in (B). There are constants $\epsilon^{\prime\prime},C^{\prime}>0$ such that

$|\chi_{\vec{y}}(v,u;z)|\leq C^{\prime}L^{\epsilon^{\prime\prime}}$ (6.1)

for all $(v,u)\in\mathcal{R}_{z}\times\mathcal{L}_{z}$ for all
$r_{1}<|\mathrm{z}|<r_{2}$.

###  6.2. Step and flat initial conditions

It turns out that Assumption 6.1 is not easy to check in general. Nevertheless
we are able to verify them for at least the classical step and flat initial
conditions. The following proposition combined with Theorem 2.4 gives the
corresponding limit theorems for dpTASEP started with step and flat initial
conditions.

######  Proposition 6.2.

  1. (i)

For step initial condition $\vec{y}_{\mathrm{step}}=(-1,-2,\cdots-N)$,
Assumption 6.1 holds with

$E_{\mathrm{step}}(\mathrm{z})=1\quad\text{and}\quad\mathrm{ch}_{\mathrm{step}}(\eta,\xi;\mathrm{z})=1.$
(6.2)

  2. (ii)

For flat initial condition $\vec{y}_{\mathrm{flat}}=(-d,\cdots,-Nd)$, where we
assume $d=L/N\in\mathbb{N}$, Assumption 6.1 holds with

$E_{\mathrm{flat}}(\mathrm{z})=(1-\mathrm{z})^{-1/4}e^{-B(\mathrm{z})}\quad\text{and}\quad\mathrm{ch}_{\mathrm{flat}}(\eta,\xi;\mathrm{z})=e^{-\mathrm{h}(\xi,\mathrm{z})-\mathrm{h}(\eta,\mathrm{z})}\eta(\eta-\xi)\mathbf{1}_{\xi=-\eta}$
(6.3)

for $0<|\mathrm{z}|<1$, where $B(\mathrm{z})$ is defined in (6.13),
$\mathrm{h}(\zeta,\mathrm{z})$ is defined in (6.14) and (6.15).

The step case is trivial since by the discussion in Remark 2.13 we have
$\mathcal{E}_{\mathrm{step}}(z)=\chi_{\mathrm{step}}(v,u;z)\equiv 1$. The
calculation for flat case is a bit more involved and we postpone the proof
till Section 7.5.

###  6.3. Formula for the limiting distribution

The following formula for the relaxation-time limiting distribution was first
obtained in [4] for step initial condition and [5] for more general initial
conditions. The formula involves
$\mathrm{C_{\mathrm{ic}}^{\mathrm{per}}}(\vec{\mathrm{z}})$ which are limits
of $\mathscr{C}_{\vec{y}}^{(L)}(\vec{z})$ and operators
$\mathrm{K}_{1}^{\mathrm{per}}$ and $\mathrm{K}_{\mathrm{ic}}^{\mathrm{per}}$
which is a limit of $\mathscr{K}_{1}^{(L)}$ and $\mathscr{K}_{\vec{y}}^{(L)}$.
The operators $\mathrm{K}_{1}^{\mathrm{per}}$ and
$\mathrm{K}_{\mathrm{ic}}^{\mathrm{per}}$ are defined on the sets

$\mathrm{S}_{1}:=\mathrm{L}_{\mathrm{z}_{1}}\cup\mathrm{R}_{\mathrm{z}_{2}}\cup\mathrm{L}_{\mathrm{z}_{3}}\cup\cdots\cup\begin{dcases}\mathrm{R}_{\mathrm{z}_{m}},&\text{if
$m$ is even},\\\ \mathrm{L}_{\mathrm{z}_{m}},&\text{if $m$ is
odd},\end{dcases}$ (6.4)

and

$\mathrm{S}_{2}:=\mathrm{R}_{\mathrm{z}_{1}}\cup\mathrm{L}_{\mathrm{z}_{2}}\cup\mathrm{R}_{\mathrm{z}_{3}}\cup\cdots\cup\begin{dcases}\mathrm{L}_{\mathrm{z}_{m}},&\text{if
$m$ is even},\\\ \mathrm{R}_{\mathrm{z}_{m}},&\text{if $m$ is
odd},\end{dcases}$ (6.5)

where $\mathrm{L}_{\mathrm{z}}$ and $\mathrm{R}_{\mathrm{z}}$ are the sets
defined in Definition 6.3. We express
$\mathbb{F}_{\mathrm{ic}}^{\mathrm{per}}$ in terms of the above terms.

######  Definition 6.3.

Given $0<|\mathrm{z}|<1$, we define the discrete sets
$\mathrm{S}_{\mathrm{z}}:=\mathrm{L}_{\mathrm{z}}\cup\mathrm{R}_{\mathrm{z}}$
where

$\mathrm{L}_{\mathrm{z}}:=\\{\xi\in\mathbb{C}:e^{-\xi^{2}/2}=\mathrm{z}\\}\cap\\{\text{Re}(\xi)<0\\},\quad\mathrm{R}_{\mathrm{z}}:=\\{\eta\in\mathbb{C}:e^{-\eta^{2}/2}=\mathrm{z}\\}\cap\\{\text{Re}(\eta)>0\\}.$
(6.6)

Figure 2. The roots for the equation $e^{-\zeta^{2}/2}=\mathrm{z}$ with
$\mathrm{z}=0.08$, the dashed lines are the corresponding level curve
$|e^{-\zeta^{2}/2}|=|\mathrm{z}|$ for the same $\mathrm{z}$.

######  Definition 6.4 (Limiting function).

Let $\mathbf{x}=(\mathrm{x}_{1},\cdots,\mathrm{x_{m}})$,
$\boldsymbol{\gamma}=(\gamma_{1},\cdots,\gamma_{m})$, and
$\boldsymbol{\tau}=(\tau_{1},\cdots,\tau_{m})$ be points in $\mathrm{R}^{m}$
such that $\mathrm{p}_{j}=(\gamma_{j},\tau_{j})\in[0,1]\times\mathbb{R}_{>0}$.
Assume that

$0<\tau_{1}\leq\cdots\leq\tau_{m}$

and that $\mathrm{x}_{i}<\mathrm{x}_{i+1}$ when $\tau_{i}=\tau_{i+1}$ for
$i=1,\cdots,m-1$. Define

$\mathbb{F}_{\mathrm{ic}}^{\mathrm{per}}(\mathrm{x}_{1},\cdots,\mathrm{x}_{m};\mathrm{p}_{1},\cdots,\mathrm{p}_{m})=\oint\cdots\oint\mathrm{C}_{\mathrm{ic}}^{\mathrm{per}}(\vec{\mathrm{z}})\mathrm{D}_{\mathrm{ic}}^{\mathrm{per}}(\vec{\mathrm{z}})\frac{{\mathrm{d}}\mathrm{z}_{m}}{2\pi{\mathrm{i}}\mathrm{z}_{m}}\cdots\frac{{\mathrm{d}}\mathrm{z}_{1}}{2\pi{\mathrm{i}}\mathrm{z}_{1}},$
(6.7)

where $\vec{\mathrm{z}}=(\mathrm{z}_{1},\cdots,\mathrm{z}_{m})$ and the
contours are nested circles satisfying
$0<|\mathrm{z}_{m}|<\cdots<|\mathrm{z}_{1}|<1$ and also,
$r_{1}<|\mathrm{z}_{1}|<r_{2}$ with $r_{1},r_{2}$ being the constants in
Assumption 6.1 (B). The first function in the integrand is given by

$\mathrm{C}_{\mathrm{ic}}^{\mathrm{per}}(\vec{\mathrm{z}})=E_{\mathrm{ic}}(\mathrm{z}_{1})\mathrm{C}_{\mathrm{step}}^{\mathrm{per}}(\vec{\mathrm{z}}).$
(6.8)

The second function is

$\mathrm{D}_{\mathrm{ic}}^{\mathrm{per}}(\vec{\mathrm{z}})=\det(I-\mathrm{K}_{1}^{\mathrm{per}}\mathrm{K}_{\mathrm{ic}}^{\mathrm{per}}),$
(6.9)

where
$\mathrm{K}_{1}^{\mathrm{per}}:\ell^{2}(\mathrm{S}_{2})\to\ell^{2}(\mathrm{S}_{1})$
and
$\mathrm{K}_{\mathrm{ic}}^{\mathrm{per}}:\ell^{2}(\mathrm{S}_{1})\to\ell^{2}(\mathrm{S}_{2})$
are given by
$\mathrm{K}_{1}^{\mathrm{per}}=\mathrm{K}_{\mathrm{step},1}^{\mathrm{per}}$
and

$\mathrm{K}_{\mathrm{ic}}^{\mathrm{per}}(\zeta,\zeta^{\prime}):=\begin{dcases}\mathrm{ch}_{\mathrm{ic}}(\zeta,\zeta^{\prime};\mathrm{z}_{1})\mathrm{K}_{\mathrm{step},2}^{\mathrm{per}}(\zeta,\zeta^{\prime}),\quad&\text{if
$\zeta\in\mathrm{R}_{\mathrm{z}_{1}}\text{ and
}\zeta^{\prime}\in\mathrm{L}_{\mathrm{z}_{1}}$,}\\\
\mathrm{K}_{\mathrm{step},2}^{\mathrm{per}}(\zeta,\zeta^{\prime}),&\text{otherwise.}\end{dcases}$

The function $\mathrm{C}_{\mathrm{step}}^{\vec{\mathrm{z}}}(\vec{\mathrm{z}})$
and kernels $\mathrm{K}_{\mathrm{step},1}^{\mathrm{per}}$ and
$\mathrm{K}_{\mathrm{step},2}^{\mathrm{per}}$ are first obtained in [4] and
the definitions will be recalled in the next two sections for completeness.

###  6.4. The factor
$\mathrm{C}_{\mathrm{step}}^{\mathrm{per}}(\vec{\mathrm{z}})$

Let $\mathrm{Li}_{s}(\mathrm{z})$ be the polylogarithm function which is
defined by

$\mathrm{Li}_{s}(\mathrm{z})=\sum_{k=1}^{\infty}\frac{\mathrm{z}^{k}}{k^{s}}=\frac{\mathrm{z}}{\Gamma(s)}\int_{0}^{\infty}\frac{t^{s-1}}{e^{t}-\mathrm{z}}dt.$
(6.10)

for $|\mathrm{z}|<1$ and $s\in\mathbb{C}$. Set

$A_{1}(\mathrm{z})=-\frac{1}{\sqrt{2\pi}}\mathrm{Li}_{3/2}(\mathrm{z})\quad\text{and}\quad
A_{2}(z)=-\frac{1}{\sqrt{2\pi}}\mathrm{Li}_{5/2}(\mathrm{z}).$ (6.11)

Let $\log\mathrm{z}$ denote the principal branch of the logarithm function
with cut $\mathbb{R}_{\leq 0}$. Set

$B(\mathrm{z},\mathrm{z}^{\prime})=\frac{\mathrm{z}\mathrm{z}^{\prime}}{2}\int\int\frac{\eta\xi\log(-\xi+\eta)}{(e^{-\xi^{2}/2}-\mathrm{z})(e^{-\eta^{2}/2}-\mathrm{z}^{\prime})}\frac{{\mathrm{d}}\xi}{2\pi{\mathrm{i}}}\frac{{\mathrm{d}}\eta}{2\pi{\mathrm{i}}}=\frac{1}{4\pi}\sum_{k,k^{\prime}\geq
1}\frac{\mathrm{z}^{k}(\mathrm{z}^{\prime})^{k^{\prime}}}{(k+k^{\prime})\sqrt{kk^{\prime}}}$
(6.12)

for $0<|\mathrm{z}|,|\mathrm{z}^{\prime}|<1$ where the integral contours are
the vertical lines $\mathrm{Re}(\xi)=\mathrm{a}$ and
$\mathrm{Re}(\eta)=\mathrm{b}$ with constants $\mathrm{a}$ and $\mathrm{b}$
satisfying
$-\sqrt{-\log|\mathrm{z}|}<\mathrm{a}<0<\mathrm{b}<\sqrt{-\log|\mathrm{z}^{\prime}|}$.
We also set $B(\mathrm{z}):=B(\mathrm{z},\mathrm{z})$. One can check that

