Multipoint distribution of periodic TASEP

Abstract

The height fluctuations of the models in the KPZ class are expected to converge to a universal process. The spatial process at equal time is known to converge to the Airy process or its variations. However, the temporal process, or more generally the two-dimensional space-time fluctuation field, is less well understood. We consider this question for the periodic TASEP (totally asymmetric simple exclusion process). For a particular initial condition, we evaluate the multitime and multilocation distribution explicitly in terms of a multiple integral involving a Fredholm determinant. We then evaluate the large-time limit in the so-called relaxation time scale.

1. Introduction

The models in the KPZ universality class are expected to have the 1:2:3 scaling for the height fluctuations, spatial correlations, and time correlations as time $t\to \infty$. This means that the scaled two-dimensional fluctuation field

$$\begin{equation} h_t(\gamma , \tau ) := \frac{H(c_1\gamma (\tau t)^{2/3}, \tau t) - \left(c_2(\tau t)+c_3(\tau t)^{2/3}\right)}{c_4(\tau t)^{1/3}} \tag{1.1} \end{equation}$$

of the height function $H(\ell , t)$, where $\ell$ is the spatial variable and $t$ is time, is believed to converge to a universal field which depends only on the initial condition.⁠¹ Here $c_1, c_2, c_3, c_4$ are model-dependent constants. Determining the limiting two-dimensional fluctuation field

¹

For some initial conditions, such as the stationary initial condition, one may need to translate the space in the characteristic direction.

✖

$$\begin{equation} (\gamma , \tau ) \mapsto h(\gamma , \tau ):= \lim _{t\to \infty } h_t(\gamma , \tau ) {\tag{1.2}} \end{equation}$$

is an outstanding question.

By now there are several results for the one-point distribution. The one-point distribution of $h(\gamma , \tau )$ for fixed $(\gamma , \tau )$ is given by random matrix distributions (Tracy-Widom distributions) or their generalizations. The convergence is proved for a quite long list of models including PNG, TASEP, ASEP, $q$-TASEP, random tilings, last passage percolations, directed polymers, the KPZ equation, and so on. See, for example, 1262641, and the review article 11. These models were studied using various integrable methods under standard initial conditions. See also the recent papers 1337 for general initial conditions.

The spatial one-dimensional process, $\gamma \mapsto h(\gamma , \tau )$ for fixed $\tau$, is also well understood. This process is given by the Airy process and its variations. However, the convergence is proved rigorously only for a smaller number of models. It was proved for the determinantal models like PNG, TASEP, last passage percolation,⁠² but not yet for other integrable models such as ASEP, $q$-TASEP, finite-temperature directed polymers, and the KPZ equation.

²

See, for example, 38910252734 for special initial conditions. See the recent paper 31 for general initial conditions for TASEP.

✖

The two-dimensional fluctuation field, $(\gamma , \tau ) \mapsto h(\gamma , \tau )$, on the other hand, is less well understood. The joint distribution is known only for the two-point distribution. In 2015, Johansson 28 considered the zero temperature Brownian semidiscrete directed polymer and computed the limit of the two-point (in time and location) distribution.⁠³ The limit is obtained in terms of rather complicated series involving the determinants of matrices whose entries contain the Airy kernel. The formula is simplified more recently in terms of a contour integral of a Fredholm determinant in 29 in which the author also extended his work to the directed last passage percolation model with geometric weights. Two other papers studied qualitative behaviors of the temporal correlations. Using a variational problem involving two independent Airy processes, Ferrari and Spohn 20 proved in 2016 the power law of the covariance in the time direction in the large- and small-time limits, $\tau _1/\tau _2\to 0$ and $\tau _1/\tau _2\to 1$. Here, $\tau _1$ and $\tau _2$ denote the scaled time parameters. De Nardis and Le Doussal 14 extended this work further and also, augmented by other physics arguments, computed the similar limits of the two-time distribution when one of the arguments is large. It is yet to be seen if one can deduce these results from the formula of Johansson.

³

There are non-rigorous physics papers for the two-time distribution of directed polymers 161718. However, another physics paper 14 indicates that the formulas in these papers are not correct.

✖

The objective of this paper is to study the two-dimensional fluctuation field of spatially periodic KPZ models. Specifically, we evaluate the multipoint distribution of the periodic TASEP (totally asymmetric simple exclusion process) and compute a large-time limit in a certain critical regime.

We denote by $L$ the period and by $N$ the number of particles per period. Set $\rho =N/L$, the average density of particles. The periodic TASEP (of period $L$ and density $\rho$) is defined by the occupation function $\eta _j(t)$ satisfying the spatial periodicity:

$$\begin{equation} \eta _j(t)=\eta _{j+L}(t),\qquad j\in \mathbb{Z}, \quad t\ge 0. \tag{1.3} \end{equation}$$

Apart from this condition, the particles follow the usual TASEP rules.

Consider the limit as $t, L, N\to \infty$ with fixed $\rho =N/L$. Since the spatial fluctuations of the usual infinite TASEP is $O(t^{2/3})$, all of the particles in the periodic TASEP are correlated when $t^{2/3}=O(L)$. We say that the periodic TASEP is in the relaxation time scale if

$$\begin{equation} t=O(L^{3/2}). \tag{1.4} \end{equation}$$

If $t\ll L^{3/2}$, we expect that the system size has negligible effect and, therefore, the system follows the KPZ dynamics. See, for example, 5. On the other hand, if $t\gg L^{3/2}$, then the system is basically in a finite system, and hence we expect the stationary dynamics. See, for example, 15. Therefore, in the relaxation time scale, we predict that the KPZ dynamics and the stationary dynamics are both present.

Even though the periodic TASEP is as natural as the infinite TASEP, the one-point distribution was obtained only recently. Over the last two years, in a physics paper 36 and, independently, in mathematics papers 430, the authors evaluated the one-point function of the height function in finite time and computed the large-time limit in the relaxation time scale. The one-point function follows the KPZ scaling $O(t^{1/3})$ but the limiting distribution is different from that of the infinite TASEP.⁠⁴ This result was obtained for the three initial conditions of periodic step, flat, and stationary. Some earlier related studies can be found in physics papers 1521222324323335, including results on the large deviation and spectral properties of the system.

⁴

The formulas obtained in 430 and 36 are similar, but different. It is yet to be checked that these formulas are the same.

✖

In this paper, we extend the analysis of the papers 430 and compute the multipoint (in time and location) distribution of the periodic TASEP with a special initial condition called the periodic step initial condition. Here we allow any number of points unlike the previous work of Johansson on the infinite TASEP. It appears that the periodicity of the model simplifies the algebraic computation compared with the infinite TASEP. In a separate paper we will consider flat and stationary initial conditions. The main results are the following:

(1)

For arbitrary initial conditions, we evaluate finite-time joint distribution functions of the periodic TASEP at multiple points in the space-time coordinates in terms of a multiple integral involving a determinant of size $N$. See Theorem 3.1 and Corollary 3.3.

(2)

For the periodic step initial condition, we simplify the determinant to a Fredholm determinant. See Theorem 4.6 and Corollary 4.7.

(3)

We compute the large-time limit of the multipoint (in the space-time coordinates) distribution in the relaxation time scale for the periodic TASEP with the periodic step initial condition. See Theorem 2.1.

One way of studying the usual infinite TASEP is the following. First, one computes the transition probability using the coordinate Bethe ansatz method. This means that we solve the Kolmogorov forward equation explicitly after replacing it (which contains complicated interactions between the particles) by the free evolution equation with certain boundary conditions. In 40, Schütz obtained the transition probability of the infinite TASEP. Second, one evaluates the marginal or joint distribution by taking a sum of the transition probabilities. It is important that the resulting expression should be suitable for the asymptotic analysis. This is achieved typically by obtaining a Fredholm determinant formula. In 38, Rákos and Schütz rederived the famous finite-time Fredholm determinant formula of Johansson 26 for the one-point distribution in the case of the step initial condition using this procedure. Subsequently, Sasamoto 39 and Borodin, Ferrari, Prähofer, and Sasamoto 9 obtained a Fredholm determinant formula for the joint distribution of multiple points with equal time. This was further extended by Borodin and Ferrari 7 to the points in spatial directions of each other. However, it was not extended to the case when the points are temporal directions of each other. The third step is to analyze the finite-time formula asymptotically using the method of steepest-descent. See 792639 and also a more recent paper 31. In the KPZ 1:2:3 scaling limit, the above algebraic formulas give only the spatial process $\gamma \mapsto h(\gamma , \tau )$.

We applied the above procedure to the one-point distribution of the periodic TASEP in 4. We obtained a formula for the transition probability, which is a periodic analogue of the formula of Schütz. Using that, we computed the finite-time one-point distribution for an arbitrary initial condition. The distribution was given by an integral of a determinant of size $N$. We then simplified the determinant to a Fredholm determinant for the cases of the step and flat initial conditions. The resulting expression was suitable for the asymptotic analysis. A similar computation for the stationary initial condition was carried out in 30.

In this paper, we extend the analysis of 430 to multipoint distributions. For general initial conditions, we evaluate the joint distribution by taking a multiple sum of the transition probabilities obtained in 4. The computation can be reduced to an evaluation of a sum involving only two arbitrary points in the space-time coordinates (with different time coordinates). The main technical result of this paper, presented in Proposition 3.4, is the evaluation of this sum in a compact form. The key point, compared with the infinite TASEP 7931, is that the points do not need to be restricted to the spatial directions.⁠⁵ The final formula is suitable for the large-time asymptotic analysis in relaxation time scale.

⁵

In the large-time limit, we add a certain restriction when the rescaled times are equal. See Theorem 2.1. The outcome of the above computation is that we find the joint distribution in terms of a multiple integral involving a determinant of size $N$. For the periodic step initial condition, we simplify the determinant further to a Fredholm determinant.

✖

If we take the period $L$ to infinity while keeping other parameters fixed, the periodic TASEP becomes the infinite TASEP. Moreover, it is easy to check (see Section 8 below) that the joint distributions of the periodic TASEP and the infinite TASEP are equal even for fixed $L$ if $L$ is large enough compared with the times. Hence, the finite-time joint distribution formula obtained in this paper (Theorem 3.1 and Corollary 3.3) in fact gives a formula of the joint distribution of the infinite TASEP; see the equations 8.7 and 8.8. This formula contains an auxiliary parameter $L$ which has no meaning in the infinite TASEP. From this observation, we find that if we take the large-time limit of our formula in the subrelaxation time scale, $t\ll L^{3/2}$, then the limit, if it exists, is the joint distribution of the two-dimensional process $h(\gamma , \tau )$ in 1.2. However, it is not clear at this moment if our formula is suitable for the asymptotic analysis in the subrelaxation time scale; the kernel of the operator in the Fredholm determinant does not seem to converge in the subrelaxation time scale while it converges in the relaxation time scale. The question of computing the limit in the subrelaxation time scale, and hence the multipoint distribution of the infinite TASEP, will be left as a later project.

