{c})}\mathscr{H}_{1}(J,J^{\prime})\mathscr{H}_{2}(J^{c},(J^{\prime})^{c}),$
(5.32)

where $\\#(J,J^{c}):=|\\{(m,n)\in(J,J^{c}):m>n\\}|$ and similar for
$\\#(J^{\prime},(J^{\prime})^{c})$. The functions
$\mathscr{H}_{1}(J,J^{\prime})$ and $\mathscr{H}_{2}(J^{c},(J^{\prime})^{c})$
are defined as follows:

$\mathscr{H}_{1}(J,J^{\prime})=\sum_{\begin{subarray}{c}\sigma:\\{1,\cdots,k\\}\to
J\\\ \sigma^{\prime}:\\{1,\cdots,k\\}\to
J^{\prime}\end{subarray}}\text{sgn}(\sigma)\text{sgn}(\sigma^{\prime})\prod_{i=2}^{k}\left(\frac{w^{\prime}_{\sigma^{\prime}(i)}(1+pw_{\sigma(i)})}{w_{\sigma(i)}(1+pw^{\prime}_{\sigma^{\prime}(i)})}\right)^{i-1}\cdot\left(-p+\frac{1}{1-\prod_{j=i}^{k}\frac{w_{\sigma^{\prime}(j)}^{\prime}+1}{w_{\sigma(j)}+1}}\right),$
(5.33)

and

$\displaystyle\mathscr{H}_{2}(J^{c},(J^{\prime})^{c})$
$\displaystyle=\sum_{\begin{subarray}{c}\pi:\\{k+1,\cdots,N\\}\to J^{c}\\\
\pi^{\prime}:\\{k+1,\cdots,N\\}\to(J^{\prime})^{c}\end{subarray}}\text{sgn}(\pi)\text{sgn}(\pi^{\prime})\frac{\prod_{j^{\prime}\in(J^{\prime})^{c}}(w_{j^{\prime}}^{\prime}+1)^{L-N}}{\prod_{j\in
J^{c}}(w_{j}+1)^{L-N}}$ (5.34)
$\displaystyle\cdot\prod_{i=k+1}^{N}\left(\frac{w^{\prime}_{\pi^{\prime}(i)}(1+pw_{\pi(i)})}{w_{\pi(i)}(1+pw^{\prime}_{\pi^{\prime}(i)})}\right)^{i-1}\cdot\left(-p+\frac{1}{1-\prod_{j=k+1}^{i}\frac{w_{\pi(j)}+1}{w_{\pi^{\prime}(j)}^{\prime}+1}}\right).$

By Lemma 5.10 below (which is of interest in its own) we have

$\mathscr{H}_{1}(J,J^{\prime})=\left[\prod_{j\in
J}(w_{j}+1)-\prod_{j^{\prime}\in
J^{\prime}}(w_{j^{\prime}}^{\prime}+1)\right]\cdot\det\left[\frac{1}{w_{j}-w_{j^{\prime}}^{\prime}}\right]_{j\in
J,j^{\prime}\in J^{\prime}}.$ (5.35)

To simplify $\mathscr{H}_{2}(J^{c},(J^{\prime})^{c})$ we use the assumption
that $\vec{w}\in(\mathcal{S}_{z})^{N}$ and
$\vec{w}^{\prime}\in(\mathcal{S}_{z^{\prime}})^{N}$. Namely for all $1\leq
i,i^{\prime}\leq N$ we have

$w_{i}^{N}(1+pw_{i})^{-N}(1+w_{i})^{L-N}=z^{L},\quad(w_{i^{\prime}}^{\prime})^{N}(1+pw_{i^{\prime}}^{\prime})^{-N}(1+w_{i^{\prime}}^{\prime})^{L-N}=(z^{\prime})^{L}.$
(5.36)

Inserting (5.36) into (5.34) we get

$\displaystyle H_{2}(J^{c},(J^{\prime})^{c})$
$\displaystyle=\sum_{\begin{subarray}{c}\pi:\\{k+1,\cdots,N\\}\to J^{c}\\\
\pi^{\prime}:\\{k+1,\cdots,N\\}\to(J^{\prime})^{c}\end{subarray}}\text{sgn}(\pi)\text{sgn}(\pi^{\prime})\frac{\prod_{j^{\prime}\in(J^{\prime})^{c}}(z^{\prime})^{L}}{\prod_{j\in
J^{c}}z^{L}}$ (5.37)
$\displaystyle\cdot\prod_{i=k+1}^{N}\left(\frac{w_{\pi(i)}(1+pw_{\pi^{\prime}(i)}^{\prime})}{w_{\pi^{\prime}(i)}^{\prime}(1+pw_{\pi(i)})}\right)^{N-i+1}\cdot\left(-p+\frac{1}{1-\prod_{j=k+1}^{i}\frac{w_{\pi(j)}+1}{w_{\pi^{\prime}(j)}^{\prime}+1}}\right).$

Now in order to apply Lemma 5.10 we reflect the permutations by defining
$\hat{\pi}:\\{1,\cdots,N-k\\}\to J^{c}$ as $\hat{\pi}(j):=\pi(N-j+1)$ for
$1\leq j\leq N$ and similarly for
$\hat{\pi}^{\prime}:\\{1,\cdots,N-k\\}\to(J^{\prime})^{c}$. Then