$B(\mathrm{z})=B(\mathrm{z},\mathrm{z})=\frac{1}{4\pi}\int_{0}^{\mathrm{z}}\frac{(\mathrm{Li}_{1/2}(\mathrm{y}))^{2}}{\mathrm{y}}{\mathrm{d}}\mathrm{y}.$
(6.13)

######  Definition 6.5.

For $\vec{\mathrm{z}}=(\mathrm{z}_{1},\cdots,\mathrm{z}_{m})$ satisfying
$0<|\mathrm{z}_{j}|<1$ and $\mathrm{z}_{j}\neq\mathrm{z}_{j+1}$ for all $j$,
we define

$\mathrm{C}_{\mathrm{step}}^{\mathrm{per}}(\vec{\mathrm{z}})=\left[\prod_{\ell=1}^{m}\frac{\mathrm{z}_{\ell}}{\mathrm{z}_{\ell}-\mathrm{z}_{\ell+1}}\right]\left[\prod_{\ell=1}^{m}\frac{e^{\mathrm{x}_{\ell}A_{1}(\mathrm{z}_{\ell})+\tau_{\ell}A_{2}(\mathrm{z}_{\ell})}}{e^{\mathrm{x}_{\ell}A_{1}(\mathrm{z}_{\ell+1})+\tau_{\ell}A_{2}(\mathrm{z}_{\ell+1})}}e^{2B(\mathrm{z}_{\ell})-2B(\mathrm{z}_{\ell+1},\mathrm{z}_{\ell})}\right],$

where we set $\mathrm{z}_{m+1}=0$.

Note that $\mathrm{C}_{\mathrm{step}}^{\mathrm{per}}(\vec{\mathrm{z}})$, and
hence $\mathrm{C}_{\mathrm{ic}}^{\mathrm{per}}(\vec{\mathrm{z}})$, depend on
$\mathrm{x}_{i}$ and $\tau_{i}$, but not the spatial parameters $\gamma_{i}$.

###  6.5. The operators $\mathrm{K}_{\mathrm{step},1}^{\mathrm{per}}$ and
$\mathrm{K}_{\mathrm{step},2}^{\mathrm{per}}$

Set

$\mathrm{h}(\zeta,\mathrm{z})=\frac{\mathrm{z}}{2\pi\mathrm{i}}\int_{\mathrm{i}\mathbb{R}}\frac{\mathrm{w}\log(\mathrm{w}-\zeta)}{e^{-\mathrm{w}^{2}/2}-\mathrm{z}}{\mathrm{d}}\mathrm{w}\qquad\text{for
$\mathrm{Re}(\zeta)<0$ and $|\mathrm{z}|<1$}$ (6.14)

and define

$\mathrm{h}(\zeta,\mathrm{z})=\mathrm{h}(-\zeta,\mathrm{z})\qquad\text{for
$\mathrm{Re}(\zeta)>0$ and $|\mathrm{z}|<1$.}$ (6.15)

For each $i$, define

$\mathrm{f}_{i}(\zeta)=\begin{dcases}e^{-\frac{1}{3}(\tau_{i}-\tau_{i-1})\zeta^{3}+\frac{1}{2}(\gamma_{i}-\gamma_{i-1})\zeta^{2}+(\mathrm{x}_{i}-\mathrm{x}_{i-1})\zeta}&\text{for
$\mathrm{Re}(\zeta)<0$,}\\\
e^{\frac{1}{3}(\tau_{i}-\tau_{i-1})\zeta^{3}-\frac{1}{2}(\gamma_{i}-\gamma_{i-1})\zeta^{2}-(\mathrm{x}_{i}-\mathrm{x}_{i-1})\zeta}&\text{for
$\mathrm{Re}(\zeta)>0$,}\end{dcases}$ (6.16)

where we set $\tau_{0}=\gamma_{0}=\mathrm{x}_{0}=0$. We also define

$\mathrm{Q}_{1}(j)=1-\frac{\mathrm{z}_{j-(-1)^{j}}}{\mathrm{z}_{j}}\quad\text{and}\quad\mathrm{Q}_{2}(j)=1-\frac{\mathrm{z}_{j+(-1)^{j}}}{\mathrm{z}_{j}},$

where we set $\mathrm{z_{0}}=\mathrm{z}_{m+1}=0$.

######  Definition 6.6.

Let $\mathrm{S}_{1}$ and $\mathrm{S}_{2}$ be the discrete sets defined in
(6.4) and (6.5). Define

$\mathrm{K}_{\mathrm{step},1}^{\mathrm{per}}:\ell^{2}(\mathrm{S}_{2})\to\ell^{2}(\mathrm{S}_{1})\quad\text{and}\quad\mathrm{K}_{\mathrm{step},2}^{\mathrm{per}}:\ell^{2}(\mathrm{S}_{1})\to\ell^{2}(\mathrm{S}_{2})$

be the operators defined by their kernels

$\mathrm{K}_{\mathrm{step},1}^{\mathrm{per}}(\zeta,\zeta^{\prime})=(\delta_{i}(j)+\delta_{i}(j+(-1)^{i}))\frac{\mathrm{f}_{i}(\zeta)e^{2\mathrm{h}(\zeta,\mathrm{z}_{i})-\mathrm{h}(\zeta,\mathrm{z}_{i-(-1)^{i}})-\mathrm{h}(\zeta^{\prime},z_{j-(-1)^{j}})}}{\zeta(\zeta-\zeta^{\prime})}\mathrm{Q}_{1}(j)$

and

$\mathrm{K}_{\mathrm{step},2}^{\mathrm{per}}(\zeta^{\prime},\zeta)=(\delta_{j}(i)+\delta_{j}(i-(-1)^{j}))\frac{\mathrm{f}_{j}(\zeta^{\prime})e^{2\mathrm{h}(\zeta^{\prime},\mathrm{z}_{j})-\mathrm{h}(\zeta^{\prime},\mathrm{z}_{j+(-1)^{j}})-\mathrm{h}(\zeta,z_{i+(-1)^{i}})}}{\zeta^{\prime}(\zeta^{\prime}-\zeta)}\mathrm{Q}_{2}(i)$

for

$\zeta\in(\mathrm{L}_{\mathrm{z}_{i}}\cup\mathrm{R}_{\mathrm{z}_{i}})\cap\mathrm{S}_{1}\quad\text{and}\quad\zeta^{\prime}\in(\mathrm{L}_{\mathrm{z}_{j}}\cup\mathrm{R}_{\mathrm{z}_{j}})\cap\mathrm{S}_{2}$

with $1\leq i,j\leq m$. Here $\mathrm{L}_{\mathrm{z}_{i}}$ and
$\mathrm{R}_{\mathrm{z}_{i}}$ are again defined in Definition 6.3.

##  7\. Proof of Theorem 2.4

In this section we discuss the proof of Theorem 2.4. The ideas are similar to
the one in [4, 5] so we omit some technical details. Clearly the theorem
follows immediately from the following two lemmas, dealing with the
asymptotics of $\mathscr{C}_{\vec{y}}^{(L)}(z)$ and
$\mathscr{D}_{\vec{y}}^{(L)}(z)$ appearing in the finite-time formula (2.6),
respectively.

######  Lemma 7.1 (Asymptotics of $\mathscr{C}_{\vec{y}}^{(L)}(\vec{z})$).

Under the same assumption as in Theorem 2.4, we have for fixed
$0<\epsilon<1/2$

$\mathscr{C}_{\vec{y}}^{(L)}(\vec{z})=\mathrm{C}_{\mathrm{ic}}^{\mathrm{per}}(\vec{\mathrm{z}})\left(1+O(L^{\epsilon-1/2})\right),\quad\mathrm{as}\
L\to\infty.$ (7.1)

The functions $\mathscr{C}_{\vec{y}}^{(L)}(z)$ and
$\mathrm{C}_{\mathrm{ic}}^{\mathrm{per}}(\mathrm{z})$ are defined in (2.23)
and (4.2) respectively, with
$z_{i}^{L}=(-1)^{N}\mathbbm{r}_{c}^{L}\mathrm{z}_{i}$, for $1\leq i\leq m$.

######  Lemma 7.2 (Asymptotics of $\mathscr{D}_{\vec{y}}^{(L)}(\vec{z})$).

Under the same assumption as in Theorem 2.4, we have the convergence

$\lim_{L\to\infty}\mathscr{D}_{\vec{y}}^{(L)}(\vec{z})=\mathrm{D}_{\mathrm{ic}}^{\mathrm{per}}(\vec{\mathrm{z}}),$
(7.2)

where the Fredholm determinants $\mathscr{D}_{\vec{y}}^{(L)}(\vec{z})$ and
$\mathrm{D}_{\mathrm{ic}}^{\mathrm{per}}(\vec{\mathrm{z}})$ are defined in
Definition 2.15 and (6.9), respectively, and the convergence is locally
uniform in $\vec{z}$. Here again $z_{i}$ and $\mathrm{z}_{i}$ are related by
the equation $z_{i}^{L}=(-1)^{N}\mathbbm{r}_{c}^{L}\mathrm{z}_{i}$.

The rest of the section are devoted to proving Lemma 7.1 and 7.2. We start
with a discussion on the asymptotic behaviors of the roots of the Bethe
polynomial $q_{z}(w)=w^{N}(1+w)^{L-N}-z^{L}(1+pw)^{N}$ under the critical re-
scaling in Section 7.1. Then in Section 7.2 we list a few lemmas discussing
the asymptotics of several products involving these roots under the critical
re-scaling. With these preparations we prove Lemma 7.1 and Lemma 7.2 in
Section 7.3 and Section 7.4. Finally in Section 7.5 we verify the Assumption
6.1 for the classical step and flat initial conditions.

###  7.1. Asympotics of the Bethe roots

We assume that the particle density $\rho:=N/L$ stays within a compact subset
of $(0,1)$ for all $L$. It turns out that in the asymptotic analysis for the
finite-time formula we have to re-scale the integral parameters $z_{i}$ so
that $|z_{i}|\to\mathbbm{r}_{c}$ in a certain rate and the main contributions
come from Bethe roots within a distance of $O(L^{-1/2})$ to $w_{c}$. More
precisely for $|z|<\mathbbm{r}_{c}$ we introduced the re-scaled integral
parameters $\mathrm{z}$ such that:

$z^{L}=(-1)^{N}\mathbbm{r}_{c}^{L}\mathrm{z},$ (7.3)

where the assumption $|z|<\mathbbm{r}_{c}$ is equivalent to $|\mathrm{z}|<1$
and we recall that

$w_{c}:=-\frac{2\rho}{1+\sqrt{1-4p\cdot\rho(1-\rho)}},\quad\mathbbm{r}_{c}:=\left(\frac{-w_{c}}{1+pw_{c}}\right)^{\rho}(1+w_{c})^{1-\rho}.$
(7.4)

Under this re-scaling the nested assumption on the integral parameters
$0<|z_{m}|<\cdots<|z_{1}|<\mathbbm{r}_{c}$ in the finite-time $m$-points
formulas becomes $0<|\mathrm{z}_{m}|<\cdots<|\mathrm{z}_{1}|<1$. From the
discussion in Section 2.3 we know the level set
$\\{w\in\mathbb{C}:|w^{N}(1+w)^{L-N}|=|z^{L}(1+pw)^{N}|\\}$ consists of two
disjoint closed contour for $|z|<\mathbbm{r}_{c}$ so we can define:

######  Definition 7.3.

Given $|z|<\mathbbm{r}_{c}$, we define two closed contours $\Lambda_{L}$ and
$\Lambda_{R}$ by