This paper is organized as follows. We state the limit theorem in Section 2. The finite-time formula for general initial conditions is in Section 3. Its simplification for the periodic step initial condition is obtained in Section 4. In Section 5, we prove Proposition 3.4, the key algebraic computation. The asymptotic analysis of the formula obtained in Section 4 is carried out in Section 6, proving the result in Section 2. We discuss some properties of the limit of the joint distribution in Section 7. In Section 8 we show that the finite-time formulas obtained in Sections 3 and 4 are also valid for infinite TASEP for all large enough $L$.

2. Limit theorem for multipoint distribution

2.1. Limit theorem

Consider the periodic TASEP of period $L$ with $N$ particles per period. We set $\rho =N/L$, the average particle density. We assume that the particles move to the right. Let $\eta _j(t)$ be the occupation function of periodic TASEP: $\eta _j(t)=1$ if the site $j$ is occupied at time $t$, otherwise $\eta _j(t)=0$, and it satisfies the periodicity $\eta _j(t)=\eta _{j+L}(t)$. We consider the periodic step initial condition defined by

$$\begin{equation} \begin{split} & \eta _{j}(0) = \begin{dcases} 1 \qquad \text{for $-N+1\le j\le 0$,}\\ 0 \qquad \text{for $1\le j\le L-N$,} \end{dcases} \\ \end{split} {\tag{2.1}} \end{equation}$$

and $\eta _{j+L}(0) =\eta _{j}(0)$.

We state the results in terms of the height function

$$\begin{equation} \mathbb{h}(\mathbf{p}), \qquad \text{where $\mathbf{p}=\ell \mathbf{e}_1+t\mathbf{e}_2=(\ell ,t)\in \mathbb{Z}\times \mathbb{R}_{\ge 0}$.} {\tag{2.2}} \end{equation}$$

Here $\mathbf{e}_1=(1,0)$ and $\mathbf{e}_2=(0,1)$ are the unit coordinate vectors in the spatial and time directions, respectively. The height function is defined by

$$\begin{equation} \mathbb{h}(\ell \mathbf{e}_1+t\mathbf{e}_2) = \begin{dcases} 2J_0(t)+\sum _{j=1}^\ell \left(1-2\eta _j(t)\right),&\ell \ge 1,\\ 2J_0(t), &\ell =0,\\ 2J_0(t)-\sum _{j=\ell +1}^0\left(1-2\eta _j(t)\right),&\ell \le -1, \end{dcases} {\tag{2.3}} \end{equation}$$

where $J_0(t)$ counts the number of particles jumping through the bond from $0$ to $1$ during the time interval $[0,t]$. The periodicity implies that

$$\begin{equation} \mathbb{h}((\ell +nL)\mathbf{e}_1+t\mathbf{e}_2)=\mathbb{h}(\ell \mathbf{e}_1+t\mathbf{e}_2)+n(L-2N) \tag{2.4} \end{equation}$$

for integers $n$.

See Figure 1 for the evolution of the density profile and Figure 2 for the limiting height function. Note that the step initial condition 2.1 generates shocks.⁠⁶. By solving the Burgers’ equation in a periodic domain, one could derive the explicit formulas of the density profile, the limiting height function and the shock location. These computations were done in 5.

⁶

These shocks are generated when faster particles from the lower density region enter the higher density region and are forced to slow down. See, for example, 1219 for the study of similar behaviors in infinite TASEP.

✖

We represent the space-time position in new coordinates. Let

$$\begin{equation} \mathbf{e}_c:=(1-2\rho )\mathbf{e}_1+\mathbf{e}_2 \tag{2.5} \end{equation}$$

be a vector parallel to the characteristic directions. If we represent $\mathbf{p}= \ell \mathbf{e}_1+t\mathbf{e}_2$ in terms of $\mathbf{e}_1$ and $\mathbf{e}_c$, then

$$\begin{equation} \mathbf{p} = s \mathbf{e}_1+t \mathbf{e}_c, \qquad \text{where $s=\ell - t(1-2\rho )$.} \tag{2.6} \end{equation}$$

Consider the region

$$\begin{equation} \begin{split} \mathbf{R}:=&\{\ell \mathbf{e}_1+t\mathbf{e}_2\in \mathbb{Z}\times \mathbb{R}_{\ge 0}: 0\le \ell -(1-2\rho ) t\le L\} \\ = &\{s \mathbf{e}_1+t \mathbf{e}_c \in \mathbb{Z}\times \mathbb{R}_{\ge 0}: 0\le s\le L\}. \end{split} \tag{2.7} \end{equation}$$

See Figure 3. Due to the periodicity, the height function in $\mathbf{R}$ determines the height function in the whole space-time plane.

The following theorem is the main asymptotic result. We take the limit as follows. We take $L,N\to \infty$ in such a way that the average density $\rho =N/L$ is fixed, or more generally $\rho$ stays in a compact subset of the interval $(0,1)$. We consider $m$ distinct points $\mathbf{p}_i = s_i \mathbf{e}_1+t_i \mathbf{e}_c$ in the space-time plane such that their temporal coordinate $t_j\to \infty$ and they satisfy the relaxation time scale $t_j=O(L^{3/2})$. The relative distances of the coordinates are scaled as in the 1:2:3 KPZ prediction: $t_i-t_j=O(L^{3/2})=O(t_i)$, $s_i=O(L)=O(t_i^{2/3})$, and the height at each point is scaled by $O(L^{1/2})=O(t_i^{1/3})$.

Theorem 2.1 (Limit of multipoint joint distribution for periodic TASEP).

Fix two constants $c_1$ and $c_2$ satisfying $0<c_1<c_2<1$. Let $N=N_L$ be a sequence of integers such that $c_1L\le N\le c_2L$ for all sufficiently large $L$. Consider the periodic TASEP of period $L$ and average particle density $\rho =\rho _L=N/L$. Assume the periodic step initial condition 2.1. Let $m$ be a positive integer. Fix $m$ points $\mathsf{p}_j=(\gamma _j,\tau _j)$, $j=1, \cdots , m$, in the region

$$\begin{equation} \mathsf{R}:=\{(\gamma ,\tau )\in \mathbb{R}\times \mathbb{R}_{>0}:0\le \gamma \le 1\}. {\tag{2.8}} \end{equation}$$

Assume that

$$\begin{equation} \tau _1<\tau _2<\cdots < \tau _m. {\tag{2.9}} \end{equation}$$

Let $\mathbf{p}_j=s_j\mathbf{e}_1+t_j\mathbf{e}_c$ be $m$ points⁠⁷ in the region $\mathbf{R}$ shown in Figure 3, where $\mathbf{e}_1=(1,0)$ and $\mathbf{e}_c=(1-2\rho ,1)$, with

⁷

Since $\mathbf{p}_j$ should have an integer value for its spatial coordinate, to be precise, we need to take the integer part of $s_j+t_j(1-2\rho )$ for the spatial coordinate. This small distinction does not change the result since the limits are uniform in the parameters $\gamma _j, \tau _j$. Therefore, we suppress the integer value notation throughout this paper.

✖

$$\begin{equation} \begin{split} s_j=\gamma _j L,\qquad t_j=\tau _j \frac{L^{3/2}}{\sqrt {\rho (1-\rho )}} . \end{split} {\tag{2.10}} \end{equation}$$

Then, for arbitrary fixed $x_1,\cdots ,x_m\in \mathbb{R}$,

$$\begin{equation} \begin{split} &\lim _{L\to \infty }\mathbb{P}\left(\bigcap _{j=1}^m\left\{\frac{\mathbb{h}(\mathbf{p}_j)-(1-2\rho )s_j-(1-2\rho +2\rho ^2)t_j}{-2\rho ^{1/2}(1-\rho )^{1/2} L^{1/2}}\le x_j\right\}\right)\\ &\qquad =\mathbb{F}(x_1,\cdots ,x_m;\mathsf{p}_1,\cdots ,\mathsf{p}_m ), \end{split} {\tag{2.11}} \end{equation}$$

where the function $\mathbb{F}$ is defined in 2.15. The convergence is locally uniform in $x_j, \tau _j$, and $\gamma _j$. If $\tau _i=\tau _{i+1}$ for some $i$, then 2.11 still holds if we assume that $x_{i}< x_{i+1}$.

Remark 2.2.

Suppose that we have arbitrary $m$ distinct points $\mathsf{p}_j=(\gamma _j,\tau _j)$ in $\mathsf{R}$. Then we may rearrange them so that $0<\tau _1\le \cdots \le \tau _m$. If $\tau _j$ are all different, we can apply the above theorem since the result holds for arbitrarily ordered $\gamma _j$. If some of the $\tau _j$ are equal, then we may rearrange the points further so that $x_j$ are ordered with those $\tau _j$, and use the theorem if $x_j$ are distinct. The only case which is not covered by the above theorem is when some of the $\tau _j$ are equal and the corresponding $x_j$ are also equal.

The case when $m=1$ was essentially obtained in our previous paper 4 (and also 36). In that paper, we considered the location of a tagged particle instead of the height function, but it is straightforward to translate the result to the height function.

Remark 2.3.

We will check that $\mathbb{F}(x_1,\cdots ,x_m; \mathrm{p}_1,\cdots ,\mathrm{p}_m)$ is periodic with respect to each of the space coordinates $\gamma _j$ in Subsection 2.2. By this spatial periodicity, we can remove the restrictions $0\le \gamma _j\le 1$ in the above theorem.

Remark 2.4.

Since we expect the KPZ dynamics in the subrelaxation scale $t_j\ll L^{3/2}$, we expect that the $\tau _j\to 0$ limit of the above result should give rise to a result for the usual infinite TASEP. Concretely, we expect that the limit

$$\begin{equation} \lim _{\tau \to 0} \mathbb{F}((\tau _1\tau )^{1/3}x_1, \cdots , (\tau _m\tau )^{1/3}x_m; (\gamma _1(\tau _1\tau )^{2/3}, \tau _1 \tau ), \cdots , (\gamma _m(\tau _m\tau )^{2/3}, \tau _m \tau )) {\tag{2.12}} \end{equation}$$

exists and it is the limit of the multitime, multilocation joint distribution of the height function $\mathfrak{H}(s,t)$ of the usual TASEP with step initial condition,

$$\begin{equation} \lim _{T \to \infty }\mathbb{P}\left(\bigcap _{j=1}^m\left\{\frac{\mathfrak{H}(\gamma _j \tau _j^{2/3}T^{2/3}, 2\tau _j T) - \tau _j T}{- \tau _j^{1/3}T^{1/3}}\le x_j\right\}\right) . \tag{2.13} \end{equation}$$

In particular, we expect that when $m=1$, 2.12 is $F_{GUE}(x_1+\gamma _1^2/4)$, the Tracy-Widom GUE distribution; we expect that when $\tau _1=\cdots =\tau _m$, 2.12 is equal to the corresponding joint distribution of the Airy$_2$ process 34 $A_2(\gamma /2)-\gamma ^2/4$; and when $m=2$, 2.12 is expected to match the two-time distribution $F_{\text{two-time}}(\gamma _1/2,x_1+\gamma _1^2/4;\gamma _2/2,x_2+\gamma _2^2/4;\tau _1^{1/3}(\tau _2-\tau _1)^{-1/3})$ obtained by Johansson 2829. See also Section 8.