$\displaystyle\mathscr{H}_{2}(J^{c},(J^{\prime})^{c})$
$\displaystyle=\sum_{\begin{subarray}{c}\hat{\pi}:\\{1,\cdots,N-k\\}\to
J^{c}\\\
\hat{\pi}^{\prime}:\\{1,\cdots,N-k\\}\to(J^{\prime})^{c}\end{subarray}}\text{sgn}(\hat{\pi})\text{sgn}(\hat{\pi}^{\prime})\frac{\prod_{j^{\prime}\in(J^{\prime})^{c}}(z^{\prime})^{L}}{\prod_{j\in
J^{c}}z^{L}}$ (5.38)
$\displaystyle\cdot\prod_{i=1}^{N-k}\left(\frac{w_{\hat{\pi}(i)}(1+pw_{\hat{\pi}^{\prime}(i)}^{\prime})}{w_{\hat{\pi}^{\prime}(i)}^{\prime}(1+pw_{\hat{\pi}(i)})}\right)^{i}\cdot\left(-p+\frac{1}{1-\prod_{j=i}^{N-k}\frac{w_{\hat{\pi}(j)}+1}{w_{\hat{\pi}^{\prime}(j)}^{\prime}+1}}\right).$

Now apply Lemma 5.10 again with the role of $\vec{w}$ and $\vec{w}^{\prime}$
exchanged we have

$\displaystyle\mathscr{H}_{2}(J^{c},(J^{\prime})^{c})$
$\displaystyle=\frac{\prod_{j^{\prime}\in(J^{\prime})^{c}}(z^{\prime})^{L}}{\prod_{j\in
J^{c}}z^{L}}\cdot\frac{\prod_{j\in
J^{c}}w_{j}/(1+pw_{j})}{\prod_{j^{\prime}\in(J^{\prime})^{c}}w_{j^{\prime}}^{\prime}/(1+pw_{j^{\prime}}^{\prime})}\cdot\left(-p+\frac{1}{1-\frac{\prod_{j\in
J^{c}}(w_{j}+1)}{\prod_{j^{\prime}\in(J^{\prime})^{c}}(w_{j^{\prime}}^{\prime}+1)}}\right)$
(5.39)
$\displaystyle\cdot\left[\prod_{j^{\prime}\in(J^{\prime})^{c}}(w_{j^{\prime}}^{\prime}+1)-\prod_{j\in
J^{c}}(w_{j}+1)\right]\cdot\det\left[\frac{1}{-w_{j}+w_{j^{\prime}}^{\prime}}\right]_{j\in
J^{c},j^{\prime}\in(J^{\prime})^{c}}.$

Here the extra factor in (5.39) comes from the fact that in (5.38) the product
starts from $i=1$ instead of $i=2$ and the exponent is $i$ instead of $i-1$
comparing to (5.40).

######  Lemma 5.10.

Let $n\in\mathbb{N}$. Given any complex numbers $w_{i}$ and
$w_{i^{\prime}}^{\prime}$, $i=1,\cdots,n$ such that $w_{i}\neq
w_{i^{\prime}}$, for all $1\leq i,i^{\prime}\leq n$. Then for any $0<p<1$ we
have

$\displaystyle\sum_{\sigma,\sigma^{\prime}\in
S_{n}}\text{sgn}(\sigma\sigma^{\prime})\prod_{j=1}^{n}\left(\frac{w^{\prime}_{\sigma^{\prime}(j)}(1+pw_{\sigma(j)})}{w_{\sigma(j)}(1+pw^{\prime}_{\sigma^{\prime}(j)})}\right)^{j-1}\prod_{j=2}^{n}\left[-p+\frac{1}{1-\prod_{i=j}^{n}\frac{w^{\prime}_{\sigma^{\prime}(i)}+1}{w_{\sigma(i)}+1}}\right]$
(5.40)
$\displaystyle=\left[\prod_{j=1}^{n}(w_{j}+1)-\prod_{j=1}^{n}(w_{j}^{\prime}+1)\right]\det\left[\frac{1}{w_{i}-w_{i^{\prime}}^{\prime}}\right]_{i,i^{\prime}=1}^{n}.$

######  Remark 5.11.

Equation (5.40) should be understood also as a Cauchy summation identity
(simpler version than Proposition 5.4, for summation over all partitions
$\lambda=(x_{1}-n+1,x_{2}-n+2,\cdots,x_{n})$ with at most $n-1$ rows) of the
symmetric funtions $\Psi_{\vec{x}}^{r}(\vec{w})/\Delta(\vec{w})$ and
$\Psi_{\vec{x}}^{\ell}(\vec{w}^{\prime})/\Delta(\vec{w}^{\prime})$, where
$\Delta(\vec{w})$ is the usual Vandermonde determinant. In fact formally we
have

$\sum_{0=x_{n}<x_{n-1}<\cdots<x_{1}}\Psi_{\vec{x}}^{r}(\vec{w})\Psi_{\vec{x}}^{\ell}(\vec{w}^{\prime})=\sum_{x_{1}=x_{2}+1}^{\infty}\cdots\sum_{x_{n-1}=1}^{\infty}\Psi_{\vec{x}}^{r}(\vec{w})\Psi_{\vec{x}}^{\ell}(\vec{w}^{\prime})=\text{LHS
of }\eqref{eq: Cauchy identity for infinite sum},$ (5.41)

assuming all the infinite geometric series converges absolutely.