$\displaystyle\Lambda_{L}$
$\displaystyle:=\\{w\in\mathbb{C}:|w^{N}(1+w)^{L-N}|=|z^{L}(1+pw)^{N}|\\}\cap\\{\mathrm{Re}(w)<w_{c}\\},$
(7.5) $\displaystyle\Lambda_{R}$
$\displaystyle:=\\{w\in\mathbb{C}:|w^{N}(1+w)^{L-N}|=|z^{L}(1+pw)^{N}|\\}\cap\\{\mathrm{Re}(w)>w_{c}\\}.$

A formal Taylor expansion at $w=w_{c}$ indicates that as $L\to\infty$, the
Bethe equation $w^{N}(1+w)^{L-N}=z^{L}(1+pw)^{N}$ converges to the equation

$e^{-\zeta^{2}/2}=\mathrm{z},$ (7.6)

where $z^{L}=(-1)^{N}\mathbbm{r}_{c}^{L}\mathrm{z}$ and

$w=w_{c}+\frac{1+\nu-2\rho}{1+\nu}\sqrt{\frac{\rho}{(1-\rho)\nu}}\zeta
L^{-1/2}:=w_{c}+c_{0}\zeta L^{-1/2},$ (7.7)

where $\nu:=\sqrt{1-4p\cdot\rho(1-\rho)}$ and
$c_{0}:=\frac{1+\nu-2\rho}{1+\nu}\sqrt{\frac{\rho}{(1-\rho)\nu}}$. The
solution of equation (7.6) is a discrete set given by
$\\{\pm\sqrt{-2\log\mathrm{z}+4k\pi i}:k\in\mathbb{Z}\\}$ for an arbitrary
choice of branches of logarithm and square root, see figure 2. Lemma 7.4 below
precisely quantifies the convergence of Bethe roots near $w=w_{c}$ to the
corresponding roots for the limiting equation.

######  Lemma 7.4.

For any $0<\epsilon<1/8$ and $|z|<\mathbbm{r}_{c}$ fixed, we define

$\mathcal{L}_{z}^{(\epsilon)}:=\mathcal{L}_{z}\cap\mathbb{D}(w_{c},c_{0}L^{-1/2+\epsilon}),$
(7.8)

where $\mathbb{D}(a,r)$ is a disc centered at $a$ with radius $r$ and $c_{0}$
is defined in (7.7). Then for the re-scaled parameter
$\mathrm{z}=(-1)^{N}z^{L}\mathbbm{r}_{c}^{-L}$ we have an injective map
$\mathcal{M}_{L,\text{left}}:\mathcal{L}_{z}^{(\epsilon)}\to\mathrm{L}_{\mathrm{z}}$
satisfying

$\left|\mathcal{M}_{L,\text{left}}(u)-L^{1/2}c_{0}^{-1}(u-w_{c})\right|\leq
L^{-1/2+3\epsilon}\log L,$ (7.9)

for all $u\in\mathcal{L}_{z}^{(\epsilon)}$ and $L$ large enough. Furthermore,
the map satisfies

$\mathrm{L}_{\mathrm{z}}\cap\mathbb{D}(0,L^{\epsilon}-1)\subset\mathcal{M}_{L,\text{left}}(\mathcal{L}_{z}^{(\epsilon)})\subset\mathrm{L}_{\mathrm{z}}\cap\mathbb{D}(0,L^{\epsilon}+1).$
(7.10)

Similar results hold if we replace $\mathcal{L}_{z}$ and
$\mathrm{L}_{\mathrm{z}}$ by $\mathcal{R}_{z}$ and $\mathrm{R}_{\mathrm{z}}$.

###### Proof.

This lemma is a minor generalization of Lemma 8.1 of [3] by allowing one extra
parameter $p$ in the Bethe equation. The proof is almost identical to the one
in [3] so we omit the details. ∎

###  7.2. Asymptotics of various products over Bethe roots

In this section we collect all the results involving limits of products of the
Bethe roots appearing in the finite-time formula that are independent of the
parameters $\vec{y}$, $a_{i}$, $t_{i}$ and $k_{i}$. The starting point is the
following simple integral formula for the sums of functions evaluated at the
left or right Bethe roots.

######  Lemma 7.5.

Let $\phi(w)$ be a function analytic in the interior and a neighborhood of
$\Lambda_{R}$. Then

$\sum_{v\in\mathcal{R}_{z}}\phi(v)=N\phi(0)+\oint_{\Sigma_{\mathrm{R}}}\frac{dw}{2\pi
i}\frac{z^{L}(1+pw)^{N}}{q_{z}(w)}\frac{\phi(w)}{J(w)},$ (7.11)

where we recall that $J(w)=\frac{w(w+1)(1+pw)}{N+Lw+p(L-N)w^{2}}$. Similarly
if $\phi(w)$ be a function analytic in the interior and a neighborhood of
$\Lambda_{L}$, then

$\sum_{u\in\mathcal{L}_{z}}\phi(u)=(L-N)\phi(-1)+\oint_{\Sigma_{\mathrm{L}}}\frac{{\mathrm{d}}w}{2\pi\mathrm{i}}\frac{z^{L}(1+pw)^{N}}{q_{z}(w)}\frac{\phi(w)}{J(w)}.$
(7.12)

Here $\Sigma_{\mathrm{L}}$ and $\Sigma_{\mathrm{R}}$ are simple closed
contours lie in the half-plane $\\{w:\mathrm{Re}(w)<w_{c}\\}$ (respectively
$\\{w:\mathrm{Re}(w)>w_{c}\\}$) with $\Lambda_{\mathrm{L}}$ (respectively
$\Lambda_{\mathrm{R}}$) inside. Taking the function $\phi$ to be constant
function $1$ implies in particular that

$\oint_{\Sigma_{\mathrm{L}}}\frac{dw}{2\pi
i}\frac{z^{L}(1+pw)^{N}}{q_{z}(w)}\frac{1}{J(w)}=\oint_{\Sigma_{\mathrm{R}}}\frac{{\mathrm{d}}w}{2\pi\mathrm{i}}\frac{z^{L}(1+pw)^{N}}{q_{z}(w)}\frac{1}{J(w)}=0.$
(7.13)

###### Proof.

A direct differentiation shows

$\displaystyle
q_{z}^{\prime}(w)=q_{z}(w)\cdot\left(\frac{N}{w}+\frac{L-N}{w+1}\right)+z^{L}(1+pw)^{N}\cdot\frac{1}{J(w)}.$

Hence by residue theorem we know

$\displaystyle\sum_{v\in\mathcal{R}_{z}}\phi(u)=\oint_{\Sigma_{R}}\frac{{\mathrm{d}}w}{2\pi\mathrm{i}}\frac{q_{z}^{\prime}(w)}{q_{z}(w)}\phi(w)=N\phi(0)+\oint_{\Sigma_{R}}\frac{{\mathrm{d}}w}{2\pi\mathrm{i}}\frac{z^{L}(1+pw)^{N}}{q_{z}(w)}\frac{1}{J(w)}\phi(w).$

The proof for (7.12) is similar. ∎

As taking logarithm transforms products into sums, the following lemma is a
direct consequence of Lemma 7.5 and method of steepest descent.

######  Lemma 7.6.

Given $|z|<\mathbbm{r}_{c}$ and
$\mathrm{z}=(-1)^{N}z^{L}\mathbbm{r}_{c}^{-L}$. Suppose $\rho=N/L$ stays in a
compact subset of $(0,1)$, then for every $0<\epsilon<1/2$ the following holds
for all large enough $L$.

  1. (i)

For $w=w_{c}+c_{0}\zeta L^{-1/2}$ with $|\zeta|\leq L^{\epsilon/4}$, where
$c_{0}=\frac{1+\nu-2\rho}{1+\nu}\sqrt{\frac{\rho}{(1-\rho)\nu}}$ and
$\nu=\sqrt{1-4p\cdot\rho(1-\rho)}$,

$\frac{z^{L}(1+pw)^{N}}{q_{z}(w)}=\frac{\mathrm{z}}{e^{-\zeta^{2}/2}-\mathrm{z}}\cdot\left(1+O(L^{-1/2+\epsilon})\right).$
(7.14)

  2. (ii)

On the other hand if $|w-w_{c}|\geq C\cdot L^{\epsilon-1/2}$ for some $C>0$,
we have for some $c>0$ and $\alpha>0$ ,

$\left|\frac{z^{L}(1+pw)^{N}}{q_{z}(w)}\right|\leq e^{-cL^{\alpha}}.$ (7.15)

  3. (iii)

For $w$ of $O(1)$ distance away from $0,-1$ and $-1/p$ we have
$\left|\frac{1}{J(w)}\right|\leq C\cdot L$ for some constant $C>0$.
Furthermore if $w=w_{c}+c_{0}\zeta L^{-1/2}$ with $|\zeta|\leq C\cdot
L^{\epsilon}$ we have

$\frac{1}{J(w)}=-c_{0}^{-1}\zeta
L^{1/2}\cdot\left(1+O(L^{\epsilon-1/2})\right).$ (7.16)

  4. (iv)

For $w=w_{c}+c_{0}\zeta L^{-1/2}$ with $|\zeta|\leq L^{\epsilon/4}$, we have

$\begin{split}\prod_{u\in\mathcal{L}_{z}}\sqrt{w-u}&=(\sqrt{w+1})^{L-N}e^{\frac{1}{2}\mathrm{h}(\zeta,\mathrm{z})}\left(1+O(L^{\epsilon-1/2}\log
L)\right)\quad\text{if $\mathrm{Re}(\zeta)\geq 0$},\\\
\prod_{v\in\mathcal{R}_{z}}\sqrt{v-w}&=(\sqrt{-w})^{N}e^{\frac{1}{2}\mathrm{h}(\zeta,\mathrm{z})}\left(1+O(L^{\epsilon-1/2}\log
L)\right)\qquad\text{if $\mathrm{Re}(\zeta)\leq 0$},\end{split}$ (7.17)

where $\mathrm{h}(\zeta,\mathrm{z})$ is the function defined in (6.14). When
$\mathrm{Re}(\zeta)=0$, $\mathrm{h}(\zeta,\mathrm{z})$ is the limit of
$\mathrm{h}(\eta,\mathrm{z})$ as $\eta\to\zeta$ from $\mathrm{Re}(\eta)>0$ for
the first case and from $\mathrm{Re}(\eta)<0$ for the second case.

  5. (v)

For $w$ of a finite distance $O(1)$ away from the trajectory
$\Lambda_{\mathrm{L}}\cup\Lambda_{\mathrm{R}}$,

$\displaystyle\prod_{u\in\mathcal{L}_{z}}\sqrt{w-u}$
$\displaystyle=(\sqrt{w+1})^{L-N}\left(1+O(L^{\epsilon-1/2})\right)\quad\text{if\
$\mathrm{Re}(w)>w_{c}$},$ (7.18)
$\displaystyle\prod_{v\in\mathcal{R}_{z}}\sqrt{v-w}$
$\displaystyle=(\sqrt{-w})^{N}\left(1+O(L^{\epsilon-1/2})\right)\qquad\text{if\
$\mathrm{Re}(w)<w_{c}$.}$

  6. (vi)

There is a constant $C>0$ such that for every $w$ satisfying $|w-w_{c}|\geq
L^{\epsilon-1/2}$,

$e^{-CL^{-\epsilon}}\leq\left|\frac{q_{z,\mathrm{L}}(w)}{(w+1)^{L-N}}\right|\leq
e^{CL^{-\epsilon}}\quad\text{if $\mathrm{Re}(w)>w_{c}$},$

and

$e^{-CL^{-\epsilon}}\leq\left|\frac{q_{z,\mathrm{R}}(w)}{w^{N}}\right|\leq
e^{CL^{-\epsilon}}\qquad\text{if $\mathrm{Re}(w)<w_{c}$.}$

  7. (vii)

We have

$\frac{\prod_{v\in\mathcal{R}_{z}}\prod_{u\in\mathcal{L}_{z^{\prime}}}\sqrt{v-u}}{\prod_{u\in\mathcal{L}_{z^{\prime}}}\left(\sqrt{-u}\right)^{N}\prod_{v\in\mathcal{R}_{z}}\left(\sqrt{v+1}\right)^{L-N}}=e^{-B(\mathrm{\mathrm{z},\mathrm{z}^{\prime}})}\left(1+O(L^{\epsilon-1/2})\right),$
(7.19)

where $B(\mathrm{\mathrm{z},\mathrm{z}^{\prime}})$ is the function defined in
(6.12).