2.2. Formula of the limit of the joint distribution

2.2.1. Definition of $\mathbb{F}$

Definition 2.5.

Fix a positive integer $m$. Let $\mathsf{p}_j=(\gamma _j, \tau _j)$ for each $j=1, \cdots , m$, where $\gamma _j\in \mathbb{R}$ and

$$\begin{equation} 0<\tau _1<\tau _2<\cdots < \tau _m. \tag{2.14} \end{equation}$$

Define, for $x_1, \cdots , x_m \in \mathbb{R}$,

$$\begin{equation} \begin{split} &\mathbb{F}\left(x_1,\cdots ,x_m;\mathrm{p}_1,\cdots ,\mathrm{p}_m\right) = \oint \cdots \oint \mathsf{C}(\mathbf{z}) \mathsf{D}(\mathbf{z}) \frac{{\mathrm{d}}z_m}{2\pi {\mathrm{i}}z_m}\cdots \frac{{\mathrm{d}}z_1}{2\pi {\mathrm{i}}z_1}, \end{split} {\tag{2.15}} \end{equation}$$

where $\mathbf{z}=(z_1, \cdots , z_m)$ and the contours are nested circles in the complex plane satisfying $0<|z_m|<\cdots <|z_1|<1$. Set $\mathbf{x}=(x_1, \cdots , x_m)$, $\boldsymbol{\tau }=(\tau _1, \cdots , \tau _m)$, and $\boldsymbol{\gamma }=(\gamma _1, \cdots , \gamma _m)$. The function $\mathsf{C}(\mathbf{z})=\mathsf{C}(\mathbf{z}; \mathbf{x}, \boldsymbol{\tau })$ is defined by 2.21 and it depends on $\mathbf{x}$ and $\boldsymbol{\tau }$ but not on $\boldsymbol{\gamma }$. The function $\mathsf{D}(\mathbf{z})=\mathsf{D}(\mathbf{z}; \mathbf{x}, \boldsymbol{\tau }, \boldsymbol{\gamma })$ depends on all $\mathbf{x}$, $\boldsymbol{\tau }$, and $\boldsymbol{\gamma }$, and it is given by the Fredholm determinant $\mathsf{D}(\mathbf{z}) =\det (1-\mathsf{K_1}\mathsf{K_2})$ defined in 2.38.

The functions in the above definition satisfy the following properties. The proofs of (P1), (P3), and (P4) are scattered in this section while (P2) is proved later in Lemma 7.1.

(P1)

For each $i$, $\mathsf{C}(\mathbf{z})$ is a meromorphic function of $z_i$ in the disk $|z_i|< 1$. It has simple poles at $z_i=z_{i+1}$ for $i=1, \cdots , m-1$.

(P2)

For each $i$, $\mathsf{D}(\mathbf{z})$ is analytic in the punctured disk $0<|z_i|<1$.

(P3)

For each $i$, $\mathsf{D}(\mathbf{z})$ does not change if we replace $\gamma _i$ by $\gamma _i+1$. Therefore, $\mathbb{F}$ is periodic, with period $1$, in the parameter $\gamma _i$ for each $i$.

(P4)

If $\tau _i=\tau _{i+1}$, the function $\mathbb{F}$ is still well-defined for $x_i< x_{i+1}$.

Remark 2.6.

It is not easy to check directly from the formula that $\mathbb{F}$ defines a joint distribution function. Nonetheless, we may check them indirectly. From the fact that $\mathbb{F}$ is a limit of a sequence of joint distribution functions, $0\le \mathbb{F}\le 1$ and $\mathbb{F}$ is a non-decreasing function of $x_k$ for each $k$. It also follows from the fact that the joint distribution can be majorized by a marginal distribution that $\mathbb{F}$ converges to $0$ if any coordinate $x_k\to -\infty$ since it was shown in (4.10) of 4 that the $m=1$ case is indeed a distribution function; see Lemma 7.5 below. The most difficult property to prove is the consistency which $\mathbb{F}$ should satisfy as a coordinate $x_k\to +\infty$. We prove this property in Section 7 by finding a probabilistic interpretation of the formula of $\mathbb{F}$ when the $z_i$-contours are not nested; see Theorem 7.3 and Proposition 7.4.

2.2.2. Definition of $\mathsf{C}(\mathbf{z})$

Let $\log z$ be the principal branch of the logarithm function with cut $\mathbb{R}_{\le 0}$. Let $\mathrm{Li}_s(z)$ be the polylogarithm function defined by

$$\begin{equation} \mathrm{Li}_s(z)= \sum _{k=1}^\infty \frac{z^k}{k^s} \quad \text{for $|z|<1$ and $s\in \mathbb{C}$.} \tag{2.16} \end{equation}$$

It has an analytic continuation using the formula

$$\begin{equation} \mathrm{Li}_s(z) = \frac{z}{\Gamma (s)} \int _0^\infty \frac{x^{s-1}}{e^x-z} \mathrm{d}x \quad \text{for $z\in \mathbb{C}\setminus [1,\infty )$ if $\Re (s)>0$.} \tag{2.17} \end{equation}$$

Set

$$\begin{equation} A_1(z)= -\frac{1}{\sqrt {2\pi }} \mathrm{Li}_{3/2}(z), \qquad A_2(z)= -\frac{1}{\sqrt {2\pi }} \mathrm{Li}_{5/2}(z). {\tag{2.18}} \end{equation}$$

For $0<|z|,|z'|<1$, set

$$\begin{equation} B(z, z')= \frac{zz'}{2} \iint \frac{\eta \xi \log (-\xi +\eta )}{(e^{-\xi ^2/2}-z)(e^{-\eta ^2/2}-z')} \frac{{\mathrm{d}}\xi }{2\pi {\mathrm{i}}}\frac{{\mathrm{d}}\eta }{2\pi {\mathrm{i}}} =\frac{1}{4\pi }\sum _{k,k'\ge 1}\frac{z^k(z')^{k'}}{(k+k')\sqrt {kk'}}, {\tag{2.19}} \end{equation}$$

where the integral contours are the vertical lines $\Re \xi =\mathfrak{a}$ and $\Re \eta =\mathfrak{b}$ with constants $\mathfrak{a}$ and $\mathfrak{b}$ satisfying $-\sqrt {-\log |z|}<\mathfrak{a}<0<\mathfrak{b}<\sqrt {-\log |z'|}$. The equality of the double integral and the series is easy to check (see (9.27)–(9.30) in 4 for a similar calculation). Note that $B(z,z')=B(z',z)$. When $z=z'$, we can also check that

$$\begin{equation} B(z):=B(z, z)=\frac{1}{4\pi }\int _0^z\frac{\left(\mathrm{Li}_{1/2}(y)\right)^2}{y}\mathrm{d}y. {\tag{2.20}} \end{equation}$$

Definition 2.7.

Define

$$\begin{equation} \begin{split} \mathsf{C}(\mathbf{z}) := \left[ \prod _{\ell =1}^{m} \frac{z_\ell }{z_\ell - z_{\ell +1}} \right] \left[ \prod _{\ell =1}^{m} \frac{e^{x_\ell A_1(z_{\ell })+ \tau _\ell A_2(z_\ell )}}{e^{x_\ell A_1(z_{\ell +1}) + \tau _\ell A_2(z_{\ell +1})}} e^{2B(z_{\ell })-2B(z_{\ell +1},z_\ell )} \right], \end{split} {\tag{2.21}} \end{equation}$$

where we set $z_{m+1}:=0$.

Since $A_1, A_2, B$ are analytic inside the unit circle, it is clear from the definition that $\mathsf{C}(\mathbf{z})$ satisfies property (P1) in Subsubsection 2.2.1.

2.2.3. Definition of $\mathsf{D}(\mathbf{z})$

The function $\mathsf{D}(\mathbf{z})$ is given by a Fredholm determinant. Before we describe the operator and the space, we first introduce a few functions.

For $|z|<1$, define the function

$$\begin{equation} \mathsf{h}(\zeta ,z) = -\frac{1}{\sqrt {2\pi }} \int _{-\infty }^{\zeta } \mathrm{Li}_{1/2} \big (z e^{(\zeta ^2-y^2)/2}\big ) \mathrm{d}y \qquad \text{for $\Re (\zeta ) < 0$} {\tag{2.22}} \end{equation}$$

and

$$\begin{equation} \mathsf{h}(\zeta ,z) = -\frac{1}{\sqrt {2\pi }} \int _{-\infty }^{-\zeta } \mathrm{Li}_{1/2} \big (z e^{(\zeta ^2-y^2)/2}\big ) \mathrm{d}y \qquad \text{for $\Re (\zeta ) > 0$.} {\tag{2.23}} \end{equation}$$

The integration contour lies in the half-plane $\Re (y)<0$, and is given by the union of the interval $(-\infty , \Re (\pm \zeta )]$ on the real axis and the line segment from $\Re (\pm \zeta )$ to $\pm \zeta$. Since $|z e^{(\zeta ^2-y^2)/2}| <1$ on the integration contour, $\mathrm{Li}_{1/2}( z e^{(\zeta ^2-y^2)/2} )$ is well-defined. Thus, we find that the integrals are well-defined using $\mathrm{Li}_{1/2}(\omega ) \sim \omega$ as $\omega \to 0$.