###### Proof of Lemma 5.10.

The proof is based on induction on $n$. The main tools are several rank-one
perturbation formulas for Cauchy determinants which we will use several times
so we collect them in a separate section, see section 5.6. For $n=1$, (5.40)
is trivial. Let $n\geq 2$ and assume (5.40) is true for all indices $\leq
n-1$. Given $\sigma,\sigma^{\prime}\in S_{n}$ we first decide two indices
$\ell=\sigma(1)$ and $\ell^{\prime}=\sigma^{\prime}(1)$ and shift the
restriction of $\sigma$ and $\sigma^{\prime}$ on $\\{2,\cdots,n\\}$ by $1$ but
still denote them by $\sigma$ and $\sigma^{\prime}$. Then

$\displaystyle\text{LHS of }\eqref{eq: Cauchy identity for infinite
sum}=\sum_{\ell,\ell^{\prime}=1}^{n}(-1)^{\ell+\ell^{\prime}}\frac{w_{\ell}(1+pw_{\ell^{\prime}}^{\prime})}{w_{\ell^{\prime}}^{\prime}(1+pw_{\ell})}\prod_{k=1}^{n}\frac{w_{k}^{\prime}(1+pw_{k})}{w_{k}(1+pw_{k}^{\prime})}\cdot\left(-p+\frac{1}{1-\frac{w_{\ell}+1}{w_{\ell^{\prime}}^{\prime}+1}\prod_{k=1}^{n}\frac{w_{k}^{\prime}+1}{w_{k}+1}}\right)$
$\displaystyle\cdot\left[\sum_{\begin{subarray}{c}\sigma:\\{1,\cdots,n-1\\}\to\\{1,\cdots,n\\}\backslash\\{\ell\\}\\\
\sigma^{\prime}:\\{1,\cdots,n-1\\}\to\\{1,\cdots,n\\}\backslash\\{\ell^{\prime}\\}\end{subarray}}\text{sgn}(\sigma)\text{sgn}(\sigma^{\prime})\prod_{i=2}^{n-1}\left(\frac{w^{\prime}_{\sigma^{\prime}(i)}(1+pw_{\sigma(i)})}{w_{\sigma(i)}(1+pw^{\prime}_{\sigma^{\prime}(i)})}\right)^{i-1}\cdot\prod_{j=2}^{n-1}\left(-p+\frac{1}{1-\prod_{i=j}^{n-1}\frac{w^{\prime}_{\sigma^{\prime}(i)}+1}{w_{\sigma(i)}+1}}\right)\right].$

By the induction hypothesis the term inside the big bracket above equals

$\left[\prod_{1\leq j\leq n,j\neq\ell}(w_{j}+1)-\prod_{1\leq j\leq
n,j\neq\ell^{\prime}}(w_{j}^{\prime}+1)\right]\det\left[\frac{1}{w_{i}-w_{i^{\prime}}^{\prime}}\right]_{\begin{subarray}{c}1\leq
i,i^{\prime}\leq n\\\ i\neq\ell,i^{\prime}\neq\ell^{\prime}\end{subarray}}.$

Hence

$\displaystyle\text{LHS of }\eqref{eq: Cauchy identity for infinite sum}$
(5.42)
$\displaystyle=(1-p)\cdot\prod_{k=1}^{n}\frac{w_{k}^{\prime}(1+w_{k})(1+pw_{k})}{w_{k}(1+pw_{k}^{\prime})}\sum_{\ell,\ell^{\prime}=1}^{n}(-1)^{\ell+\ell^{\prime}}\frac{w_{\ell}(1+pw_{\ell^{\prime}}^{\prime})}{w_{\ell^{\prime}}^{\prime}(1+w_{\ell})(1+pw_{\ell})}\det\left[\frac{1}{w_{i}-w_{i^{\prime}}^{\prime}}\right]_{\begin{subarray}{c}1\leq
i,i^{\prime}\leq n\\\ i\neq\ell,i^{\prime}\neq\ell^{\prime}\end{subarray}}$
$\displaystyle+p\cdot\prod_{k=1}^{n}\frac{w_{k}^{\prime}(1+w_{k}^{\prime})(1+pw_{k})}{w_{k}(1+pw_{k}^{\prime})}\sum_{\ell,\ell^{\prime}=1}^{n}(-1)^{\ell+\ell^{\prime}}\frac{w_{\ell}(1+pw_{\ell^{\prime}}^{\prime})}{w_{\ell^{\prime}}^{\prime}(1+w_{\ell^{\prime}}^{\prime})(1+pw_{\ell})}\det\left[\frac{1}{w_{i}-w_{i^{\prime}}^{\prime}}\right]_{\begin{subarray}{c}1\leq
i,i^{\prime}\leq n\\\ i\neq\ell,i^{\prime}\neq\ell^{\prime}\end{subarray}}$
$\displaystyle:=(1-p)\cdot(-1)^{n}A(-1)\cdot\frac{B(0)A(-1/p)}{A(0)B(-1/p)}\cdot
D_{1}+p\cdot(-1)^{n}B(-1)\cdot\frac{B(0)A(-1/p)}{A(0)B(-1/p)}\cdot D_{2}.$

Here we set $A(z):=\prod_{i=1}^{n}(z-w_{i})$ and
$B(z):=\prod_{i=1}^{n}(z-w_{i}^{\prime})$ and $D_{1}$, $D_{2}$ represent the
first and second sum over $\ell,\ell^{\prime}$ respectively. Now by (5.58) and
Lemma 5.15 we have