###### Proof.

These estimates are again one-parameter generalizations of Lemma 8.2 and Lemma
8.4 of [3]. The main difference is due to the extra parameter $p$ in the Bethe
polynomial $q_{z}(w)=w^{N}(1+w)^{L-N}-z^{L}(1+pw)^{N}$, the proper critical
point for steepest descent analysis now is at
$w_{c}=-\frac{2\rho}{1+\sqrt{1-4p\cdot\rho(1-\rho)}}$, which comes from the
larger root of the quadratic equation $p(L-N)w^{2}+Lw+N=0$, as opposed to
$w=-\rho$ for the $p=0$ degeneration discussed in [3]. A standard steepest
descent analysis using integral representations obtained in Lemma 7.5 with
critical point $w_{c}$ yields all the estimates. We omit the details. ∎

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

Now we are ready to prove Lemma 7.1. Recall that (see Definition 2.14),
$\mathscr{C}_{\vec{y}}^{(L)}(z)=\mathcal{E}_{\vec{y}}(z_{1})\mathscr{C}_{\mathrm{step}}^{(L)}(z)$
where
$\mathcal{E}_{\vec{y}}(z_{1})=E_{\mathrm{ic}}(\mathrm{z}_{1})\left(1+O(L^{\epsilon-1/2})\right)$
as $L\to\infty$ due to Assumption 6.1. Hence it suffices to prove

$\mathscr{C}_{\mathrm{step}}^{(L)}(\vec{z})=\mathrm{C}_{\mathrm{step}}^{\mathrm{per}}(\vec{\mathrm{z}})\left(1+O(L^{\epsilon-1/2})\right),\quad\mathrm{as}\
L\to\infty$ (7.20)

where $\mathrm{C}_{\mathrm{step}}^{\mathrm{per}}(\vec{\mathrm{z}})$ is defined
in Definition 6.5 and $\mathscr{C}_{\mathrm{step}}^{(L)}(\vec{z})$ is defined
as follows (see Definition 2.14):

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

Here
$E_{\ell}(z):=\prod_{u\in\mathcal{L}_{z}}(-u)^{-k_{\ell}}\prod_{v\in\mathcal{R}_{z}}(v+1)^{-a_{\ell}-k_{\ell}}(pv+1)^{t_{\ell}-k_{\ell}}$
with $E_{0}(z):=1$. Under the re-scaling
$z_{\ell}^{L}=(-1)^{N}\mathbbm{r}_{c}^{L}\mathrm{z}_{\ell}$ we clearly have
$\prod_{\ell=2}^{m}\frac{z_{\ell-1}^{L}}{z_{\ell-1}^{L}-z_{\ell}^{L}}=\prod_{\ell=2}^{N}\frac{\mathrm{z}_{\ell-1}}{\mathrm{z}_{\ell-1}-\mathrm{z}_{\ell}}$.
On the other hand by Lemma 7.6 (iv)

$\displaystyle\frac{\prod_{u\in\mathcal{L}_{z_{\ell}}}(-u)^{N}\prod_{v\in\mathcal{R}_{z_{\ell}}}(v+1)^{L-N}}{\Delta(\mathcal{R}_{z_{\ell}};\mathcal{L}_{z_{\ell}})}=e^{B(\mathrm{z}_{\ell})}\left(1+O(L^{\epsilon-1/2})\right),$
$\displaystyle\frac{\Delta(\mathcal{R}_{z_{\ell}};\mathcal{L}_{z_{\ell-1}})}{\prod_{u\in\mathcal{L}_{z_{\ell-1}}}(-u)^{N}\prod_{v\in\mathcal{R}_{z_{\ell}}}(v+1)^{L-N}}=e^{-B(\mathrm{z}_{\ell},\mathrm{z}_{\ell-1})}\left(1+O(L^{\epsilon-1/2})\right).$

Hence (7.20) follows immediately once we establish the following lemma on the
asymptotics of $E_{\ell}(z)$.

######  Lemma 7.7.

Let
$E(z)=E(z;a,k,t):=\prod_{u\in\mathcal{L}_{z}}(-u)^{-k}\prod_{v\in\mathcal{R}_{z}}(v+1)^{-a-k}(pv+1)^{t-k}$
where $a,k\in\mathbb{Z}$ and $t\in\mathbb{N}$ are given parameters. Then for
$z^{L}=(-1)^{N}\mathbbm{r}_{c}^{L}\mathrm{z}$ with $0<|\mathrm{z}|<1$ and the
parameters satisfying

$t=c_{1}\tau L^{3/2}+O(1),\quad a=c_{2}t+\gamma L+O(1),\quad
k=c_{3}t+c_{4}\gamma L+c_{5}\mathrm{x}L^{1/2}+O(1),$ (7.21)

we have for $L$ large enough and fixed $0<\epsilon<1/2$

$E(z)=e^{xA_{1}(\mathrm{z})+\tau
A_{2}(\mathrm{z})}\cdot\left(1+O(L^{\epsilon-1/2})\right).$ (7.22)

where $A_{1}(\mathrm{z})$ and $A_{2}(\mathrm{z})$ are scaled polylogarithm
functions as in (6.11). The constants $c_{i}$ are the same as in (2.10) and
the re-scaled parameters are chosen such that $\tau>0$, $\gamma\in[0,1]$ and
$\mathrm{x}\in\mathbb{R}$.

###### Proof.

Note $\log
E(z)=\sum_{u\in\mathcal{L}_{z}}(-k)\log(-u)+\sum_{v\in\mathcal{R}_{z}}\left[(-a-k+1)\log(v+1)+(t-k+1)\log(1+pv)\right]$.
Apply Lemma 7.5 to the two sums over left and right Bethe roots and deform
both contours $\Sigma_{\mathrm{L}}$ and $\Sigma_{\mathrm{R}}$ to the vertical
line with real part $w_{c}=-\frac{2\rho}{1+\sqrt{1-4p\rho(1-\rho)}}$ we have

$\log
E(z)=\int_{w_{c}-\mathrm{i}\infty}^{w_{c}+\mathrm{i}\infty}\frac{{\mathrm{d}}w}{2\pi\mathrm{i}}\frac{z^{L}(1+pw)^{N}}{q_{z}(w)}\frac{1}{J(w)}\left(G(w)-G(w_{c})\right),$
(7.23)

where $G(w)=(-k)\log(-w)+(a+k)\log(w+1)+(-t+k)\log(1+pw)$. Note that in (7.23)
we have used the fact discussed in Lemma 7.5 that

$\oint_{\Sigma_{\mathrm{L}}}\frac{{\mathrm{d}}w}{2\pi\mathrm{i}}\frac{z^{L}(1+pw)^{N}}{q_{z}(w)}\frac{1}{J(w)}G(w_{c})=\oint_{\Sigma_{\mathrm{R}}}\frac{{\mathrm{d}}w}{2\pi\mathrm{i}}\frac{z^{L}(1+pw)^{N}}{q_{z}(w)}\frac{1}{J(w)}G(w_{c})=0.$

Now a Taylor expansion at $w=w_{c}$ shows

$G(w)-G(w_{c})=G^{\prime}(w_{c})(w-w_{c})+\frac{G^{\prime\prime}(w_{c})}{2}(w-w_{c})^{2}+\frac{G^{\prime\prime\prime}(w_{c})}{6}(w-w_{c})^{3}+O\left(G^{(4)}(w_{c})\cdot(w-w_{c})^{4}\right).$

Set $w=w_{c}+c_{0}\zeta L^{-1/2}$ where $c_{0}$ as in (7.7) and assume the
parameters are re-scaled as in (7.21). After a tedious but straightforward
calculation we obtain for $|\zeta|\leq L^{\epsilon/4}$ with $0<\epsilon<1/2$

$G(w)-G(w_{c})=-\mathrm{x}\zeta-\frac{\gamma}{2}\zeta^{2}+\frac{\tau}{3}\zeta^{3}+O(L^{\epsilon-1/2}).$
(7.24)

Splitting the integral representation for $\log E(\mathrm{z})$ into two parts
with $|\zeta|\leq L^{\epsilon/4}$ and $|\zeta|>L^{\epsilon/4}$ and using the
estimates for $\frac{z^{L}(1+pw)^{N}}{q_{z}(w)}$ and $\frac{1}{J(w)}$ obtained
in Lemma 7.6 we see

$\displaystyle\log
E(z)=\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\frac{{\mathrm{d}}\zeta}{2\pi\mathrm{i}}\frac{\mathrm{z}}{e^{-\zeta^{2}/2}-\mathrm{z}}\cdot\left(\mathrm{x}\zeta^{2}+\frac{\gamma}{2}\zeta^{3}-\frac{\tau}{3}\zeta^{4}\right)\cdot\left(1+O(L^{\epsilon-1/2})\right)+O(e^{-cL^{\alpha}}),$
(7.25)

for some constants $c,\alpha>0$. Now (7.22) follows from integral
representations of polylogarithm (6.10). ∎

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

Next we discuss the asymptotics of the Fredholm determinant part
$\mathscr{D}_{\vec{y}}(z)$. Note first that by a standard series expansion of
Fredholm determinants we have

$\mathscr{D}_{\vec{y}}^{(L)}(\mathrm{z})=\sum_{\vec{n}\in(\mathbb{Z}_{\geq
0})^{m}}\frac{1}{(n_{1}!\cdots
n_{m}!)^{2}}\mathscr{D}_{\vec{y},\vec{n}}^{(L)}(\vec{z}),$ (7.26)

where $\vec{n}=(n_{1},\cdots,n_{m})$ and

$\mathscr{D}_{\vec{y},\vec{n}}^{(L)}(\vec{z})=(-1)^{|\vec{n}|}\sum_{\begin{subarray}{c}U^{(\ell)\in(\mathcal{L}_{z_{\ell}})^{n_{\ell}}}\\\
V^{(\ell)\in(\mathcal{R}_{z_{\ell}})^{n_{\ell}}}\\\
\ell=1,\cdots,m\end{subarray}}\det[\mathscr{K}_{1}^{(L)}(w_{i},w_{j}^{\prime})]_{i,j=1}^{|\vec{n}|}\det[\mathscr{K}_{\vec{y}}^{(L)}(w_{i}^{\prime},w_{j})]_{i,j=1}^{|\vec{n}|},$
(7.27)

where $U^{(\ell)}=(u^{(\ell)}_{1},\cdots u^{(\ell)}_{n_{\ell}})$ and
$V^{(\ell)}=(v^{(\ell)}_{1},\cdots v^{(\ell)}_{n_{\ell}})$ and

$w_{i}=\begin{cases}u^{(\ell)}_{k}\quad\mathrm{if}\
k=n_{1}+\cdots+n_{\ell-1}+k\quad\text{for integer $k\leq n_{\ell}$ with $\ell$
odd },\\\ v^{(\ell)}_{k}\quad\mathrm{if}\
k=n_{1}+\cdots+n_{\ell-1}+k\quad\text{for integer $k\leq n_{\ell}$ with $\ell$
even },\end{cases}$ (7.28)

and

$w_{i}^{\prime}=\begin{cases}v^{(\ell)}_{k}\quad\mathrm{if}\
k=n_{1}+\cdots+n_{\ell-1}+k\quad\text{for integer $k\leq n_{\ell}$ with $\ell$
odd },\\\ u^{(\ell)}_{k}\quad\mathrm{if}\
k=n_{1}+\cdots+n_{\ell-1}+k\quad\text{for integer $k\leq n_{\ell}$ with $\ell$
even }.\end{cases}$ (7.29)

A similar series expansion holds for the limiting Fredholm determinant
$\mathrm{D}_{\mathrm{ic}}^{\mathrm{per}}(\vec{\mathrm{z}})$ with
$\mathcal{L}_{z_{\ell}}$ and $\mathcal{R}_{z_{\ell}}$ replaced by the limiting
roots $\mathrm{L}_{\mathrm{z}_{\ell}}$ and $\mathrm{R}_{\mathrm{z}_{\ell}}$
and the kernels replaced by the limiting kernels
$\mathrm{K}_{1}^{\mathrm{per}}$ and $\mathrm{K}_{\mathrm{ic}}^{\mathrm{per}}$
defined in Definition 6.6.