Observe the symmetry

$$\begin{equation} \mathsf{h}(\zeta , z) =\mathsf{h}(-\zeta ,z) \quad \text{for $\Re (\zeta )< 0$.} {\tag{2.24}} \end{equation}$$

We also have

$$\begin{equation} \mathsf{h}(\zeta ,z)= \int _{-\mathrm{i}\infty }^{\mathrm{i}\infty } \frac{\log (1-ze^{\omega ^2/2})}{\omega -\zeta }\frac{\mathrm{d}\omega }{2\pi \mathrm{i}} \quad \text{for $\Re (\zeta )< 0$.} {\tag{2.25}} \end{equation}$$

This identity can be obtained by the power series expansion and using the fact that $\frac{1}{\sqrt {2\pi }} \int _{-\infty }^u e^{-\omega ^2/2} \mathrm{d}\omega = \int _{-\mathrm{i}\infty }^{\mathrm{i}\infty } \frac{e^{(-u^2+\omega ^2)/2}}{\omega -u} \frac{\mathrm{d}\omega }{2\pi \mathrm{i}}$ for $u$ with $\arg (u)\in (3\pi /4, 5\pi /4)$; see (4.8) of 4. From 2.24 and 2.25, we find that

$$\begin{equation} \mathsf{h}(\zeta ,z)= O(\zeta ^{-1})\quad \text{as $\zeta \to \infty $ in the region $\left|\arg (\zeta )\pm \frac{\pi }{2}\right|>\epsilon $} {\tag{2.26}} \end{equation}$$

for any fixed $z$ satisfying $|z|<1$.

Let $\mathbf{x}$, $\boldsymbol{\tau }$, and $\boldsymbol{\gamma }$ be the parameters in Definition 2.5. We set

$$\begin{equation} \mathsf{f}_i(\zeta ) :=\begin{dcases} e^{-\frac{1}{3} (\tau _i-\tau _{i-1}) \zeta ^3 + \frac{1}{2} (\gamma _i-\gamma _{i-1}) \zeta ^2 + (x_i-x_{i-1}) \zeta } \quad &\text{for $\Re (\zeta )<0$}, \\ e^{\frac{1}{3} (\tau _i-\tau _{i-1}) \zeta ^3 - \frac{1}{2} (\gamma _i-\gamma _{i-1}) \zeta ^2 - (x_i-x_{i-1}) \zeta } \quad & \text{for $\Re (\zeta )>0$} \end{dcases} {\tag{2.27}} \end{equation}$$

for $i=1, \cdots , m$, where we set $\tau _0=\gamma _0=x_0=0$.

Now we describe the space and the operators. For a non-zero complex number $z$, consider the roots $\zeta$ of the equation $e^{-\zeta ^2/2}=z$. The roots are on the contour $\Re (\zeta ^2)=-2\log |z|$. It is easy to check that if $0<|z|<1$, the contour $\Re (\zeta ^2)=-2\log |z|$ consists of two disjoint components, one in $\Re (\zeta )>0$ and the other in $\Re (\zeta )<0$. See Figure 4. The asymptotes of the contours are the straight lines of slope $\pm 1$. For $0<|z|<1$, we define the discrete sets

$$\begin{equation} \begin{split} \mathsf{L}_{z} &:= \{ \zeta \in \mathbb{C}: e^{-\zeta ^2/2}=z\} \cap \{\Re (\zeta )<0\}, \\ \mathsf{R}_{z} &:= \{ \zeta \in \mathbb{C}: e^{-\zeta ^2/2}=z\} \cap \{\Re (\zeta )>0\}. \end{split} {\tag{2.28}} \end{equation}$$

For distinct complex numbers $z_1, \cdots , z_m$ satisfying $0<|z_i|<1$, define the sets

$$\begin{equation} \mathsf{S_1}:=\mathsf{L}_{z_1}\cup \mathsf{R}_{z_2}\cup \mathsf{L}_{z_3}\cup \cdots \cup \begin{cases} \mathsf{R}_{z_m} & \text{if $m$ is even,} \\ \mathsf{L}_{z_m} & \text{if $m$ is odd,} \end{cases} {\tag{2.29}} \end{equation}$$

and

$$\begin{equation} \mathsf{S_2}:=\mathsf{R}_{z_1}\cup \mathsf{L}_{z_2}\cup \mathsf{R}_{z_3}\cup \cdots \cup \begin{cases} \mathsf{L}_{z_m} & \text{if $m$ is even,} \\ \mathsf{R}_{z_m}& \text{if $m$ is odd.} \end{cases} {\tag{2.30}} \end{equation}$$

See Figure 5. Now we define two operators

$$\begin{equation} \mathsf{K_1}:\ell ^2(\mathsf{S_2})\to \ell ^2(\mathsf{S_1}),\qquad \mathsf{K_2}:\ell ^2(\mathsf{S_1})\to \ell ^2(\mathsf{S_2}) \tag{2.31} \end{equation}$$

by kernels as follows. If

$$\begin{equation} \text{$\zeta \in (\mathsf{L}_{z_i}\cup \mathsf{R}_{z_i}) \cap \mathsf{S_1}$ and $\zeta '\in (\mathsf{L}_{z_j}\cup \mathsf{R}_{z_j})\cap \mathsf{S_2}$} \tag{2.32} \end{equation}$$ for some $i,j\in \{1, \cdots , m\}$, then we set

$$\begin{equation} \begin{split} \mathsf{K_1}(\zeta ,\zeta ')&=(\delta _i(j)+\delta _i(j+(-1)^i)) \frac{ \mathsf{f}_i(\zeta ) e^{2\mathsf{h}(\zeta , z_i)-\mathsf{h}(\zeta , z_{i-(-1)^{i}}) -\mathsf{h}(\zeta ', z_{j-(-1)^{j}})}}{\zeta (\zeta -\zeta ')}\mathsf{Q_1}(j) . \end{split} {\tag{2.33}} \end{equation}$$

Similarly, if

$$\begin{equation} \text{$\zeta \in (\mathsf{L}_{z_i}\cup \mathsf{R}_{z_i})\cap \mathsf{S_2}$ and $\zeta '\in (\mathsf{L}_{z_j}\cup \mathsf{R}_{z_j})\cap \mathsf{S_1}$} \tag{2.34} \end{equation}$$

for some $i,j\in \{1, \cdots , m\}$, then we set

$$\begin{equation} \begin{split} \mathsf{K_2}(\zeta ,\zeta ')&=(\delta _i(j)+\delta _i(j-(-1)^i)) \frac{\mathsf{f}_i(\zeta ) e^{2\mathsf{h}(\zeta , z_i)-\mathsf{h}(\zeta , z_{i+(-1)^{i}}) -\mathsf{h}(\zeta ', z_{j+(-1)^{j}})}}{\zeta (\zeta -\zeta ')} \mathsf{Q_2}(j). \end{split} {\tag{2.35}} \end{equation}$$

Here the delta function $\delta _i(k)=1$ if $k=i$ or $0$ otherwise. We also set $z_0=z_{m+1}=0$ so that

$$\begin{equation} e^{-\mathsf{h}(\zeta , z_{0})}=e^{-\mathsf{h}(\zeta , z_{m+1})}= 1, \tag{2.36} \end{equation}$$

and the functions $\mathsf{Q_1}(j)$ and $\mathsf{Q_2}(j)$ are defined by

$$\begin{equation} \text{$\mathsf{Q_1}(j)= 1- \frac{z_{j-(-1)^j}}{z_j}$ and $\mathsf{Q_2}(j) = 1-\frac{z_{j+(-1)^j}}{z_j}$.} {\tag{2.37}} \end{equation}$$

Definition 2.8.

Define

$$\begin{equation} \mathsf{D}(\mathbf{z}):=\det \left(1-\mathsf{K_1}\mathsf{K_2}\right) {\tag{2.38}} \end{equation}$$

for $\mathbf{z}=(z_1, \cdots , z_m)$, where $0<|z_i|<1$ and $z_i$ are distinct.

In this definition, we temporarily assumed that $z_i$ are distinct in order to ensure that the term $\zeta -\zeta '$ in the denominators in 2.33 and 2.35 does not vanish. However, as we stated in (P2) in Subsubsection 2.2.1, $\mathsf{D}(\mathbf{z})$ is still well-defined when the $z_i$ are equal. See Lemma 7.1.

The definition of $\mathsf{L}_{z}$ and $\mathsf{R}_{z}$ implies that $|\arg (\zeta )|\to 3\pi /4$ as $|\zeta |\to \infty$ along $\zeta \in \mathsf{L}_z$ and $|\arg (\zeta )|\to \pi /4$ as $|\zeta |\to \infty$ along $\zeta \in \mathsf{R}_z$. Hence, due to the cubic term $\zeta ^3$ in 2.27, $\mathsf{f}_i(\zeta )\to 0$ super exponentially as $|\zeta |\to \infty$ on the set $\mathsf{L}_{z}\cup \mathsf{R}_{z}$ if $\tau _1<\cdots <\tau _m$. Hence, using the property 2.26 of $\mathsf{h}$, we see that the kernels decay super-exponentially fast as $|\zeta |,|\zeta '|\to \infty$ on the spaces. Therefore, the Fredholm determinant is well-defined if $\tau _1<\cdots <\tau _m$.

We now check property (P4). If $\tau _i=\tau _{i+1}$, the exponent of $\mathsf{f}_i$ has no cubic term $\zeta ^3$. The quadratic term contributes to $O(1)$ since $|e^{-\zeta ^2/2}|=|z_i|$ for $\zeta \in \mathsf{L}_{z_i}\cup \mathsf{R}_{z_i}$, and hence $|e^{c\zeta ^2}|=O(1)$. On the other hand, the linear term in the exponent of $\mathsf{f}_i$ has a negative real part if $x_i< x_{i+1}$. Hence, if $\tau _i=\tau _{i+1}$ and $x_i< x_{i+1}$, then $|\mathsf{f}_i(\zeta )|\to 0$ exponentially as $\zeta \to \infty$ along $\zeta \in \mathsf{S_1}\cup \mathsf{S_2}$ and hence the kernel decays exponentially fast as $|\zeta |,|\zeta '|\to \infty$ on the spaces. Therefore, the Fredholm determinant is still well-defined if $\tau _i=\tau _{i+1}$ and $x_i< x_{i+1}$. This proves (P4).

2.3. Matrix kernel formula of $\mathsf{K_1}$ and $\mathsf{K_2}$

Due to the delta functions, $\mathsf{K_1}(\zeta , \zeta ') \neq 0$ only when

$$\begin{equation} \text{$\zeta \in \mathsf{L}_{z_{2\ell -1}}\cup \mathsf{R}_{z_{2\ell }}$ and $\zeta '\in \mathsf{R}_{z_{2\ell -1}}\cup \mathsf{L}_{z_{2\ell }}$} \tag{2.39} \end{equation}$$

for some integer $\ell$, and similarly $\mathsf{K_2}(\zeta , \zeta ')\neq 0$ only when

$$\begin{equation} \text{$\zeta \in \mathsf{L}_{z_{2\ell }}\cup \mathsf{R}_{z_{2\ell +1}}$ and $\zeta '\in \mathsf{R}_{z_{2\ell }}\cup \mathsf{L}_{z_{2\ell +1}}$} \tag{2.40} \end{equation}$$

for some integer $\ell$. Thus, if we represent the kernels as $m\times m$ matrix kernels, then they have $2\times 2$ block structures.