$D_{1}=\frac{1}{1-p}\cdot\left[\frac{A(0)B(-1/p)}{B(0)A(-1/p)}-\frac{B(-1/p)}{A(-1/p)}+p\cdot\frac{B(-1)}{A(-1)}-\frac{A(0)B(-1)}{B(0)A(-1)}+1-p\right]\cdot\det\left[\frac{1}{w_{i}-w_{i^{\prime}}^{\prime}}\right]_{i,i^{\prime}=1}^{n}$
(5.43)

and

$D_{2}=\frac{1}{p}\cdot\left[\frac{A(0)}{B(0)}-\frac{B(-1/p)A(0)}{A(-1/p)B(0)}+(p-1)\cdot\frac{A(-1)}{B(-1)}+\frac{A(-1)B(-1/p)}{B(-1)A(-1/p)}-p\right]\cdot\det\left[\frac{1}{w_{i}-w_{i^{\prime}}^{\prime}}\right]_{i,i^{\prime}=1}^{n}.$
(5.44)

Inserting (5.43) and (5.44) into (5.42), after necessary cancellation we
obtain

$\displaystyle\text{LHS of }\eqref{eq: Cauchy identity for infinite sum}$
$\displaystyle=\left[(-1)^{n}A(-1)-(-1)^{n}B(-1)\right]\cdot\det\left[\frac{1}{w_{i}-w_{i^{\prime}}^{\prime}}\right]_{i,i^{\prime}=1}^{n}$
(5.45)
$\displaystyle=\left[\prod_{j=1}^{n}(1+w_{j})-\prod_{j=1}^{n}(1+w_{j}^{\prime})\right]\det\left[\frac{1}{w_{i}-w_{i^{\prime}}^{\prime}}\right]_{i,i^{\prime}=1}^{n}.$

This completes the proof of Lemma 5.10. ∎

Finally inserting (5.35) and (5.39) into (5.32) and apply Lemma 5.12 below we
obtain

$\displaystyle\hat{S}_{N}$
$\displaystyle=\sum_{\begin{subarray}{c}J,J^{\prime}\subset\\{1,\cdots,N\\}\\\
|J|=|J^{\prime}|\end{subarray}}(-1)^{\\#(J,J^{c})+\\#(J^{\prime},(J^{\prime})^{c})}H_{1}(J,J^{\prime})H_{2}(J^{c},(J^{\prime})^{c})$
(5.46)
$\displaystyle=p\cdot\prod_{j=1}^{N}(w_{j}+1)\left(\det\left[\hat{C}(i,i^{\prime})\right]_{i,i^{\prime}=1}^{N}-\det\left[\hat{C}(i,i^{\prime})-\frac{1}{w_{i}+1}\right]_{i,i^{\prime}=1}^{N}\right)$
$\displaystyle-(1-p)\cdot\prod_{j=1}^{N}(w_{j}^{\prime}+1)\left(\det\left[\hat{C}(i,i^{\prime})\right]_{i,i^{\prime}=1}^{N}-\det\left[\hat{C}(i,i^{\prime})+\frac{1}{w_{i^{\prime}}^{\prime}+1}\right]_{i,i^{\prime}=1}^{N}\right),$

where

$\hat{C}(i,i^{\prime})=\frac{1}{w_{i}-w_{i^{\prime}}^{\prime}}-\left(\frac{z^{\prime}}{z}\right)^{L}\frac{w_{i}(1+pw_{i^{\prime}}^{\prime})}{w_{i^{\prime}}^{\prime}(1+pw_{i})}\frac{1}{w_{i}-w_{i^{\prime}}^{\prime}}.$
(5.47)

Note that in the first equality of (5.46) we add an extra term corresponding
to $|J|=|J^{\prime}|=0$ comparing to (5.32) which is harmless since the
summand is $0$ in this case.

######  Lemma 5.12 (Lemma 5.9 of [4]).

For two $n\times n$ matrices $A$ and $B$,

$\sum_{\begin{subarray}{c}J,J^{\prime}\subset\\{1,\cdots,n\\}\\\
|J|=|J^{\prime}|\end{subarray}}(-1)^{\\#(J,J^{c})+\\#(J^{\prime},(J^{\prime})^{c})}\det[A(i,i^{\prime})]_{i\in
J,i^{\prime}\in J^{\prime}}\det[B(i,i^{\prime})]_{i\in
J^{c},i^{\prime}\in(J^{\prime})^{c}}=\det[A+B]_{1\leq i,i^{\prime}\leq n}.$
(5.48)

###  5.5. Proof of Proposition 5.4: Step 3

In this section we further simplify equation (5.46) to conclude the proof of
Proposition 5.4. We compute $\det[\hat{C}(i,i^{\prime})]$ first. For
notational convenience we set $(z^{\prime}/z)^{L}:=\mu$. Note that

$\displaystyle\hat{C}(i,i^{\prime})$
$\displaystyle=\frac{1}{w_{i}-w_{i^{\prime}}^{\prime}}-\mu\cdot\frac{w_{i}(1+pw_{i^{\prime}}^{\prime})}{w_{i^{\prime}}^{\prime}(1+pw_{i})}\frac{1}{w_{i}-w_{i^{\prime}}^{\prime}}$
$\displaystyle=\left(1-\mu\right)\frac{1}{w_{i}-w_{i^{\prime}}^{\prime}}-\mu\cdot\frac{1}{w_{i^{\prime}}^{\prime}(1+pw_{i})}.$