$\mathrm{D}_{\mathrm{ic}}^{\mathrm{per}}(\vec{\mathrm{z}})=\sum_{\vec{n}\in(\mathbb{Z}_{\geq
0})^{m}}\frac{1}{(n_{1}!\cdots
n_{m}!)^{2}}\mathrm{D}_{\mathrm{ic},\vec{n}}^{\mathrm{per}}(\vec{\mathrm{z}}).$
(7.30)

We will prove the convergence of each of these
$\mathscr{D}_{\vec{y},\vec{n}}^{(L)}(\vec{z})$ as well as some exponential
bounds.

######  Lemma 7.8.

Under the same assumption in Theorem 2.4, for every fixed
$\vec{n}\in(\mathbb{Z}_{\geq 0})^{m}$, we have

  1. (i)

$\mathscr{D}_{\vec{y},\vec{n}}^{(L)}(\vec{z})\to\mathrm{D}_{\mathrm{ic},\vec{n}}^{\mathrm{per}}(\mathrm{z})$
as $L\to\infty$.

  2. (ii)

There exists constant $C>0$ such that
$|\mathscr{D}_{\vec{y},\vec{n}}^{(L)}(\vec{z})|\leq C^{|\vec{n}|}$ for all $L$
large enough.

It is clear that Lemma 7.2 follows immediately from Lemma 7.8 by dominated
convergence theorem. To prove Lemma 7.8 we will prove the convergence of the
kernels after proper conjugation for points inside the critical region as well
as exponential decay estimates for the kernel at points outside the critical
region. The conjugation is as follows: we replace $\mathscr{K}_{1}^{(L)}$ and
$\mathscr{K}_{\vec{y}}^{(L)}$ defined in Definition 2.15 by
$\tilde{\mathscr{K}}_{1}^{(L)}$ and $\tilde{\mathscr{K}}_{\vec{y}}^{(L)}$
where

$\displaystyle\tilde{\mathscr{K}}_{1}^{(L)}(w,w^{\prime}):=-\left(\delta_{i}(j)+\delta_{i}(j+(-1)^{i})\right)\frac{J(w)\sqrt{\tilde{f}_{i}(w)}\sqrt{\tilde{f}_{j}(w^{\prime})}(H_{z_{i}}(w))^{2}}{H_{z_{i-(-1)^{i}}}(w)H_{z_{j-(-1)^{j}}}(w^{\prime})(w-w^{\prime})}Q_{1}(j),$
(7.31)
$\displaystyle\tilde{\mathscr{K}}_{\vec{y}}^{(L)}(w,w^{\prime}):=\left(\delta_{j}(i)+\delta_{j}(i-(-1)^{j})\right)\frac{J(w^{\prime})\sqrt{\tilde{f}_{j}(w^{\prime})}\sqrt{\tilde{f}_{i}(w)}(H_{z_{j}}(w^{\prime}))^{2}}{H_{z_{j+(-1)^{j}}}(w^{\prime})H_{z_{i+(-1)^{i}}}(w)(w^{\prime}-w)}Q_{2}(i)\Lambda(i,w,w^{\prime}),$

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

$\tilde{f}_{i}(w):=\begin{cases}&\frac{F_{i}(w)F_{i-1}(w_{c})}{F_{i-1}(w)F_{i}(w_{c})}\quad\text{for
}\mathrm{Re}(w)<w_{c},\\\
&\frac{F_{i}(w)F_{i-1}(w_{c})}{F_{i-1}(w)F_{i}(w_{c})}\quad\text{for
}\mathrm{Re}(w)<w_{c}.\end{cases}$ (7.32)

We define the square root to be $\sqrt{w}=r^{1/2}e^{i\theta/2}$ for
$w=re^{i\theta}$ with $-\pi<\theta\leq\pi$. Note that the product of
determinants will always be continuous even though the square root function is
not since every $(\tilde{f}_{i})^{1/2}$ is multiplied twice. We change the
limiting kernels in a similar way by replacing $\mathrm{f_{i}}$ or
$\mathrm{f}_{j}$ in the kernels with
$\sqrt{\mathrm{f_{i}}}\sqrt{\mathrm{f_{j}}}$ and denote them as
$\tilde{\mathrm{K}}_{1}^{\mathrm{per}}$ and
$\tilde{\mathrm{K}}_{\mathrm{ic}}^{\mathrm{per}}$. We have the following
asymptotics for the conjugated kernels, which easily implies Lemma 7.8.

######  Lemma 7.9.

Fix $0<\epsilon<1/(1+2m)$. Let

$\Omega=\Omega_{L}:=\bigg{\\{}w\in\mathbb{C}:|w-w_{c}|\leq
c_{0}^{-1}L^{-1/2+\epsilon/4}\bigg{\\}}$ (7.33)

be a disk centered at $w_{c}$. Under the same assumption in Theorem 2.4 we
have

  1. (i)

As $L\to\infty$, uniformly for $w\in\mathscr{S}_{1}\cap\Omega$ and
$w^{\prime}\in\mathscr{S}_{2}\cap\Omega$

$|\tilde{\mathscr{K}}_{1}^{(L)}(w,w^{\prime})|=|\tilde{\mathrm{K}}_{1}^{\mathrm{per}}(\zeta,\zeta^{\prime})|+O(L^{\epsilon-1/2}\log
L),\quad|\tilde{\mathscr{K}}_{\vec{y}}^{(L)}(w,w^{\prime})|=|\tilde{\mathrm{K}}_{\mathrm{ic}}^{\mathrm{per}}(\zeta,\zeta^{\prime})|+O(L^{\epsilon-1/2}\log
L),$ (7.34)

where $\zeta\in\mathrm{S}_{1},\zeta^{\prime}\in\mathrm{S}_{2}$ are the image
of $w,w^{\prime}$ under either the map $\mathcal{M}_{L,\text{left}}$ or
$\mathcal{M}_{L,\text{right}}$ in Lemma 7.4.

  2. (ii)

As $L\to\infty$, for $w_{i}\in\mathscr{S}_{1}\cap\Omega$ and
$w_{i}^{\prime}\in\mathscr{S}_{2}\cap\Omega$,

$\det\left[\tilde{\mathscr{K}}_{1}^{(L)}(w_{i},w^{\prime}_{j})\right]_{i,j=1}^{|\vec{n}|}\to\det\left[\tilde{\mathrm{K}}_{1}^{\mathrm{per}}(\zeta_{i},\zeta^{\prime}_{j})\right]_{i,j=1}^{|\vec{n}|},\quad\det\left[\tilde{\mathscr{K}}_{\vec{y}}^{(L)}(w_{i},w^{\prime}_{j})\right]_{i,j=1}^{|\vec{n}|}\to\det\left[\tilde{\mathrm{K}}_{\mathrm{ic}}^{\mathrm{per}}(\zeta_{i},\zeta^{\prime}_{j})\right]_{i,j=1}^{|\vec{n}|},$

for each $\vec{n}\in(\mathbb{Z}_{\geq 0})^{m}$, where
$\zeta\in\mathrm{S}_{1},\zeta^{\prime}\in\mathrm{S}_{2}$ are the image of
$w,w^{\prime}$ under either the map $\mathcal{M}_{L,\text{left}}$ or
$\mathcal{M}_{L,\text{right}}$ in Lemma 7.4.

  3. (iii)

There are positive constants $c$ and $\alpha$ such that

$|\tilde{\mathscr{K}}_{1}^{(L)}(w,w^{\prime})|=O(e^{-cL^{\alpha}}),\qquad|\tilde{\mathscr{K}}_{\vec{y}}^{(L)}(w^{\prime},w)|=O(e^{-cL^{\alpha}})$
(7.35)

as $L\to\infty$, uniformly for $w\in\mathscr{S}_{1}\cap\Omega^{c}$ and
$w^{\prime}\in\mathscr{S}_{2}$, and also for
$w^{\prime}\in\mathscr{S}_{2}\cap\Omega^{c}$ and $w\in\mathscr{S}_{1}$.

###### Proof.

Due to the structure of the kernel (7.31), the lemma is proved once we
establish the corresponding asymptotics and tail estimates for the functions
$J(w)$, $H_{z_{i}}(w)$ and $\tilde{f}_{i}(w)$. For $J(w)$ and $H_{z}(w)$ these
have already been discussed in Lemma 7.6 (iii),(iv) and (vi). The needed
estimates for $\tilde{f}_{i}(w)$ is summarized in the following Lemma 7.10,
the proof is similar to Lemma 7.7 so we omit the details. ∎

######  Lemma 7.10.

Under the same assumption as in Theorem 2.4 for the parameters
$a_{i},k_{i},t_{i}$, for $w=w_{c}+c_{0}\zeta L^{-1/2}$ with
$w\in\mathcal{L}_{z}\cup\mathcal{R}_{z}$ we have for fixed $0<\epsilon<1/2$

$\tilde{f}_{j}(w)=\begin{cases}\mathrm{f}_{j}(\zeta)\left(1+O(L^{\epsilon-1/2})\right)\quad&\text{if
}\ |\zeta|\leq L^{\epsilon/4}\\\ O(e^{-cL^{3\epsilon/4}})\quad&\text{if }\
|\zeta|\geq L^{\epsilon/4}\end{cases}$ (7.36)

for $1\leq j\leq m$, where $\mathrm{f}_{j}(\zeta)$ is defined in (6.16).

###  7.5. Proof of Theorem 6.2

In this section we verify that the flat initial condition satisfy Assumption
6.1 with the limiting functions $E_{\mathrm{flat}}$ and
$\mathrm{ch}_{\mathrm{flat}}$ given by (6.3). We start with a product formula
for the pre-limit functions $\mathcal{E}_{\mathrm{flat}}(z)$ and
$\chi_{\mathrm{flat}}(v,u;z)$.

######  Lemma 7.11.

Recall the global energy function $\mathcal{E}_{\vec{y}}(z)$ and
characteristic function $\chi_{\vec{y}}(v,u;z)$ defined in Definition 2.12.
For flat initial condition $\vec{y}=(-d,-2d,\cdots,-Nd)$ with
$d=L/N\in\mathbb{N}$ we have

  1. (i)

With the standard square root function $\sqrt{w}$ with branch cut
$\mathbb{R}_{\leq 0}$,

$\mathcal{E}_{\mathrm{flat}}(z)=\frac{\prod_{v\in\mathcal{R}_{z}}\left(\sqrt{v+1}\right)^{2-d}}{\prod_{v\in\mathcal{R}_{z}}\sqrt{p(d-1)v^{2}+dv+1}}\left[\frac{\prod_{v\in\mathcal{R}_{z}}\prod_{u\in\mathcal{L}_{z}}\sqrt{v-u}}{\prod_{u\in\mathcal{L}_{z}}\left(\sqrt{-u}\right)^{N}\prod_{v\in\mathcal{R}_{z}}\left(\sqrt{v+1}\right)^{L-N}}\right].$
(7.37)

  2. (ii)

For $v\in\mathcal{R}_{z}$ and $u\in\mathcal{L}_{z}$,

$\chi_{\mathrm{flat}}(v,u;z)=\begin{cases}\frac{1}{J(v)}\frac{(v+1)^{L-N}u^{N}}{q_{z,\mathrm{L}}(v)q_{z,\mathrm{R}}(u)}\frac{1+pv}{1+pu}(u-v)\quad&\text{if
}\ \frac{u(u+1)^{d-1}}{1+pu}=\frac{v(v+1)^{d-1}}{1+pv},\\\
0&\text{otherwise},\end{cases}$ (7.38)

where $J(w)=\frac{w(w+1)(1+pw)}{p(L-N)w^{2}+Lw+N}$,
$q_{z,\mathrm{L}}(w):=\prod_{u\in\mathcal{L}_{z}}(w-u)$ and
$q_{z,\mathrm{R}}(w):=\prod_{v\in\mathcal{R}_{z}}(w-v)$.