For example, consider the case when $m=5$. Let us use $\xi _i$ and $\eta _i$ to represent variables in $\mathsf{L}_{z_i}$ and $\mathsf{R}_{z_i}$, respectively:

$$\begin{equation} \xi _i\in \mathsf{L}_{z_i}, \qquad \eta _i\in \mathsf{R}_{z_i}. \tag{2.41} \end{equation}$$

The matrix kernels are given by

$$\begin{equation} \mathsf{K_1}= \begin{bmatrix} \mathsf{k}(\xi _1, \eta _1) & \mathsf{k}(\xi _1, \xi _2) \\ \mathsf{k}(\eta _2, \eta _1) & \mathsf{k}(\eta _2, \xi _2) \\ && \mathsf{k}(\xi _3, \eta _3) & \mathsf{k}(\xi _3, \xi _4) \\ && \mathsf{k}(\eta _4, \eta _3) & \mathsf{k}(\eta _4, \xi _4) \\ &&&& \mathsf{k}(\xi _5, \eta _5) \end{bmatrix} \tag{2.42} \end{equation}$$

and

$$\begin{equation} \mathsf{K_2}= \begin{bmatrix} \mathsf{k}(\eta _1, \xi _1) \\ & \mathsf{k}(\xi _2, \eta _2) & \mathsf{k}(\xi _2, \xi _3) \\ & \mathsf{k}(\eta _3, \eta _2) & \mathsf{k}(\eta _3, \xi _3) \\ &&& \mathsf{k}(\xi _4, \eta _4) & \mathsf{k}(\xi _4, \xi _5) \\ &&& \mathsf{k}(\eta _5, \eta _4) & \mathsf{k}(\eta _5, \xi _5) \end{bmatrix}, \tag{2.43} \end{equation}$$

where the empty entries are zeros and the function $\mathsf{k}$ is given below. When $m$ is odd, the structure is similar. On the other hand, when $m$ is even, $\mathsf{K_1}$ consists only of $2\times 2$ blocks and $\mathsf{K_2}$ contains an additional non-zero $1\times 1$ block at the bottom right corner.

We now define $\mathsf{k}$. For $1\le i\le m-1$, writing

$$\begin{equation} \xi =\xi _i, \quad \eta =\eta _i, \quad \xi '=\xi _{i+1}, \quad \eta '=\eta _{i+1}, \tag{2.44} \end{equation}$$

we define

$$\begin{equation} \begin{split} \begin{bmatrix} \mathsf{k}(\xi , \eta ) & \mathsf{k}(\xi , \xi ') \\ \mathsf{k}(\eta ', \eta ) & \mathsf{k}(\eta ', \xi ') \end{bmatrix} = &\begin{bmatrix} \mathsf{f}_i(\xi ) \\ & \mathsf{f}_{i+1}(\eta ') \end{bmatrix} \begin{bmatrix} \frac{e^{2\mathsf{h}(\xi ,z_i)}}{\xi e^{\mathsf{h}(\xi , z_{i+1})}} \\ & \frac{e^{2\mathsf{h}(\eta ',z_{i+1})}}{\eta ' e^{\mathsf{h}(\eta ', z_i)} } \end{bmatrix} \begin{bmatrix} \frac{1}{\xi -\eta } & \frac{1}{\xi -\xi '} \\ \frac{1}{\eta '-\eta } & \frac{1}{\eta '-\xi '} \end{bmatrix} \\ &\times \begin{bmatrix} \frac{1}{e^{\mathsf{h}(\eta , z_{i+1})}} \\ & \frac{1}{e^{\mathsf{h}(\xi ', z_i)}} \end{bmatrix} \begin{bmatrix} 1-\frac{z_{i+1}}{z_i} \\ & 1-\frac{z_{i}}{z_{i+1}} \end{bmatrix}. \end{split} {\tag{2.45}} \end{equation}$$

This means that

$$\begin{equation} \mathsf{k}(\xi _i, \eta _i)= \mathsf{f}_i(\xi _i) \frac{e^{2\mathsf{h}(\xi _i,z_i)}}{\xi _i e^{\mathsf{h}(\xi _i, z_{i+1})}} \frac{1}{\xi _i-\eta _i} \frac{1}{e^{\mathsf{h}(\eta _i, z_{i+1})}} \left( 1-\frac{z_{i+1}}{z_i} \right), \tag{2.46} \end{equation}$$

$$\begin{equation} \mathsf{k}(\xi _i, \xi _{i+1})= \mathsf{f}_i(\xi _i) \frac{e^{2\mathsf{h}(\xi _i,z_i)}}{\xi _i e^{\mathsf{h}(\xi _i, z_{i+1})}} \frac{1}{\xi _i-\xi _{i+1}} \frac{1}{e^{\mathsf{h}(\xi _{i+1}, z_i)}} \left( 1-\frac{z_{i}}{z_{i+1}} \right), \tag{2.47} \end{equation}$$ and so on. The term $\mathsf{k}(\xi _m, \eta _m)$ is defined by the $(1,1)$ entry of 2.45 with $i=m$, where we set $z_{m+1}=0$. The term $\mathsf{k}(\eta _1, \xi _1)$ is defined by the $(2,2)$ entry of 2.45 with $i=0$, where we set $z_{0}=0$.

2.4. Series formulas for $\mathsf{D}(\mathbf{z})$

We present two series formulas for the function $\mathsf{D}(\mathbf{z})$. The first one 2.52 is the series expansion of Fredholm determinant using the block structure of the matrix kernel. The second formula 2.53 is obtained after evaluating the finite determinants in 2.52 explicitly.

To simplify formulas, we introduce the following notation.

Definition 2.9 (Notational conventions).

For complex vectors $\mathsf{W}=(\mathsf{w}_1, \cdots , \mathsf{w}_n)$ and $\mathsf{W}'=(\mathsf{w}'_1,\cdots ,\mathsf{w}'_{n'})$, we set

$$\begin{equation} \Delta (\mathsf{W})=\prod _{i<j}(\mathsf{w}_j-\mathsf{w}_i) = \det \left[ \mathsf{w}_i^{j-1}\right], \qquad \Delta (\mathsf{W};\mathsf{W}')=\prod _{\substack{1\le i\le n\\ 1\le i'\le n'}}(\mathsf{w}_i-\mathsf{w}'_{i'}). {\tag{2.48}} \end{equation}$$

For a function $h$ of single variable, we write

$$\begin{equation} h(\mathsf{W}) = \prod _{i=1}^n h(\mathsf{w}_i) . {\tag{2.49}} \end{equation}$$

We also use the notation

$$\begin{equation} \Delta (S;S')=\prod _{\substack{s\in S\\ s'\in S'}}(s-s'), \qquad f(S)= \prod _{s\in S} f(s) \tag{2.50} \end{equation}$$

for finite sets $S$ and $S'$.

The next lemma follows from a general result whose proof is given in Subsection 4.3 below.

Lemma 2.10 (Series formulas for $\mathsf{D}(\mathbf{z})$).

We have

$$\begin{equation} \mathsf{D}(\mathbf{z})= \sum _{\mathbf{n}\in (\mathbb{Z}_{\ge 0})^m} \frac{1}{(\mathbf{n}!)^2} \mathsf{D}_\mathbf{n}(\mathbf{z}) {\tag{2.51}} \end{equation}$$

with $\mathbf{n}!= \prod _{\ell =1}^m n_\ell !$ for $\mathbf{n}=(n_1, \cdots , n_m)$, where $\mathsf{D}_{\mathbf{n}}(\mathbf{z})$ can be expressed in the following two ways.

(i)

We have, for $\mathbf{n}=(n_1, \cdots , n_m)$,$$\begin{equation} \mathsf{D}_{\mathbf{n}}(\mathbf{z}) =(-1)^{|\mathbf{n}|}\sum _{\substack{\mathsf{U}^{(\ell )}\in (\mathsf{L}_{z_\ell })^{n_\ell } \\ \mathsf{V}^{(\ell )}\in (\mathsf{R}_{z_\ell })^{n_\ell } \\ \ell =1,\cdots ,m}} \det \left[\mathsf{K_1}(\zeta _i, \zeta '_j)\right]_{i,j=1}^{|\mathbf{n}|} \det \left[\mathsf{K_2}(\zeta '_i, \zeta _j)\right]_{i,j=1}^{|\mathbf{n}|}, {\tag{2.52}} \end{equation}$$

where $\mathbf{\mathsf{U}}\!=\!(\mathsf{U}^{(1)}, \cdots , \mathsf{U}^{(m)})$, $\mathbf{\mathsf{V}}\!=\! (\mathsf{V}^{(1)}, \cdots , \mathsf{V}^{(m)})$ with $\mathsf{U}^{(\ell )}\!=\!(\mathsf{u}_1^{(\ell )}, \cdots , \mathsf{u}_{n_\ell }^{(\ell )})$, $\mathsf{V}^{(\ell )}=(\mathsf{v}_1^{(\ell )}, \cdots , \mathsf{v}_{n_\ell }^{(\ell )})$, and where$$\begin{equation} \zeta _i=\begin{dcases} \mathsf{u}_k^{(\ell )}& \text{if $i=n_1 +\cdots +n_{\ell -1}+k$ for some $k\le n_\ell $ with odd integer $\ell $,}\\ \mathsf{v}_k^{(\ell )}& \text{if $i=n_1 +\cdots +n_{\ell -1}+k$ for some $k\le n_\ell $ with even integer $\ell $,} \end{dcases} {\tag{2.53}} \end{equation}$$

and$$\begin{equation} \zeta '_i=\begin{dcases} \mathsf{v}_k^{(\ell )}& \text{if $i=n_1+\cdots +n_{\ell -1}+k$ for some $k\le n_\ell $ with odd integer $\ell $,}\\ \mathsf{u}_k^{(\ell )}& \text{if $i=n_1+\cdots +n_{\ell -1}+k$ for some $k\le n_\ell $ with even integer $\ell $.} \end{dcases} \tag{2.54} \end{equation}$$

Here, we set $n_0=0$.