Hence by Lemma 5.14 and Lemma 5.15 we have

$\displaystyle\det[\hat{C}(i,i^{\prime})]_{i,i^{\prime}=1}^{N}$
$\displaystyle=(1-\mu)^{N}\det\left[\frac{1}{w_{i}-w_{i^{\prime}}^{\prime}}-\frac{\mu}{1-\mu}\frac{1}{w_{i^{\prime}}^{\prime}(1+pw_{i})}\right]_{i,i^{\prime}=1}^{N}$
(5.49)
$\displaystyle=(1-\mu)^{N}\det[C(i,i^{\prime})]_{i,i^{\prime}=1}^{N}\cdot\left(1-\frac{\mu}{1-\mu}\left(\frac{B(-1/p)A(0)}{A(-1/p)B(0)}-1\right)\right)$
$\displaystyle=(1-\mu)^{N-1}\det[C(i,i^{\prime})]_{i,i^{\prime}=1}^{N}\cdot\left(1-\mu\cdot\frac{B(-1/p)A(0)}{A(-1/p)B(0)}\right).$

Here $C(i,i^{\prime})=\frac{1}{w_{i}-w_{i^{\prime}}^{\prime}}$ and $A(0)$ is
the evaluation at $z=0$ of the polynomial $A(z):=\prod_{j=1}^{N}(z-w_{j})$.
The other terms involving $A(\cdot)$ and $B(\cdot)$ are defined in a similar
way with $B(z):=\prod_{j=1}^{N}(z-w_{j}^{\prime})$. Now by Lemma 5.13 we have

$\det\left[\hat{C}(i,i^{\prime})\right]-\det\left[\hat{C}(i,i^{\prime})-\frac{1}{w_{i}+1}\right]=\sum_{k,\ell=1}^{N}(-1)^{\ell+k}\det[\hat{C}^{\ell,k}]\cdot\frac{1}{w_{\ell}+1},$
(5.50)

where $\hat{C}^{\ell,k}$ is the matrix obtained by removing row $\ell$ and
column $k$ from $\hat{C}$. Similarly

$\det\left[\hat{C}(i,i^{\prime})\right]-\det\left[\hat{C}(i,i^{\prime})+\frac{1}{w_{i^{\prime}}^{\prime}+1}\right]=-\sum_{k,\ell=1}^{N}(-1)^{\ell+k}\det[\hat{C}^{\ell,k}]\cdot\frac{1}{w_{k}^{\prime}+1}.$
(5.51)

Now since $\hat{C}^{\ell,k}$ has the same entry as $\hat{C}$ except for
lacking row $\ell$ and column $k$, by (5.49) we have

$\det[\hat{C}^{\ell,k}]=(1-\mu)^{N-2}\det[C^{\ell,k}]\cdot\left(1-\mu\cdot\frac{B(-1/p)A(0)}{A(-1/p)B(0)}\cdot\frac{w_{k}^{\prime}(1+pw_{\ell})}{w_{\ell}(1+pw_{k}^{\prime})}\right).$
(5.52)

Where $C^{\ell,k}$ is obtained from removing row $\ell$ and column $k$ from
the $N\times N$ Cauchy matrix $C$ with
$C(i,i^{\prime})=\frac{1}{w_{i}-w_{i^{\prime}}^{\prime}}$ and
$A(z):=\prod_{j=1}^{N}(z-w_{j})$ and
$B(z):=\prod_{j=1}^{N}(z-w_{j}^{\prime})$. Inserting (5.52) into (5.50) we see

$\displaystyle\det\left[\hat{C}(i,i^{\prime})\right]-\det\left[\hat{C}(i,i^{\prime})-\frac{1}{w_{i}+1}\right]=(1-\mu)^{N-2}\sum_{k,\ell=1}^{N}(-1)^{\ell+k}\det[C^{\ell,k}]\cdot\frac{1}{w_{\ell}+1}$
(5.53)
$\displaystyle-\mu(1-\mu)^{N-2}\cdot\frac{B(-1/p)A(0)}{A(-1/p)B(0)}\sum_{k,\ell=1}^{N}(-1)^{\ell+k}\det[C^{\ell,k}]\cdot\frac{w_{k}^{\prime}(1+pw_{\ell})}{w_{\ell}(1+w_{\ell})(1+pw_{k}^{\prime})}.$

Similarly

$\displaystyle\det\left[\hat{C}(i,i^{\prime})\right]-\det\left[\hat{C}(i,i^{\prime})+\frac{1}{w_{i^{\prime}}^{\prime}+1}\right]=-(1-\mu)^{N-2}\sum_{k,\ell=1}^{N}(-1)^{\ell+k}\det[C^{\ell,k}]\cdot\frac{1}{w_{k}^{\prime}+1}$
(5.54)
$\displaystyle+\mu(1-\mu)^{N-2}\cdot\frac{B(-1/p)A(0)}{A(-1/p)B(0)}\sum_{k,\ell=1}^{N}(-1)^{\ell+k}\det[C^{\ell,k}]\cdot\frac{w_{k}^{\prime}(1+pw_{\ell})}{w_{\ell}(1+w_{k^{\prime}}^{\prime})(1+pw_{k}^{\prime})}.$