###### Proof.

The proof is similar to the proof of Lemma 10.2 and Lemma 10.3 in [5]. The key
observation is the existence of a $d-1$ to $1$ map $\mathcal{M}$ from
$\mathcal{L}_{z}$ to $\mathcal{R}_{z}$ satisfying that if $v=\mathcal{M}(u)$
for $u\in\mathcal{L}_{z}$, then
$\frac{u(u+1)^{d-1}}{1+pu}=\frac{v(v+1)^{d-1}}{1+pv}$. Using this relation we
can express the global energy and characteristic functions in terms of
products over the Bethe roots. We omit the details. ∎

Combining Lemma 7.11 with the asymptotics obtained in Lemma 7.6 we can now
prove Theorem 6.2.

###### Proof of Theorem 6.2.

For fixed $0<\epsilon<1/2$ by Lemma 7.6 (viii) we have

$\frac{\prod_{v\in\mathcal{R}_{z}}\prod_{u\in\mathcal{L}_{z^{\prime}}}\sqrt{v-u}}{\prod_{u\in\mathcal{L}_{z^{\prime}}}\left(\sqrt{-u}\right)^{N}\prod_{v\in\mathcal{R}_{z}}\left(\sqrt{v+1}\right)^{L-N}}=e^{-B(\mathrm{\mathrm{z},\mathrm{z}^{\prime}})}\left(1+O(L^{\epsilon-1/2})\right).$
(7.39)

Similarly by Lemma 7.6 (v) we have

$\prod_{v\in\mathcal{R}_{z}}\left(\sqrt{v+1}\right)^{2-d}=1+O(L^{\epsilon-1/2}).$
(7.40)

On the other hand we have $p(d-1)v^{2}+dv+1=p(d-1)(v-w_{c})(w-w_{c}^{*})$
where

$w_{c}=\frac{-2\rho}{1+\sqrt{1-4p\rho(1-\rho)}}\quad\text{and}\quad
w_{c}^{*}=\frac{-2\rho}{1-\sqrt{1-4p\rho(1-\rho)}}.$

Hence by Lemma 7.6 (iv) and (v) we have

$\displaystyle\prod_{v\in\mathcal{R}_{z}}\sqrt{p(d-1)v^{2}+dv+1}$
$\displaystyle=\prod_{v\in\mathcal{R}_{z}}\sqrt{v-w_{c}}\cdot\prod_{v\in\mathcal{R}_{z}}\sqrt{p(d-1)(w-w_{c}^{*})}$
(7.41)
$\displaystyle=\left(\sqrt{-w_{c}}\right)^{N}e^{\frac{1}{2}h(0,\mathrm{z})}\cdot\left(\sqrt{p(d-1)}\cdot\sqrt{-w_{c}^{*}}\right)^{N}\cdot\left(1+O(L^{\epsilon-1/2})\right)$
$\displaystyle=(1-\mathrm{z})^{1/4}\left(1+O(L^{\epsilon-1/2}\right),$

where we used the fact that $w_{c}\cdot w_{c}^{*}=\frac{1}{p(d-1)}$ and also
$e^{h(0,\mathrm{z})}=(1-\mathrm{z})^{1/2}$ which follows from (6.14).
Combining (7.39), (7.40) and (7.41) we conclude that
$\mathcal{E}_{\mathrm{flat}}(z)=E_{\mathrm{flat}}(\mathrm{z})(1+O(L^{\epsilon-1/2}))$
as $L\to\infty$.

The argument for the characteristic function part is quite similar. To verify
part (B) of Assumption 6.1 we note that given $0<\epsilon<1/8$, for
$u\in\mathcal{L}_{z}^{(\epsilon)}$ and $v\in\mathcal{R}_{z}^{(\epsilon)}$ as
defined in Lemma 7.4 we have

$\frac{q_{z,\mathrm{L}}(v)}{(1+v)^{L-N}}=e^{h(\eta,\mathrm{z})}\cdot\left(1+O(L^{4\epsilon-1/2})\right),\quad\frac{q_{z,\mathrm{R}}(u)}{u^{N}}=e^{h(\xi,\mathrm{z})}\cdot\left(1+O(L^{4\epsilon-1/2})\right),$
(7.42)

where $\xi=\mathcal{M}_{L,\mathrm{left}}(u)$ and
$\eta=\mathcal{M}_{L,\mathrm{right}}(v)$ with the injective maps
$\mathcal{M}_{L,\mathrm{left}}$ and $\mathcal{M}_{L,\mathrm{right}}$ defined
in Lemma 7.4 satisfying $\left|\xi-c_{0}^{-1}L^{1/2}(u-w_{c})\right|\leq
L^{-1/2+3\epsilon}\log L$, $\quad\left|\eta-
c_{0}^{-1}L^{1/2}(v-w_{c})\right|\leq L^{-1/2+3\epsilon}\log L$. This then
implies that

$u-v=c_{0}L^{-1/2}(\xi-\eta)\cdot\left(1+O(L^{3\epsilon-1/2}\log L)\right).$
(7.43)

A straightforward Taylor expansion combined with the injectivity of
$\mathcal{M}_{L,\mathrm{left}}$ and $\mathcal{M}_{L,\mathrm{right}}$ shows

$\mathbf{1}_{\frac{u(u+1)^{d-1}}{1+pu}=\frac{v(v+1)^{d-1}}{1+pv}}=\mathbf{1}_{\xi^{2}=\eta^{2}}=\mathbf{1}_{\xi=-\eta}.$
(7.44)

Finally by Lemma 7.6 (iii) we have

$\frac{1}{J(v)}=-c_{0}^{-1}\eta L^{1/2}\cdot(1+O(L^{\epsilon-1/2})).$ (7.45)

Combining (7.42), (7.43), (7.44) and (7.45) we conclude that

$\chi_{\mathrm{flat}}(v,u;z)=\mathrm{ch}_{\mathrm{flat}}(\eta,\xi;\mathrm{z})+O(L^{4\epsilon-1/2}),\quad\text{as}\
L\to\infty.$ (7.46)

Finally part (C) in Assumption 6.1 is clearly true since by Lemma 7.6 every
factor in for $\chi_{\mathrm{flat}}(v,u;z)$ is $O(1)$ except $\frac{1}{J(v)}$
which is $O(L)$. Thus $|\chi_{\mathrm{flat}}(v,u;z)|\leq C\cdot L$ and part
(C) of Assumption 6.1 is satisfied. ∎

##  8\. Multi-time distribution for discrete time parallel TASEP on
$\mathbb{Z}$

So far we have been focusing on the so-called relaxation time scale when the
period $L$ is comparable to $t^{2/3}$ and the height fluctuation is critically
affected by the boundary conditions. In this section we discuss another
important regime, the sub-relaxation time scale $L\gg t^{2/3}$ when the
particles are not affected by the boundary and are expected to behave the same
as when the underlying space has infinite-volume.

###  8.1. Discrete time parallel TASEP with large period

Instead of taking the re-scaled time parameter $\tau\to 0$ in the relaxation-
time limiting distribution formula, we will start with the pre-limit joint
distribution formula (2.6). The following elementary proposition confirms the
intuition that for fixed spatial and time parameters $a_{i},k_{i}$ and
$t_{i}$, when $L$ is large enough, the joint distribution
$\mathbb{P}_{\vec{y}}^{(L)}(\cap_{i=1}^{m}\\{x_{k_{i}}(t_{i})\geq a_{i}\\})$
agrees with the one when the underlying space is the whole integer lattice
$\mathbb{Z}$, hence should be independent of $L$.

######  Proposition 8.1.

Given $\vec{y}=(y_{1},\cdots,y_{N})\in\mathcal{X}_{N}$. Let $m$ be a positive
integer and $k_{i}$ be integers in $\\{1,\cdots,N\\}$, for $1\leq i\leq m$.
Then for any integers $a_{i}$, $1\leq i\leq m$, as long as $L$ is large enough
such that

$L>y_{1}-y_{N}\quad\text{and }\
L>\max\\{a_{1}+k_{1},\cdots,a_{m}+k_{m}\\}-y_{N},$ (8.1)

we have

$\mathbb{P}_{\vec{y}}^{(L)}\left(\bigcap_{\ell=1}^{m}\left\\{x_{k_{\ell}}(t_{\ell})\geq
a_{\ell}\right\\}\right)=\mathbb{P}_{\vec{y}}^{(\infty)}\left(\bigcap_{\ell=1}^{m}\left\\{x_{k_{\ell}}(t_{\ell})\geq
a_{\ell}\right\\}\right),$ (8.2)

for any positive integers $t_{1},\cdots,t_{m}$. Here
$\mathbb{P}_{\vec{y}}^{(L)}$ is the probability with respect to discrete time
parallel TASEP on a periodic domain with period $L$ and
$\mathbb{P}_{\vec{y}}^{(\infty)}$ is the probability with respect to an
independent discrete time parallel TASEP on the infinite lattice $\mathbb{Z}$,
both starting with the initial condition satisfying $x_{i}(0)=y_{i}$ for
$1\leq i\leq N$.

###### Proof.

The proof is almost identical to the similar statement for continuous time
TASEP as in Lemma 8.1 of [4] where the particle ordering is reversed. We omit
the details. ∎

In principle for every $L$ satisfying (8.1), the left-hand side of (8.2) which
has an expression as contour integrals of Fredholm determinant given by (2.6)
gives a formula for the multi-point joint distribution of discrete time
parallel TASEP on $\mathbb{Z}$ for fixed parameters $a_{i},k_{i}$ and $t_{i}$.
The main obstacle here is the dependence on the extra parameter $L$ for the
left-hand side of (8.2). The key idea to get rid of the dependence on $L$ was
already illustrated in Proposition 3.4 (iii) and can be summarized as follows:

  1. (1)

By rewriting the sum over the Bethe roots as contour integrals using residue
theorem we get an analytic continuation of the integrand(for the outer
integral with respect to $z$) to $\\{|z|>0\\}$.

  2. (2)

Under proper assumption on the parameter $L$(in fact exactly (8.1)), one can
further show the integrand is analytic at $z=0$, thus the outer integral with
respect to $z$ equals the evaluation of the integrand at $z=0$, which turns
out to be independent of $L$.