(ii)

We also have$$\begin{equation} \mathsf{D}_\mathbf{n}(\mathbf{z}) =\sum _{\substack{\mathsf{U}^{(\ell )}\in \left(\mathsf{L}_{z_\ell }\right)^{n_\ell } \\ \mathsf{V}^{(\ell )}\in \left(\mathsf{R}_{z_\ell }\right)^{n_\ell } \\ \ell =1,\cdots ,m}} \mathsf{d}_{\mathbf{n}, \mathbf{z}} (\mathbf{\mathsf{U}}, \mathbf{\mathsf{V}}) {\tag{2.55}} \end{equation}$$

with$$\begin{equation} \begin{split} \mathsf{d}_{\mathbf{n}, \mathbf{z}} (\mathbf{\mathsf{U}}, \mathbf{\mathsf{V}}) :=& \left[ \prod _{\ell =1}^m \frac{\Delta (\mathsf{U}^{(\ell )})^2\Delta (\mathsf{V}^{(\ell )})^2}{\Delta (\mathsf{U}^{(\ell )};\mathsf{V}^{(\ell )})^2} \hat{\mathsf{f}}_\ell (\mathsf{U}^{(\ell )}) \hat{\mathsf{f}}_\ell (\mathsf{V}^{(\ell )})\right]\\ &\times \left[ \prod _{\ell =2}^m \frac{\Delta (\mathsf{U}^{(\ell )};\mathsf{V}^{(\ell -1)})\Delta (\mathsf{V}^{(\ell )};\mathsf{U}^{(\ell -1)}) e^{-\mathsf{h}(\mathsf{V}^{(\ell )}, z_{\ell -1}) - \mathsf{h}(\mathsf{V}^{(\ell -1)}, z_{\ell })} }{\Delta (\mathsf{U}^{(\ell )};\mathsf{U}^{(\ell -1)})\Delta (\mathsf{V}^{(\ell )};\mathsf{V}^{(\ell -1)}) e^{\mathsf{h}(\mathsf{U}^{(\ell )}, z_{\ell -1}) + \mathsf{h}(\mathsf{U}^{(\ell -1)}, z_{\ell })} } \right. \\ &\qquad \left. \left(1- \frac{z_{\ell -1}}{z_{\ell }} \right)^{n_\ell } \left(1-\frac{z_\ell }{z_{\ell -1}} \right)^{n_{\ell -1}} \right], \end{split} {\tag{2.56}} \end{equation}$$

where$$\begin{equation} \begin{split} \hat{\mathsf{f}}_\ell (\zeta )&:=\frac{1}{\zeta } \mathsf{f}_\ell (\zeta ) e^{2\mathsf{h}(\zeta , z_\ell )}. \end{split} {\tag{2.57}} \end{equation}$$

Recall 2.22, 2.23, and 2.27 for the definition of $\mathsf{h}$ and $\mathsf{f}_j$.

Property (P3) in Subsubsection 2.2.1 follows easily from 2.56. Note that $\gamma _i$ only appears in the factor $\hat{\mathsf{f}}_\ell (\mathsf{U}^{(\ell )})\hat{\mathsf{f}}_\ell (\mathsf{V}^{(\ell )})$ for $\ell =i$ or $i+1$. If we replace $\gamma _i$ by $\gamma _i+1$, then $\mathsf{f}_i(\zeta )$ and $\mathsf{f}_{i+1}(\zeta )$ are changed by $z_i^{-1}\mathsf{f}_i(\zeta )$ and $z_{i+1}\mathsf{f}_{i+1}(\zeta )$ if $\Re (\zeta )<0$, or by $z_i\mathsf{f}_i(\zeta )$ and $z_{i+1}^{-1}\mathsf{f}_{i+1}(\zeta )$ if $\Re (\zeta )>0$. But $\mathsf{U}^{(\ell )}$ has the same number of components as $\mathsf{V}^{(\ell )}$ for each $\ell$. Therefore $\hat{\mathsf{f}}_\ell (\mathsf{U}^{(\ell )})\hat{\mathsf{f}}_\ell (\mathsf{V}^{(\ell )})$ does not change. We can also check (P3) from the original Fredholm determinant formula.

The analyticity property (P2) is proved in Lemma 7.1 later using the series formula.

3. Joint distribution function for the general initial condition

We obtain the limit theorem of the previous section from a finite-time formula of the joint distribution. In this section, we describe a formula of the finite-time joint distribution for an arbitrary initial condition. We simplify the formula further in the next section for the case of the periodic step initial condition.

We state the results in terms of particle locations instead of the height function used in the previous section. It is easy to convert one to another; see 6.5. The particle locations are denoted by $\mathbb{x}_i(t)$, where

$$\begin{equation} \cdots < \mathbb{x}_0(t)<\mathbb{x}_1(t)<\mathbb{x}_2(t)< \cdots . \tag{3.1} \end{equation}$$

Due to the periodicity of the system, we have $\mathbb{x}_i(t)=\mathbb{x}_{i+nN}(t)-nL$ for all integers $n$.

The periodic TASEP can be described if we keep track of $N$ consecutive particles, say $\mathbb{x}_1(t)<\cdots <\mathbb{x}_N(t)$. If we focus only on these particles, they follow the usual TASEP rules plus the extra condition that $\mathbb{x}_N(t)<\mathbb{x}_1(t)+L$ for all $t$. Define the configuration space

$$\begin{equation} \mathcal{X}_N(L) = \{(x_1,x_2,\cdots ,x_N)\in \mathbb{Z}^N: x_1 <x_2 <\cdots <x_N <x_1+L \}. \tag{3.2} \end{equation}$$

We call the process of the $N$ particles TASEP in $\mathcal{X}_N(L)$. We use the same notation $\mathbb{x}_i(t)$, $i=1, \cdots , N$, to denote the particle locations in the TASEP in $\mathcal{X}_N(L)$. We state the result for the TASEP in $\mathcal{X}_N(L)$ first and then for the periodic TASEP as a corollary.

For $z\in \mathbb{C}$, consider the polynomial of degree $L$ given by

$$\begin{equation} q_z(w) = w^N(w+1)^{L-N}-z^L. {\tag{3.3}} \end{equation}$$

Denote the set of the roots by

$$\begin{equation} \mathcal{R}_z = \{w\in \mathbb{C}: q_z(w)=0\}. {\tag{3.4}} \end{equation}$$

The roots are on the level set $|w^N(w+1)^{L-N}|=|z|^L$. It is straightforward to check the following properties of the level set. Set

$$\begin{equation} \mathbb{r}_0:= \rho ^\rho (1-\rho )^{1-\rho }, {\tag{3.5}} \end{equation}$$

where, as before, $\rho =N/L$. The level set becomes larger as $|z|$ increases; see Figure 6. If $0<|z|< \mathbb{r}_0$, the level set consists of two closed contours, one in $\Re (w)<-\rho$ enclosing the point $w=-1$ and the other in $\Re (w)>-\rho$ enclosing the point $w=0$. When $|z|=\mathbb{r}_0$, the level set has a self-intersection at $w=-\rho$. If $|z|>\mathbb{r}_0$, then the level set is a connected closed contour. Now consider the set of roots $\mathcal{R}_z$. Note that if $z\neq 0$, then $-1, 0\notin \mathcal{R}_z$. It is also easy to check that if a non-zero $z$ satisfies $z^L\neq \mathbb{r}_0^L$, then the roots of $q_z(w)$ are all simple. On the other hand, if $z^L=\mathbb{r}_0^L$, then there is a double root at $w=-\rho$ and the remaining $L-2$ roots are simple. For the results in this section, we take $z$ to be any non-zero complex number. But in the next section, we restrict $0<|z|<\mathbb{r}_0$.

Theorem 3.1 (Joint distribution of TASEP in $\mathcal{X}_N(L)$ for the general initial condition).

Consider the TASEP in $\mathcal{X}_N(L)$. Let $Y=(y_1, \cdots , y_N) \in \mathcal{X}_N(L)$ and assume that $(\mathbb{x}_1(0), \cdots , \mathbb{x}_N(0))=Y$. Fix a positive integer $m$. Let $(k_1, t_1), \cdots , (k_m, t_m)$ be $m$ distinct points in $\{1, \cdots , N\} \times [0, \infty )$. Assume that $0\le t_1\le \cdots \le t_m$. Let $a_i\in \mathbb{Z}$ for $1\le i\le m$. Then

$$\begin{equation} \begin{split} &\mathbb{P}_Y\left(\mathbb{x}_{k_1}(t_1)\ge a_1,\cdots , \mathbb{x}_{k_m}(t_m)\ge a_m\right) \\ &\qquad = \oint \cdots \oint \mathcal{C}(\mathbf{z}, \mathbf{k}) \mathcal{D}_Y(\mathbf{z}, \mathbf{k}, \mathbf{a}, \mathbf{t}) \frac{{\mathrm{d}}z_m}{2\pi {\mathrm{i}}z_m}\cdots \frac{{\mathrm{d}}z_1}{2\pi {\mathrm{i}}z_1}, \end{split} {\tag{3.6}} \end{equation}$$

where the contours are nested circles in the complex plane satisfying $0<|z_m|<\cdots <|z_1|$. Here $\mathbf{z}=(z_1, \cdots , z_m)$, $\mathbf{k}=(k_1, \cdots , k_m)$, $\mathbf{a}=(a_1, \cdots , a_m)$, and $\mathbf{t}=(t_1, \cdots , t_m)$. The functions in the integrand are

$$\begin{equation} \begin{split} \mathcal{C}(\mathbf{z}, \mathbf{k}) &= (-1)^{(k_m-1)(N+1)}z_1^{(k_1-1)L} \prod _{\ell =2}^m \Bigg [ z_\ell ^{(k_\ell -k_{\ell -1})L}\bigg (\bigg (\frac{z_\ell }{z_{\ell -1}}\bigg )^L-1 \bigg )^{N-1}\Bigg ] \end{split} {\tag{3.7}} \end{equation}$$

and

$$\begin{equation} \begin{split} \mathcal{D}_Y(\mathbf{z}, \mathbf{k}, \mathbf{a}, \mathbf{t}) = \det \left[ \sum _{\substack{w_1\in \mathcal{R}_{z_1} \\ \cdots \\ w_m\in \mathcal{R}_{z_m}}} \frac{w_1^i \left(w_1+1\right)^{y_i-i}w_m^{-j}}{ \prod _{\ell =2}^m (w_\ell -w_{\ell -1}) } \prod _{\ell =1}^m \mathcal{G}_\ell (w_\ell ) \right]_{i,j=1}^N \end{split} {\tag{3.8}} \end{equation}$$

with

$$\begin{equation} \mathcal{G}_\ell (w) = \frac{w(w+1)}{L(w+\rho )} \frac{w^{-k_\ell }(w+1)^{-a_\ell +k_\ell }e^{t_\ell w}}{w^{-k_{\ell -1}}(w+1)^{-a_{\ell -1}+k_{\ell -1}}e^{t_{\ell -1}w}}, \tag{3.9} \end{equation}$$

where we set $t_0=k_0=a_0=0$.