Now by (5.59) and Lemma 5.15 we have

$\displaystyle\sum_{k,\ell=1}^{N}(-1)^{\ell+k}\det[C^{\ell,k}]\cdot\frac{1}{w_{\ell}+1}=\det[C]\cdot\left(1-\frac{B(-1)}{A(-1)}\right),$
(5.55)
$\displaystyle\sum_{k,\ell=1}^{N}(-1)^{\ell+k}\det[C^{\ell,k}]\cdot\frac{1}{w_{k}^{\prime}+1}=\det[C]\cdot\left(\frac{A(-1)}{B(-1)}-1\right),$
$\displaystyle\sum_{k,\ell=1}^{N}(-1)^{\ell+k}\det[C^{\ell,k}]\cdot\frac{w_{k}^{\prime}(1+pw_{\ell})}{w_{\ell}(1+w_{\ell})(1+pw_{k}^{\prime})}$
$\displaystyle=-\frac{1}{p}\det[C]\cdot\left[\left(1-\frac{A(-1/p)}{B(-1/p)}\right)\frac{B(0)}{A(0)}+\left(p-1+\frac{A(-1/p)}{B(-1/p)}\right)\frac{B(-1)}{A(-1)}-p\right],$
$\displaystyle\sum_{k,\ell=1}^{N}(-1)^{\ell+k}\det[C^{\ell,k}]\cdot\frac{w_{k}^{\prime}(1+pw_{\ell})}{w_{\ell}(1+w_{\ell})(1+pw_{k}^{\prime})}$
$\displaystyle=-\frac{1}{1-p}\det[C]\cdot\left[\left(\frac{B(0)}{A(0)}-1\right)\frac{A(-1/p)}{B(-1/p)}+\left(p-\frac{B(0)}{A(0)}\right)\frac{A(-1)}{B(-1)}+1-p\right].$

Inserting (5.55) into (5.53) and (5.54) and combine with (5.46), after some
tedious simplification we conclude that

$\hat{S}_{N}=(1-\mu)^{N}\left(\prod_{j=1}^{N}(w_{j}+1)-\prod_{j=1}^{N}(w_{j}^{\prime}+1)\right)\cdot\det\left[\frac{1}{w_{i}-w_{i^{\prime}}^{\prime}}\right]_{i,i^{\prime}=1}^{N}.$
(5.56)

This completes the proof of Proposition 5.4.

###  5.6. Perturbation formulas for Cauchy determinants

In this section we collect all the elementary linear algebra facts needed in
the proof of Proposition 5.4. Some of them have already been discussed in [4].
First we state a general linear algebra lemma on rank-one perturbations:

######  Lemma 5.13.

Let $D=[D_{ij}]_{i,j=1}^{n}$ be an $n\times n$ matrix. Then for any function
$f,g:\mathbb{C}\to\mathbb{C}$ and complex numbers $x_{1},\cdots,x_{n}$,
$y_{1},\cdots,y_{n}$ we have

$\displaystyle\det\left[D_{ij}+f(x_{i})g(y_{j})\right]_{i,j=1}^{n}=\det[D]+\sum_{k,\ell=1}^{n}(-1)^{\ell+k}\det[D^{\ell,k}]g(y_{k})f(x_{\ell}),$
(5.57)

where $D^{\ell,k}$ is obtained by removing row $\ell$ and column $k$ from $D$.

###### Proof.

For $D$ invertible by rank-one property and Cramer’s rule we have

$\displaystyle\det\left[D_{ij}+f(x_{i})g(y_{j})\right]_{i,j=1}^{n}$
$\displaystyle=\det[D]\left(1+\sum_{k,\ell=1}^{n}g(y_{k})(D^{-1})_{k,\ell}f(x_{\ell})\right)$
$\displaystyle=\det[D]+\sum_{k,\ell=1}^{n}(-1)^{\ell+k}\det[D^{\ell,k}]g(y_{k})f(x_{\ell}).$

For general matrix $D$ we pick $\\{\epsilon_{k}\\}_{k=1}^{\infty}$ such that
$\epsilon_{k}\to 0$ as $k\to\infty$ and $D+\epsilon_{k}I_{n}$ are invertible
for all $\epsilon_{k}$. Now apply the above argument for $D+\epsilon_{k}I_{n}$
and let $k\to\infty$. ∎

Next we specialize to the case for $C$ be a Cauchy matrix when the minors can
be explicitly calculated:

######  Lemma 5.14.

Assume further that the matrix $C$ is a Cauchy matrix with $(i,j)$-th entry
$\frac{1}{x_{i}-y_{j}}$ for distinct complex numbers $x_{1},\cdots,x_{n}$ and
$y_{1},\cdots,y_{n}$. Then we further have

$\det\left[C_{ij}+f(x_{i})g(y_{j})\right]_{i,j=1}^{n}=\det[C]\cdot\left(1-\sum_{k,\ell=1}^{n}\frac{f(x_{\ell})B(x_{\ell})A(y_{k})g(y_{k})}{(x_{\ell}-y_{k})A^{\prime}(x_{\ell})B^{\prime}(y_{k})}\right).$
(5.58)

Here $A(z):=\prod_{i=1}^{n}(z-x_{i})$ and $B(z):=\prod_{i=1}^{n}(z-y_{i})$ are
monic polynomials with roots at $x_{i}$’s and $y_{i}$’s.