Now as in the above discussion, we would like to send $z_{1},\cdots,z_{m}$ all
to $0$ in (2.6) while still keep the nested relation
$0<|z_{m}|<\cdots<|z_{1}|<\mathbbm{r}_{c}$. For this we change the integral
variables as follows: for given $L$ satisfying (8.1), we set

$\theta_{i}:=\frac{z_{i+1}^{L}}{z_{i}^{L}},\quad\text{for }0\leq i\leq m-1,$
(8.3)

where $z_{0}:=1$ and $0<|z_{m}|<\cdots<|z_{1}|<\mathbbm{r}_{c}$. Then the
corresponding relations between $\theta_{i}$’s are precisely
$0<|\theta_{0}|<\mathbbm{r}_{c}^{L}:=\mathbf{r}_{c}$ and $0<|\theta_{i}|<1$
for $1\leq i\leq m-1$. After this change of variable, the finite-time multi-
point joint distribution formula (2.6) can be rewritten as

$\mathbb{P}_{\mathrm{step}}^{(L)}\left(\bigcap_{\ell=1}^{m}\left\\{x_{k_{\ell}}(t_{\ell})\geq
a_{\ell}\right\\}\right)=\oint\cdots\oint\hat{\mathscr{C}}_{\mathrm{step}}^{(L)}(\theta_{0},\cdots,\theta_{m-1})\hat{\mathscr{D}}_{\mathrm{step}}^{(L)}(\theta_{0},\cdots,\theta_{m-1})\frac{d\theta_{0}}{2\pi\mathrm{i}\theta_{0}}\cdots\frac{d\theta_{m-1}}{2\pi\mathrm{i}\theta_{m-1}},$
(8.4)

where the integral contours are circle centered at origin with radius smaller
than $\mathbf{r}_{c}$ for $\theta_{0}$ and circles centered at origin with
radii less than $1$ for $\theta_{i}$, $1\leq i\leq m-1$. Here the integrand is
defined such that

$\hat{\mathscr{C}}_{\mathrm{step}}^{(L)}(\theta_{0},\cdots,\theta_{m-1}):=\mathscr{C}_{\mathrm{step}}^{(L)}(z_{1},\cdots,z_{m}),\quad\hat{\mathscr{D}}_{\mathrm{step}}^{(L)}(\theta_{0},\cdots,\theta_{m-1}):=\mathscr{D}_{\mathrm{step}}^{(L)}(z_{1},\cdots,z_{m}),$
(8.5)

for any $z_{1},\cdots,z_{m}$ satisfying the relations
$z_{i}^{L}=\prod_{j=0}^{i-1}\theta_{j}$ for $1\leq i\leq m$. Note that by
definition (see Section 2.5),
$\mathscr{C}_{\mathrm{step}}^{(L)}(z_{1},\cdots,z_{m})$ and
$\mathscr{D}_{\mathrm{step}}^{(L)}(z_{1},\cdots,z_{m})$ only depend on
$z_{1}^{L},\cdots,z_{m}^{L}$ so $\hat{\mathscr{C}}^{(L)}$ and
$\hat{\mathscr{D}}^{(L)}$ are well-defined functions analytic in
$\mathbb{D}_{0}(\mathbf{r}_{c})\times(\mathbb{D}_{0}(1))^{m-1}$, where
$\mathbb{D}_{0}(r)$ is the disk centered at $0$ with radius $r$ with the
origin removed.

###  8.2. Proof of Theorem 2.5

As discussed above, Theorem 2.5 follows immediately once we establish the
analyticity of
$\hat{\mathscr{C}}_{\vec{y}}^{(L)}(\theta_{0},\cdots,\theta_{m-1})$ and
$\hat{\mathscr{D}}_{\vec{y}}^{(L)}(\theta_{0},\cdots,\theta_{m-1})$ at
$\theta_{0}=0$ so that we can get rid of the integral with respect to
$\theta_{0}$ whose value equals the integrand evaluated at $\theta_{0}=0$. The
precise statement needed are summarized in the following two lemmas which will
be proved in the next two subsections.

######  Lemma 8.2.

The function
$\hat{\mathscr{C}}_{\mathrm{step}}^{(L)}(\theta_{0},\cdots,\theta_{m-1})$ is
in fact analytic in $\mathbb{D}(\mathbf{r}_{c})\times(\mathbb{D}(1))^{m-1}$,
where $\mathbb{D}(r)$ is a disk centered at $0$ with radius $r$. Moreover we
have

$\hat{\mathscr{C}}_{\mathrm{step}}^{(L)}(0,\theta_{1},\cdots,\theta_{m-1})=\prod_{\ell=1}^{m-1}\frac{1}{1-\theta_{\ell}},$
(8.6)

######  Lemma 8.3.

For large enough integer $L$ satisfying (8.1), the function
$\hat{\mathscr{D}}_{\vec{y}}^{(L)}(\theta_{0},\cdots,\theta_{m-1})$ is
analytic in $\mathbb{D}(\mathbf{r}_{c})\times(\mathbb{D}_{0}(1))^{m-1}$.
Moreover we have for any
$(\theta_{1},\cdots,\theta_{m-1})\in(\mathbb{D}_{0}(1))^{m-1}$

$\hat{\mathscr{D}}_{\mathrm{step}}^{(L)}(0,\theta_{1},\cdots,\theta_{m-1})=\mathcal{D}_{\mathrm{step}}^{(\infty)}(\theta_{1},\cdots,\theta_{m-1}),$
(8.7)

where $\mathcal{D}_{\mathrm{step}}^{(\infty)}(\theta_{1},\cdots,\theta_{m-1})$
is defined in Section 2.6.

Combing Lemma 8.2 and Lemma 8.3 we conclude that

$\displaystyle\mathbb{P}_{\mathrm{step}}^{(L)}\left(\bigcap_{\ell=1}^{m}\left\\{x_{k_{\ell}}(t_{\ell})\geq
a_{\ell}\right\\}\right)$
$\displaystyle=\oint\cdots\oint\hat{\mathscr{C}}_{\mathrm{step}}(0,\theta_{1},\cdots,\theta_{m-1})\hat{\mathscr{D}}_{\mathrm{step}}(0,\theta_{1},\cdots,\theta_{m-1})\frac{d\theta_{1}}{2\pi\mathrm{i}\theta_{1}}\cdots\frac{d\theta_{m-1}}{2\pi\mathrm{i}\theta_{m-1}}$
$\displaystyle=\oint\cdots\oint\left[\prod_{\ell=1}^{m-1}\frac{1}{1-\theta_{\ell}}\right]\mathcal{D}_{\mathrm{step}}^{(\infty)}(\theta_{1},\cdots,\theta_{m-1})\frac{d\theta_{1}}{2\pi\mathrm{i}\theta_{1}}\cdots\frac{d\theta_{m-1}}{2\pi\mathrm{i}\theta_{m-1}}.$

Finally using (8.2) we complete the proof of Theorem 2.5.

###  8.3. Proof of Lemma 8.2

By the definition of the function $\mathscr{C}_{\mathrm{step}}^{(L)}$ (see
Definition 2.14) we know

$\displaystyle\hat{\mathscr{C}}_{\mathrm{step}}^{(L)}(\theta_{0},\theta_{1},\cdots,\theta_{m-1})$
$\displaystyle=\left[\prod_{\ell=1}^{m-1}\frac{1}{1-\theta_{\ell}}\right]\left[\prod_{\ell=1}^{m}\frac{\prod_{u\in\mathcal{L}_{z_{\ell}}}(-u)^{N}\prod_{v\in\mathcal{R}_{z_{\ell}}}(v+1)^{L-N}}{\Delta(\mathcal{R}_{z_{\ell}};\mathcal{L}_{z_{\ell}})}\right]$
(8.8)
$\displaystyle\cdot\left[\prod_{\ell=1}^{m}\frac{E_{\ell}(z_{\ell})}{E_{\ell-1}(z_{\ell})}\right]\left[\prod_{\ell=2}^{m}\frac{\Delta(\mathcal{R}_{z_{\ell}};\mathcal{L}_{z_{\ell-1}})}{\prod_{u\in\mathcal{L}_{z_{\ell-1}}}(-u)^{N}\prod_{v\in\mathcal{R}_{z_{\ell}}}(v+1)^{L-N}}\right]$
$\displaystyle:=\left[\prod_{\ell=1}^{m-1}\frac{1}{1-\theta_{\ell}}\right]\mathscr{A}_{1}(z_{1},\cdots,z_{m})\mathscr{A}_{2}(z_{1},\cdots,z_{m})\mathscr{A}_{3}(z_{1},\cdots,z_{m}).$

where
$E_{\ell}(z)=\prod_{u\in\mathcal{L}_{z}}(-u)^{-k_{\ell}}\prod_{v\in\mathcal{R}_{z}}(v+1)^{-a_{\ell}-k_{\ell}}(pv+1)^{t_{\ell}-k_{\ell}}$
and the functions $\mathscr{A}_{i}$ represents the products inside each
brackets. The parameters $\theta_{i}$ and $z_{i}$ are related by the equations
$z_{i}^{L}=\prod_{j=0}^{i-1}\theta_{j}$ for $1\leq i\leq m$. Now if we send
$\theta_{0}\to 0$, then all the $z_{i}$’s will converge to $0$. As a result,
all the roots in $\mathcal{L}_{z_{i}}$ converge to $-1$ while all the roots in
$\mathcal{R}_{z_{i}}$ converge to $0$, for all $1\leq i\leq m$. It is then
easy to see that for $j=1,2,3$

$\lim_{\begin{subarray}{c}z_{i}\to 0\\\
i=1,\cdots,m\end{subarray}}\mathscr{A}_{j}(z_{1},\cdots,z_{m})=1$ (8.9)

is well-defined and hence
$\hat{\mathscr{C}}_{\mathrm{step}}^{(L)}(\theta_{0},\theta_{1},\cdots,\theta_{m-1})$
can be analytically continued to the point $\theta_{0}=0$ with

$\hat{\mathscr{C}}_{\mathrm{step}}^{(L)}(0,\theta_{1},\cdots,\theta_{m-1})=\prod_{\ell=1}^{m-1}\frac{1}{1-\theta_{\ell}}.$
(8.10)

###  8.4. Proof of Lemma 8.3

This Lemma is an analogue of Lemma 5.2 of [15] and the proof is similar so we
omit most of the details. What we want to prove is an identity between two
Fredholm determinants, with the first one acting on discrete sets consisting
of roots of certain polynomials and the other one acting on continuous
contours. By looking at the standard series expansions of both Fredholm
determinants, it suffices to show that every term in the two series expansions
matches. These type of identities between multiple sums over roots of certain
algebraic equations and multiple contour integrals can be understood as a
highly nontrivial consequence of residue theorem, which is stated and proved
with enough generality for our purpose in Proposition 4.4 of [15]. We take
$q(w)=w^{N}(w+1)^{L-N}(1+pw)^{-N}$ in the proposition and follow the same
argument as in Lemma 5.2 of [15].

###  8.5. KPZ scaling limit

In this final section we briefly discuss the large-time asymptotics for the
multi-time distribution of discrete time parallel TASEP on $\mathbb{Z}$ under
the $1:2:3$ KPZ scaling. For completeness we first recall the definition of
the limiting distribution function $\mathbb{F}_{\mathrm{step}}^{\mathrm{kpz}}$
which essentially agrees with the function $F_{\mathrm{step}}$ appearing in
Theorem 2.20 of [15] with a slight different choices of the parameters. The
structure of $\mathbb{F}_{\mathrm{step}}^{\mathrm{kpz}}$ is similar to the
finite-time distribution as contour integrals of a Fredholm determinant
$\mathrm{D}_{\mathrm{step}}^{\mathrm{kpz}}(\theta_{1},\cdots,\theta_{m-1})$.