Remark 3.2.

The limiting joint distribution $\mathbb{F}$ in the previous section was not defined for all parameters: when $\tau _i=\tau _{i+1}$, we need to put the restriction $x_i<x_{i+1}$. See property (P4) in Subsubsection 2.2.1. The finite-time joint distribution does not require such restrictions. The sums in the entries of the determinant $\mathcal{D}_Y(\mathbf{z}, \mathbf{k}, \mathbf{a}, \mathbf{t})$ are over finite sets, and hence there is no issue with the convergence. Therefore, the right-hand side of 3.6 is well-defined for all real numbers $t_i$ and integers $a_i$ and $k_i$.

Corollary 3.3 (Joint distribution of periodic TASEP for the general initial condition).

Consider the periodic TASEP with a general initial condition determined by $Y=(y_1, \cdots , y_N) \in \mathcal{X}_N(L)$ and its periodic translations; $\mathbb{x}_{j+n N}(0)=y_j+n L$ for all $n\in \mathbb{Z}$ and $j=1,\cdots ,N$. Then 3.6 holds for all $k_i\in \mathbb{Z}$ without the restriction that $k_i\in \{1, \cdots , N\}$.

Proof.

The particles in the periodic TASEP satisfy $\mathbb{x}_{j+nN}(t)=\mathbb{x}_{j}(t)+nL$ for every integer $n$. Hence if $k_\ell$ is not between $1$ and $N$, we may translate it. This amounts to changing $k_\ell$ to $k_\ell +nN$ and $a_\ell$ to $a_\ell +nL$ for some integer $n$. Hence it is enough to show that the right-hand side of 3.6 is invariant under these changes. Under these changes, the term $\mathcal{C}(\mathbf{z}, \mathbf{k})$ is multiplied by the factor $z_\ell ^{nNL}z_{\ell +1}^{-nNL}$ if $1\le \ell \le m-1$ and by $z_m^{nNL}$ if $\ell =m$. On the other hand, $\mathcal{G}_\ell (w_\ell )$ produces the multiplicative factor $w_\ell ^{-nN}(w_\ell +1)^{-nL+nN}$ which is $z_\ell ^{-nL}$ by 3.4. Taking this factor outside the determinant 3.8, we cancels out the factor $z_\ell ^{nNL}$ from $\mathcal{C}(\mathbf{z}, \mathbf{k})$. Similarly $\mathcal{G}_{\ell +1}(w_{\ell +1})$ produces a factor which cancel out $z_{\ell +1}^{-nNL}$ if $1\le \ell \le m-1$.

■

Before we prove the theorem, let us comment on the analytic property of the integrand in the formula 3.6. The function $\mathcal{C}(\mathbf{z}, \mathbf{k})$ is clearly analytic in each $z_\ell \neq 0$. Consider the function $\mathcal{D}_Y(\mathbf{z}, \mathbf{k}, \mathbf{a}, \mathbf{t})$. Note that

$$\begin{equation} \frac{\mathrm{d}}{\mathrm{d}w} q_z(w) = \frac{L(w+\rho )}{w(w+1)} w^N (w+1)^{L-N}. \tag{3.10} \end{equation}$$

Hence, if $F(w)$ is an analytic function of $w$ in $\mathbb{C}\setminus \{-1,0\}$ and

$$\begin{equation*} f(w)= F(w)w^N(w+1)^{L-N}, \end{equation*}$$

then

$$\begin{equation} \begin{split} &\sum _{w\in \mathcal{R}_z} F(w) \frac{w(w+1)}{L(w+\rho )} \\ &= \frac{1}{2\pi \mathrm{i}} \oint _{|w|=r} \frac{f(w)}{q_z(w)} \mathrm{d}w- \frac{1}{2\pi \mathrm{i}} \oint _{|w+1|=\epsilon _1} \frac{f(w)}{q_z(w)} \mathrm{d}w - \frac{1}{2\pi \mathrm{i}} \oint _{|w|=\epsilon _2} \frac{f(w)}{q_z(w)} \mathrm{d}w \end{split} {\tag{3.11}} \end{equation}$$

for any $\epsilon _1, \epsilon _2>0$ and $r> \max \{\epsilon _1+1, \epsilon _2\}$ such that all roots of $q_z(w)$ lie in the region $\{w: \epsilon _2<|w|<r, |w+1|>\epsilon _1\}$. Note that $q_z(w)$ is an entire function of $z$ for each $w$. Since we may take $r$ arbitrarily large and $\epsilon _1, \epsilon _2$ arbitrarily small and positive, the right-hand side of 3.11 defines an analytic function of $z\neq 0$. Now the entries of the determinant in 3.8 are of the form

$$\begin{equation} \sum _{\substack{w_1\in \mathcal{R}_{z_1} \\ \cdots \\ w_m\in \mathcal{R}_{z_m}}} F(w_1, \cdots , w_m) \prod _{\ell =1}^m \frac{w_\ell (w_\ell +1)}{L(w_\ell +\rho )} {\tag{3.12}} \end{equation}$$

for a function $F(w_1, \cdots , w_m)$ which is analytic in each variable in $\mathbb{C}\setminus \{-1, 0\}$ as long as $w_\ell \neq w_{\ell -1}$ for all $\ell =2, \cdots , m$. The last condition is due to the factor $\prod _{\ell =2}^m (w_\ell -w_{\ell -1})$ in the denominator. Note that if $w_\ell =w_{\ell -1}$, then $z_\ell = z_{\ell -1}$. Hence by using 3.11 $m$ times, each entry of 3.8, and hence $\mathcal{D}_Y(\mathbf{z}, \mathbf{k}, \mathbf{a}, \mathbf{t})$, is an analytic function of each $z_\ell \neq 0$ in the region where all $z_\ell$ are distinct.

When $m=1$, the product in 3.7 is set to be $1$ and the formula 3.6 in this case was obtained in Proposition 6.1 in 4. For $m\ge 2$, as we mentioned in the Introduction, we prove 3.6 by taking a multiple sum of the transition probability. The main new technical result is a summation formula and we summarize it in Proposition 3.4 below.

The transition probability was obtained in Proposition 5.1 of 4. Denote by $\mathbb{P}_X(X';t)$ the transition probability from $X=(x_1, \cdots , x_N) \in \mathcal{X}_N(L)$ to $X'=(x'_1, \cdots , x'_N)\in \mathcal{X}_N(L)$ in time $t$,

$$\begin{equation} \mathbb{P}_X(X';t) = \oint \det \left[ \frac{1}{L} \sum _{w\in \mathcal{R}_z} \frac{w^{j-i+1}(w+1)^{-x'_i+x_j+i-j}e^{tw}}{w+\rho } \right]_{i,j=1}^N \frac{{\mathrm{d}}z}{2\pi {\mathrm{i}}z}, {\tag{3.13}} \end{equation}$$

where the integral is over any simple closed contour in $|z|>0$ which contains $0$ inside. The integrand is an analytic function of $z$ for $z\neq 0$ by using 3.11.

Proof of Theorem 3.1.

It is enough to consider $m\ge 2$. It is also sufficient to consider the case when the times are distinct, $t_1<\cdots <t_m$, because both sides of 3.6 are continuous functions of $t_1, \cdots , t_m$. Note that 3.8 involves only finite sums.

Denoting by $X^{(\ell )}=(x_1^{(\ell )}, \cdots , x_N^{(\ell )})$ the configuration of the particles at time $t_\ell$, the joint distribution function on the left-hand side of 3.6 is equal to

$$\begin{equation} \sum _{\substack{X^{(\ell )}\in \mathcal{X}_N(L) \cap \{x_{k_\ell }^{(\ell )}\ge a_\ell \} \\ \ell =1,\cdots , m}} \mathbb{P}_Y(X^{(1)};t_1)\mathbb{P}_{X^{(1)}}(X^{(2)},t_2-t_1)\cdots \mathbb{P}_{X^{(m-1)}}(X^{(m)};t_m-t_{m-1}). {\tag{3.14}} \end{equation}$$

Applying the Cauchy-Binet formula to 3.13, we have

$$\begin{equation} \mathbb{P}_X(X';t)=\oint \sum _{\substack{W\in (\mathcal{R}_{z})^N}} \mathcal{L}_X(W) \mathcal{R}_{X'}(W) \mathcal{Q}(W;t) \frac{{\mathrm{d}}z}{2\pi {\mathrm{i}}z}, {\tag{3.15}} \end{equation}$$

where, for $W=(w_1,\cdots ,w_N)\in \mathbb{C}$,

$$\begin{equation} \mathcal{L}_X(W)=\det \left[w_i^{j}(w_i+1)^{x_j-j}\right]_{i,j=1}^N, \, \, \mathcal{R}_{X'}(W)=\det \left[w_i^{-j}(w_i+1)^{-x'_j+j}\right]_{i,j=1}^N, \tag{3.16} \end{equation}$$

and

$$\begin{equation} \mathcal{Q}(W;t)=\frac{1}{N!L^N}\prod _{i=1}^N\frac{w_ie^{tw_i}}{w_i+\rho }. {\tag{3.17}} \end{equation}$$

Here the factor $N!$ in the denominator comes from the Cauchy-Binet formula; it will eventually disappear since we will apply the Cauchy-Binet identity backward again at the end of the proof.