###### Proof.

For Cauchy matrix $C$ we have

$\det[C]=\frac{\prod_{1\leq i<j\leq n}(x_{i}-x_{j})(y_{i}-y_{j})}{\prod_{1\leq
i,j\leq n}(x_{i}-y_{j})}.$

Note that $C^{\ell,k}$ is also a Cauchy matrix so we have

$\det[C^{\ell,k}]=\frac{\prod_{\begin{subarray}{c}1\leq i<j\leq n\\\
i\neq\ell,j\neq
k\end{subarray}}(x_{i}-x_{j})(y_{i}-y_{j})}{\prod_{\begin{subarray}{c}1\leq
i,j\leq n\\\ i\neq\ell,j\neq k\end{subarray}}(x_{i}-y_{j})}.$

Hence

$\frac{\det[C^{\ell,k}]}{\det[C]}=(-1)^{\ell+k+1}\frac{B(x_{\ell})A(y_{k})}{(x_{\ell}-y_{k})A^{\prime}(x_{\ell})B^{\prime}(y_{k})}.$
(5.59)

Now (5.57) and (5.59) imply (5.58). ∎

For special choices of $f$ and $g$, (5.58) can be further simplified using
residue theorem. We list here all the situations encountered in the proof of
Proposition 5.4.

######  Lemma 5.15.

Given distinct complex numbers $x_{1},\cdots,x_{n}$ and $y_{1},\cdots,y_{n}$.
Let $C$ be the Cauchy matrix with $(i,j)$-th entry $\frac{1}{x_{i}-y_{j}}$ and
$A(z)=\prod_{i=1}^{n}(z-x_{i})$ and $B(z)=\prod_{i=1}^{n}(z-y_{i})$. Then

  1. (1)

For $f(x)=\frac{x}{(1+x)(1+px)}$ and $g(y)=\frac{1+py}{y}$ we have
$\sum_{k,\ell=1}^{n}\frac{f(x_{\ell})B(x_{\ell})A(y_{k})g(y_{k})}{(x_{\ell}-y_{k})A^{\prime}(x_{\ell})B^{\prime}(y_{k})}$
equals

$-\frac{1}{1-p}\cdot\left[\left(\frac{A(0)}{B(0)}-1\right)\frac{B(-1/p)}{A(-1/p)}+\left(p-\frac{A(0)}{B(0)}\right)\frac{B(-1)}{A(-1)}+1-p\right].$
(5.60)

  2. (2)

For $f(x)=\frac{x}{1+px}$ and $g(y)=\frac{1+py}{y(1+y)}$ we have
$\sum_{k,\ell=1}^{n}\frac{f(x_{\ell})B(x_{\ell})A(y_{k})g(y_{k})}{(x_{\ell}-y_{k})A^{\prime}(x_{\ell})B^{\prime}(y_{k})}$
equals

$-\frac{1}{p}\cdot\left[\left(1-\frac{B(-1/p)}{A(-1/p)}\right)\frac{A(0)}{B(0)}+\left(p-1+\frac{B(-1/p)}{A(-1/p)}\right)\frac{A(-1)}{B(-1)}-p\right].$
(5.61)

  3. (3)

For $f(x)=\frac{1}{1+x}$ and $g(y)=1$ we have

$\sum_{k,\ell=1}^{n}\frac{f(x_{\ell})B(x_{\ell})A(y_{k})g(y_{k})}{(x_{\ell}-y_{k})A^{\prime}(x_{\ell})B^{\prime}(y_{k})}=\frac{B(-1)}{A(-1)}-1.$
(5.62)

###### Proof.

We will only prove part $(1)$, the arguments for the other parts are similar.
For $f(z)=\frac{z}{(1+z)(1+pz)}$ and $g(\xi)=\frac{1+p\xi}{\xi}$ consider the
double contour integral

$\oint_{|z|=R}\oint_{|\xi|=r}\frac{{\mathrm{d}}\xi}{2\pi\mathrm{i}}\frac{{\mathrm{d}}z}{2\pi\mathrm{i}}f(z)g(\xi)\frac{B(z)A(\xi)}{(z-\xi)A(z)B(\xi)}.$

Where $R>r$ are both large enough so that all the possible poles of the
integrand are inside the integral contours. Now since for fixed $r$ the
integrand is of order $O(R^{-2})$, the double integral goes to $0$ as
$R\to\infty$. Thus for all $R$ large enough the double contour integral equals
$0$. On the other hand by residue theorem we have

$\displaystyle 0$
$\displaystyle=\oint_{|z|=R}\oint_{|\xi|=r}\frac{{\mathrm{d}}\xi}{2\pi\mathrm{i}}\frac{{\mathrm{d}}z}{2\pi\mathrm{i}}f(z)g(\xi)\frac{B(z)A(\xi)}{(z-\xi)A(z)B(\xi)}$
(5.63)
$\displaystyle=\oint_{|z|=R}\frac{{\mathrm{d}}z}{2\pi\mathrm{i}}\frac{f(z)}{z}\frac{B(z)A(0)}{A(z)B(0)}+\sum_{k=1}^{n}\oint_{|z|=R}\frac{{\mathrm{d}}z}{2\pi\mathrm{i}}f(z)g(y_{k})\frac{B(z)A(y_{k})}{(z-y_{k})A(z)B^{\prime}(y_{k})}$
$\displaystyle=\frac{1}{1-p}\sum_{k=1}^{n}\frac{A(y_{k})}{B^{\prime}(y_{k})}\left(\frac{B(-1)}{A(-1)}\frac{1+py_{k}}{y_{k}(1+y_{k})}-\frac{B(-1/p)}{A(-1/p)}\frac{1}{y_{k}}\right)$
$\displaystyle+\sum_{k,\ell=1}^{n}f(x_{\ell})g(y_{k})\frac{B(x_{\ell})A(y_{k})}{(x_{\ell}-y_{k})A^{\prime}(x_{\ell})B^{\prime}(y_{k})}.$