####  8.5.1. Spaces of the operators

Given integer $m\geq 1$. We first define the contour in the complex plane

$\Gamma_{1,\mathrm{R}}:=\\{w=m+re^{\frac{\pi\mathrm{i}}{3}}:r\geq
0\\}\cup\\{w=m+re^{\frac{-\pi\mathrm{i}}{3}}:r\geq 0\\},$

oriented from $e^{-\frac{\pi\mathrm{i}}{3}}\infty$ to
$e^{\frac{\pi\mathrm{i}}{3}}\infty$. Then for $2\leq j\leq m$, we define the
contours

$\Gamma_{j,\mathrm{R}}^{\pm}:=\Gamma_{1,\mathrm{R}}\pm(j-1)=\\{w\in\mathbb{C}:w=\tilde{w}\pm(j-1)\text{
for some }\tilde{w}\in\Gamma_{1,\mathrm{R}}\\}.$

The left contours are defined simply as the reflections of the right contours

$\Gamma_{j,\mathrm{L}}^{\pm}:=-\Gamma_{j,\mathrm{R}}^{\pm},\quad\text{for
}j=2,\cdots,m,\quad\text{ and
}\quad\Gamma_{1,\mathrm{L}}:=-\Gamma_{1,\mathrm{R}},$

with orientation from $e^{-\frac{2\pi\mathrm{i}}{3}}\infty$ to
$e^{\frac{2\pi\mathrm{i}}{3}}\infty$. We further denote

$\Gamma_{j,\mathrm{L}}:=\Sigma_{j,\mathrm{L}}^{+}\cup\Gamma_{\ell,\mathrm{L}}^{-},\qquad\Gamma_{j,\mathrm{R}}:=\Gamma_{j,\mathrm{R}}^{+}\cup\Gamma_{j,\mathrm{R}}^{-},\qquad\Gamma_{j}=\Gamma_{j,\mathrm{L}}\cup\Gamma_{j,\mathrm{R}},\qquad
j=1,\cdots,m,$

and

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

and

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

####  8.5.2. Operators $\mathrm{K}_{1}^{\mathrm{kpz}}$ and
$\mathrm{K}_{\mathrm{step}}^{\mathrm{kpz}}$

Now we introduce the operators $\mathrm{K}_{1}^{\mathrm{kpz}}$ and
$\mathrm{K}_{\mathrm{step}}^{\mathrm{kpz}}$ to define
$\mathrm{D}_{\mathrm{step}}^{\mathrm{kpz}}$ in Theorem 2.7.

######  Definition 8.4.

We define

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

where two operators

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

are defined by the kernels

$\mathrm{K}_{1}^{\mathrm{kpz}}(\zeta,\zeta^{\prime}):=\left(\delta_{i}(j)+\delta_{i}(j+(-1)^{i})\right)\frac{\mathrm{f}_{i}(\zeta)}{\zeta-\zeta^{\prime}}Q_{1}^{(\infty)}(j)P_{j}(\zeta^{\prime})\quad\text{for
}\zeta^{\prime}\in\Gamma_{j}\cap\mathbb{S}_{2},\
\zeta\in\Gamma_{i}\cap\mathbb{S}_{1},$ (8.11)

and

$\mathrm{K}_{\mathrm{step}}^{\mathrm{kpz}}(\zeta^{\prime},\zeta):=\left(\delta_{j}(i)+\delta_{j}(i-(-1)^{j})\right)\frac{\mathrm{f}_{j}(\zeta^{\prime})}{\zeta^{\prime}-\zeta}Q_{2}^{(\infty)}(i)P_{i}(\zeta)\quad\text{for
}\zeta\in\Gamma_{i}\cap\mathbb{S}_{1},\
\zeta^{\prime}\in\Gamma_{j}\cap\mathbb{S}_{2},$ (8.12)

where $1\leq i,j\leq m$. Here the function $\mathrm{f}_{i}(\zeta)$ is the same
as the one appearing in the kernels for periodic case defined in (6.16) and
$Q_{1}^{(\infty)}$ and $Q_{2}^{(\infty)}$ are defined in (2.39) and (2.40). We
also set

$P_{j}(\zeta):=\begin{cases}\frac{1}{1-\theta_{j-1}},\quad&\zeta\in\Gamma_{j,\mathrm{L}}^{+}\cup\Gamma_{j,\mathrm{R}}^{+},\quad
j=2,\cdots,m\\\
\frac{-\theta_{j-1}}{1-\theta_{j-1}},\quad&w\in\Gamma_{j,\mathrm{L}}^{-}\cup\Gamma_{j,\mathrm{R}}^{-},\quad
j=2,\cdots,m\\\
1,\quad&w\in\Gamma_{1,\mathrm{L}}\cup\Gamma_{1,\mathrm{R}}.\end{cases}$ (8.13)

###### Proof of Theorem 2.7.

The proof is a standard steepest descent analysis and most of the key
ingredients are similar to Lemma 7.8 so we only provide the essential
calculations. First it is not difficult to see that for any nonzero constants
$c_{1},\cdots,c_{m}$, the Fredholm determinant part
$\mathcal{D}_{\mathrm{step}}^{(\infty)}$ is invariant if we replace the
functions $F_{i}$ appearing in the kernels by $\tilde{F}_{i}:=c_{i}F_{i}$ for
$1\leq i\leq m$. For our purpose we set $c_{i}=\frac{1}{F_{i}(w_{c})}$ where
$w_{c}=-\frac{1}{1+\sqrt{q}}$. We point out that this agrees with the critical
point for periodic cases defined in (2.5) when $\rho=1/2$ so the notation is
consistent. To ensure the kernels have sufficiently fast decay on each
variable we make the same conjugation as in (7.31) by setting

$\displaystyle\tilde{\mathcal{K}}_{1}^{(\infty)}(w,w^{\prime}):=\left(\delta_{i}(j)+\delta_{i}(j+(-1)^{i})\right)\frac{\sqrt{\tilde{f}_{i}(w)}\sqrt{\tilde{f}_{j}(w^{\prime})}}{w-w^{\prime}}Q_{1}^{(\infty)}(j)P_{j}(w^{\prime}),$
$\displaystyle\tilde{\mathcal{K}}_{\mathrm{step}}^{(\infty)}(w,w^{\prime}):=\left(\delta_{j}(i)+\delta_{j}(i+(-1)^{j})\right)\frac{\sqrt{\tilde{f}_{j}(w^{\prime})}\sqrt{\tilde{f}_{i}(w)}}{w-w^{\prime}}Q_{1}^{(\infty)}(j)P_{i}(w),$

for $w^{\prime}\in\Sigma_{j}\cap\mathcal{S}_{2},\
w\in\Sigma_{i}\cap\mathcal{S}_{1}$. Here $\tilde{f}_{i}$ is obtained by
replacing $F_{i}$ with $\tilde{F}_{i}$ in (2.35). We also conjugate the
limiting kernels in a similar way:

$\displaystyle\tilde{\mathrm{K}}_{1}^{\mathrm{kpz}}(\zeta,\zeta^{\prime}):=\left(\delta_{i}(j)+\delta_{i}(j+(-1)^{i})\right)\frac{\sqrt{\mathrm{f}_{i}(\zeta)}\sqrt{\mathrm{f}_{j}(\zeta^{\prime})}}{\zeta-\zeta^{\prime}}Q_{1}^{(\infty)}(j)P_{j}(\zeta^{\prime}),$
$\displaystyle\tilde{\mathrm{K}}_{\mathrm{step}}^{\mathrm{kpz}}(\zeta,\zeta^{\prime}):=\left(\delta_{j}(i)+\delta_{j}(i+(-1)^{j})\right)\frac{\sqrt{\mathrm{f}_{j}(\zeta^{\prime})}\sqrt{\mathrm{f}_{i}(\zeta)}}{\zeta-\zeta^{\prime}}Q_{1}^{(\infty)}(j)P_{i}(\zeta),$

for $\zeta^{\prime}\in\Gamma_{j}\cap\mathbb{S}_{2},\
\zeta\in\Gamma_{i}\cap\mathbb{S}_{1}$. These conjugations do not change the
Fredholm determinants.

Now a straightforward Taylor expansion shows that for
$w=w_{c}+\frac{q^{1/4}}{1+\sqrt{q}}\zeta T^{-1/3}$ with $|\zeta|\leq
T^{\epsilon/4}$ for $0<\epsilon<1/3$ we have

$\displaystyle\frac{F_{i}(w)}{F_{i}(w_{c})}$
$\displaystyle=\exp\left(k_{i}\log\left(\frac{w}{w_{c}}\right)-(a_{i}+k_{i})\log\left(\frac{w+1}{w_{c}+1}\right)+(t_{i}-k_{i})\log\left(\frac{pw+1}{pw_{c}+1}\right)\right)$
(8.14)
$\displaystyle=\exp\left(-\frac{1}{3}\tau_{i}\zeta^{3}+\frac{1}{2}\gamma_{i}\zeta^{2}+\mathrm{x}_{i}\zeta+O(T^{\epsilon-\frac{1}{3}})\right),$

where the parameters $a_{i},k_{i}$ and $t_{i}$ satisfies (2.13). On the other
for $w$ far away from $w_{c}$, $\frac{F_{i}(w)}{F_{i}(w_{c})}$ decays or grows
exponentially fast. Hence it is straightforward to show that for
$w=-\frac{1}{1+\sqrt{q}}+\frac{q^{1/4}}{1+\sqrt{q}}\zeta T^{-1/3}$ and
$0<\epsilon<1/3$ we have

$\tilde{f}_{j}(w)=\begin{cases}\mathrm{f}_{j}(\zeta)(1+O(T^{\epsilon-1/3}))\quad&\text{if
}|\zeta|\leq T^{\epsilon/4},\\\ O(e^{-cT^{\epsilon/2}})\quad&\text{if
}|\zeta|\geq T^{\epsilon/4}.\end{cases}$ (8.15)

Now we deform the contours $\Sigma_{j}$ to be sufficiently close to the
critical points $w_{c}=-\frac{1}{1+\sqrt{q}}$ such that locally they look like
the limiting contours $\Gamma_{j}$. More precisely we deform
$\Sigma_{j,\mathrm{R}}^{+}$ such that

$\displaystyle\Sigma_{j,\mathrm{R}}^{+}\cap\\{w\in\mathbb{C}:|w-w_{c}|\leq\frac{1+\sqrt{q}}{q^{1/4}}T^{\epsilon/4-1/3}\\}$
$\displaystyle=\\{w_{c}+T^{-1/3}(m+j-1+re^{\frac{\pi\mathrm{i}}{3}}):0\leq
r\leq T^{\epsilon/4}\\}$
$\displaystyle\cup\\{w_{c}+T^{-1/3}(m+j-1+re^{-\frac{\pi\mathrm{i}}{3}}):0\leq
r\leq T^{\epsilon/4}\\},$

and similar for other contours. Then by (8.15) it is straightforward to show
that

  1. (1)

For each $n\in\mathbb{N}$ and $0<|\theta_{i}|<1$ we have

$\lim_{T\to\infty}\mathrm{Tr}\left(\tilde{\mathcal{K}}_{1}^{(\infty)}\tilde{\mathcal{K}}_{\mathrm{step}}^{(\infty)}\right)^{n}=\mathrm{Tr}\left(\tilde{\mathrm{K}}_{1}^{\mathrm{kpz}}\tilde{\mathrm{K}}_{\mathrm{step}}^{\mathrm{kpz}}\right)^{n}.$

  2. (2)

There exists constant $C>0$ independent of $T$ and $n$ such that for all
$n\in\mathbb{N}$

$\left|\oint_{\mathcal{S}_{1}}\cdots\oint_{\mathcal{S}_{1}}\det\left[\left(\tilde{\mathcal{K}}_{1}^{(\infty)}\tilde{\mathcal{K}}_{\mathrm{step}}^{(\infty)}\right)(w_{i},w_{j})\right]_{i,j=1}^{n}\frac{{\mathrm{d}}w_{1}}{2\pi\mathrm{i}}\cdots\frac{{\mathrm{d}}w_{n}}{2\pi\mathrm{i}}\right|\leq
C^{n}.$

Now (1) and (2) immediately implies
$\mathcal{D}_{\mathrm{step}}^{(\infty)}(\theta_{1},\cdots,\theta_{m-1})\text{
converges to
}\mathrm{D}_{\mathrm{step}}^{(\mathrm{kpz}}(\theta_{1},\cdots,\theta_{m-1})$
locally uniformly for $0<|\theta_{i}|<1$ as $T\to\infty$, thus proving Theorem
2.7. ∎

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Yuchen Liao, University of Michigan, Department of Mathematics, 530 Church
Street, Ann Arbor, MI 48109, USA

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