Where in the first equality the first contour integral is $O(R^{-2})$ hence
$0$ for $R$ large enough. The single sum over $1\leq k\leq n$ can be obtained
as the residue terms for the following single contour integral:

$\oint_{|\xi|=r}\frac{{\mathrm{d}}\xi}{2\pi\mathrm{i}}\frac{1+p\xi}{\xi(1+\xi)}\frac{A(\xi)}{B(\xi)}=\frac{A(0)}{B(0)}-(1-p)\frac{A(-1)}{B(-1)}+\sum_{k=1}^{n}\frac{A(y_{k})}{B^{\prime}(y_{k})}\frac{1+py_{k}}{y_{k}(1+y_{k})}.$

Here $r$ is large enough so that $|\xi|=r$ contains all possible poles of the
integrand inside. On the other hand by considering the residue at $\infty$ we
have $\oint_{|\xi|=r}\frac{d\xi}{2\pi
i}\frac{1+p\xi}{\xi(1+\xi)}\frac{A(\xi)}{B(\xi)}=p$. Hence

$\sum_{k=1}^{n}\frac{A(y_{k})}{B^{\prime}(y_{k})}\frac{1+py_{k}}{y_{k}(1+y_{k})}=p-\frac{A(0)}{B(0)}+(1-p)\frac{A(-1)}{B(-1)}.$
(5.64)

Similar residue analysis on the contour integral
$\oint_{|\xi|=r}\frac{{\mathrm{d}}\xi}{2\pi\mathrm{i}}\frac{1}{\xi}\frac{A(\xi)}{B(\xi)}$
gives

$\sum_{k=1}^{n}\frac{A(y_{k})}{B^{\prime}(y_{k})}\frac{1}{y_{k}}=1-\frac{A(0)}{B(0)}.$
(5.65)

Inserting (5.64) and (5.65) into (5.63) we obtain

$\displaystyle\sum_{k,\ell=1}^{n}f(x_{\ell})g(y_{k})\frac{B(x_{\ell})A(y_{k})}{(x_{\ell}-y_{k})A^{\prime}(x_{\ell})B^{\prime}(y_{k})}$
(5.66)
$\displaystyle=-\frac{1}{1-p}\cdot\left[\left(p-\frac{A(0)}{B(0)}\right)\frac{B(-1)}{A(-1)}+1-p-\left(1-\frac{A(0)}{B(0)}\right)\frac{B(-1/p)}{A(-1/p)}\right].$

This completes the proof of part (1). ∎

##  6\. Large-time asymptotics under relaxation time scale

In this section we discuss the large time limit of the multi-point
distribution for dpTASEP($L,N,\vec{y}$) under the relaxation time scale
$t=O(L^{3/2})$. In Theorem 2.4 we state the limit theorem for general initial
condition satisfying certain conditions. Below is the precise assumptions on
the initial conditions we need:

###  6.1. Assumptions on the initial condition

We now state the assumptions on the sequence of the initial conditions
$\vec{y}(L)$ under which we prove the limit theorem. The conditions are in
terms of the global energy function and the characteristic function defined in
Definition 2.12.

######  Assumption 6.1.

We assume that the sequence of the initial profiles $\vec{y}=\vec{y}(L)$
satisfies the following three conditions as $L\to\infty$.

  1. (A)

(Convergence of global energy) There exist a constant $r\in(0,1)$ and a non-
zero function $E_{\mathrm{ic}}(\mathrm{z})$ such that for every
$0<\epsilon<1/2$,

$\mathcal{E}_{\vec{y}}(z)=E_{\mathrm{ic}}(\mathrm{z})\left(1+O(L^{\epsilon-1/2})\right)$

uniformly for $|\mathrm{z}|<{r}$ as $L\to\infty$.

  2. (B)

(Convergence of characteristic function) There exist constants
$0<r_{1}<r_{2}<1$ and a function
$\mathrm{ch}_{\mathrm{ic}}(\eta,\xi;\mathrm{z})$ such that for every
$0<\epsilon<1/8$,

$\chi_{\vec{y}}(v,u;z)=\mathrm{ch}_{\mathrm{ic}}(\eta,\xi;\mathrm{z})+O(L^{4\epsilon-1/2})$

uniformly for $r_{1}<|\mathrm{z}|<r_{2}$, $u\in\mathcal{L}_{z}^{(\epsilon)}$
and $v\in\mathcal{R}_{z}^{(\epsilon)}$ as $L\to\infty$ where

$\xi=\mathcal{M}_{L,\mathrm{left}}(u)\in\mathrm{L}_{\mathrm{z}}\quad\text{and}\quad\eta=\mathcal{M}_{L,\mathrm{right}}(v)\in\mathrm{R}_{\mathrm{z}}$

are the images under the maps defined in Lemma 7.4.

  3. (C)

(Tail estimates of characteristic function) Let $r_{1}$ and $r_{2}$ be same as