We insert 3.15 into 3.14 and interchange the order of the sums and the integrals. Assuming that the series converges absolutely so that the interchange is possible, the joint distribution is equal to

$$\begin{equation} \begin{split} &\oint \frac{{\mathrm{d}}z_1}{2\pi {\mathrm{i}}z_1}\cdots \oint \frac{{\mathrm{d}}z_m}{2\pi {\mathrm{i}}z_m} \sum _{\substack{W^{(1)}\in (\mathcal{R}_{z_1})^N \\ \cdots \\ W^{(m)}\in (\mathcal{R}_{z_m})^N}} \mathcal{P}(W^{(1)}, \cdots , W^{(m}) \prod _{\ell =1}^m\mathcal{Q}(W^{(\ell )};t_\ell -t_{\ell -1}), \end{split} {\tag{3.18}} \end{equation}$$

where $W^{(\ell )}=(w_1^{(\ell )}, \cdots , w_N^{(\ell )})$ and

$$\begin{equation} \begin{split} &\mathcal{P}(W^{(1)}, \cdots , W^{(m}) \\ & \quad = \mathcal{L}_Y (W^{(1)}) \left[ \prod _{\ell =1}^{m-1} \mathcal{H}_{k_\ell ,a_\ell } (W^{(\ell )};W^{(\ell +1)}) \right] \bigg [ \sum _{\substack{X \in \mathcal{X}_N(L)\\ x_{k_m}\ge a_m}} \mathcal{R}_{X}(W^{(m)}) \bigg ] . \end{split} {\tag{3.19}} \end{equation}$$

Here we set

$$\begin{equation} \mathcal{H}_{k,a}(W;W') :=\sum _{\substack{X\in \mathcal{X}_N(L)\cap \{x_{k}\ge a\} }} \mathcal{R}_{X}(W) \mathcal{L}_{X}(W') {\tag{3.20}} \end{equation}$$

for a pair of complex vectors $W=(w_1,\cdots ,w_N)$ and $W'=(w'_{1},\cdots ,w'_N)$. Let us now show that it is possible to exchange the sums and integrals if we take the $z_i$-contours properly. We first consider the convergence of 3.20 and the sum in 3.19. Note that, shifting the summation variable $X$ to $X-(b, \cdots , b)$,

$$\begin{equation} \begin{split} &\sum _{\substack{X\in \mathcal{X}_N(L)\cap \{x_{k}= b\} }} \mathcal{R}_{X}(W) \mathcal{L}_{X}(W') \\ &\qquad =\left[\sum _{\substack{Y\in \mathcal{X}_N(X)}\cap \{y_k=0\}} \mathcal{R}_{Y}(W) \mathcal{L}_{Y}(W') \right] \left(\prod _{j=1}^N\frac{w'_j+1}{w_j+1}\right)^{b}. \end{split} \tag{3.21} \end{equation}$$

The right-hand side of 3.20 is the sum of the above formula over $b\ge a$. Hence 3.20 converges absolutely and the convergence is uniform for $w_i, w_i'$ if $\prod _{j=1}^N \big | \frac{w'_j+1}{w_j+1} \big |$ is in a compact subset of $[0,1)$. Similarly, the sum of $\mathcal{R}_{X}(W^{(m)})$ in 3.19 converges if $\prod _{j=1}^N |w_j^{(m)}+1|>1$. Therefore, 3.19 converges absolutely if the intermediate variables $W^{(\ell )}=(w_1^{(\ell )}, \cdots , w_N^{(\ell )})$ satisfy

$$\begin{equation} \prod _{j=1}^N|w_j^{(1)}+1| > \prod _{j=1}^N|w_j^{(2)}+1| >\cdots > \prod _{j=1}^N|w_j^{(m)}+1| > 1. {\tag{3.22}} \end{equation}$$

We now show that it is possible to choose the contours of $z_i$ so that 3.22 is achieved. Since $W^{(\ell )}\in (R_{z_\ell })^N$, $w_j^{(\ell )}$ satisfies the equation $|w^N(w+1)^{L-N}|=|z_\ell ^L|$. Hence $|w_j^{(\ell )}|=|z_j|+O(1)$ as $|z_j|\to \infty$. Therefore, if we take the contours $|z_\ell |=r_\ell$ where $r_1>\cdots >r_m>0$ and $r_\ell -r_{\ell +1}$ are large enough (where $r_{m+1}:=0$), then 3.22 is satisfied. Thus, 3.20 and the sum in 3.19 converge absolutely. It is easy to see that the convergences are uniform. Hence we can exchange the sums and integrals, and therefore, the joint distribution is indeed given by 3.18 if we take the contours of $z_i$ to be large nested circles.

We simplify 3.18. The terms $\mathcal{H}_{k_\ell ,a_\ell } (W^{(\ell )};W^{(\ell +1)})$ are evaluated in Proposition 3.4 below. Note that since the $z_i$-contours are the large nested circles, we have 3.22, and hence the assumptions in Proposition 3.4 are satisfied. On the other hand, the sum of $\mathcal{R}_{X}(W^{(m)})$ in 3.18 was computed in 4. Lemma 6.1 in 4 implies that for $W=(w_1,\cdots ,w_N)\in \mathcal{R}_z^N$,

$$\begin{equation} \begin{split} &\sum _{\substack{X \in \mathcal{X}_N(L)\\ x_{k}= a}} \mathcal{R}_{X}(W) =(-1)^{(k-1)(N+1)} z^{(k-1)L} \\ & \qquad \times \left[ 1- \prod _{j=1}^N (w_{j}+1)^{-1} \right] \left[ \prod _{j=1}^N w_j ^{-k} (w_{j}+1)^{-a+k+1} \right] \det \left[ w_j^{-i}\right]_{i,j=1}^N. \end{split} {\tag{3.23}} \end{equation}$$

Hence, from the geometric series, for $W=(w_1,\cdots ,w_N)\in \mathcal{R}_z^N$,

$$\begin{equation} \begin{split} &\sum _{\substack{X \in \mathcal{X}_N(L)\\ x_{k}\ge a}}\mathcal{R}_{X}(W) \\ &\qquad = (-1)^{(k-1)(N+1)} z^{(k-1)L} \left[ \prod _{j=1}^N w_j ^{-k} (w_{j}+1)^{-a+k+1} \right] \det \left[ w_j^{-i}\right]_{i,j=1}^N \end{split} {\tag{3.24}} \end{equation}$$

if $\prod _{j=1}^N|w_j+1|>1$. The last condition is satisfied for $W=W^{(m)}$. We thus find that 3.19 is equal to an explicit factor times a product of $m-1$ Cauchy determinants times a Vandermonde determinant. By using the Cauchy-Binet identity $m$ times, we obtain 3.6 assuming that the $z_i$-contours are large nested circles.

Finally, using the analyticity of the integrand on the right-hand side of 3.6, which was discussed before the start of this proof, we can deform the contours of $z_i$ to any nested circles, not necessarily large circles. This completes the proof.

■

The main technical part of this section is the following summation formula. We prove it in Section 5.

Proposition 3.4.

Let $z$ and $z'$ be two non-zero complex numbers satisfying $z^L\ne (z')^L$. Let $W=(w_1,\cdots ,w_N)\in (\mathcal{R}_z)^N$ and $W'=(w'_1,\cdots ,w'_N)\in (\mathcal{R}_{z'})^N$. Suppose that $\prod _{j=1}^N|w'_j+1|<\prod _{j=1}^N|w_j+1|$. Consider $\mathcal{H}_{k,a}(W;W')$ defined in 3.20. Then for any $1\le k\le N$ and integer $a$,

$$\begin{equation} \begin{split} &\mathcal{H}_{k,a}(W;W')\\ &= \left(\frac{z}{z'}\right)^{(k-1)L}\left(1-\left(\frac{z'}{z}\right)^L\right)^{N-1} \left[ \prod _{j=1}^N \frac{w_j^{-k}(w_j+1)^{-a+k+1}}{(w'_j)^{-k}(w'_j+1)^{-a+k}} \right] \det \left[\frac{1}{w_i-w'_{i'}}\right]_{i,i'=1}^N. \end{split} {\tag{3.25}} \end{equation}$$

4. Periodic step initial condition

We now assume the following periodic step condition:

$$\begin{equation} \text{$\mathbb{x}_{i+nN}(0)=i-N+nL$ for $1\le i\le N$ and $n\in \mathbb{Z}$.} {\tag{4.1}} \end{equation}$$

In the previous section, we obtained a formula for general initial conditions. In this section, we find a simpler formula for the periodic step initial condition which is suitable for the asymptotic analysis. We express $\mathcal{D}_Y(\mathbf{z}, \mathbf{k}, \mathbf{a}, \mathbf{t})$ as a Fredholm determinant times a simple factor. The result is described in terms of two functions $C(\mathbf{z})$ and $D(\mathbf{z})$. We first define them and then state the result.

Throughout this section, we fix a positive integer $m$, and fix parameters $k_1,\cdots , k_m$, $a_1,\cdots ,a_m$, $t_1,\cdots ,t_m$ as in the previous section.

4.1. Definitions

Recall the function $q_z(w) = w^N(w+1)^{L-N}-z^L$ for complex $z$ in 3.3 and the set of its roots

$$\begin{equation} \mathcal{R}_z = \{w\in \mathbb{C}: q_z(w)=0\} \tag{4.2} \end{equation}$$

in 3.4. Set

$$\begin{equation} \mathbb{r}_0:= \rho ^\rho (1-\rho )^{1-\rho }, \qquad \rho =N/L, \tag{4.3} \end{equation}$$

as in 3.5. We discussed in the previous section that if $0<|z|<\mathbb{r}_0$, then the contour $|q_z(w)|=0$ consists of two closed contours, one in $\Re (w)<-\rho$ enclosing the point $w=-1$ and the other in $\Re (w)>-\rho$ enclosing the point $w=0$. Now, for $0<|z|<\mathbb{r}_0$, set

$$\begin{equation} \mathrm{L}_{z}=\{w\in \mathcal{R}_{z}:\Re (w)<-\rho \},\qquad \mathrm{R}_{z}=\{w\in \mathcal{R}_{z}:\Re (w)>-\rho \}. {\tag{4.4}} \end{equation}$$

It is not difficult to check that

$$\begin{equation} |\mathrm{L}_z|=L-N, \qquad |\mathrm{R}_z|=N. \tag{4.5} \end{equation}$$

See the left picture in Figure 6 in Section 3. (Note that if $z=0$, then the roots are $w=-1$ with multiplicity $L-N$ and $w=0$ with multiplicity $N$.) From the definitions, we have

$$\begin{equation} \mathcal{R}_z= \mathrm{L}_z\cup \mathrm{R}_z. \tag{4.6} \end{equation}$$

In Theorem 3.1, we took the contours of $z_i$ as nested circles of arbitrary sizes. In this section, we assume that the circles satisfy

$$\begin{equation} 0<|z_m|<\cdots <|z_1|<\mathbb{r}_0 . {\tag{4.7}} \end{equation}$$

Hence $\mathrm{L}_{z_i}$ and $\mathrm{R}_{z_i}$ are all well-defined.

We define two functions $C(\mathbf{z})$ and $D(\mathbf{z})$ of $\mathbf{z}=(z_1, \cdots , z_m)$, both of which depend on the parameters $k_i, t_i, a_i$. The first one is the following. Recall the notational convention introduced in Definition 2.9. For example, $\Delta (\mathrm{R}_{z};\mathrm{L}_{z})=\prod _{v\in \mathrm{R}_z} \prod _{u\in \mathrm{L}_z} (v-u)$.

Definition 4.1.

Define

$$\begin{equation} \begin{split} C(\mathbf{z}) &=\left[\prod _{\ell =1}^m\frac{\mathrm{E}_\ell (z_\ell )}{\mathrm{E}_{\ell -1}(z_{\ell })}\right]\left[\prod _{\ell =1}^m\frac{\prod _{u\in \mathrm{L}_{z_\ell }}(-u)^{N}\prod _{v\in \mathrm{R}_{z_\ell }}(v+1)^{L-N}}{\Delta (\mathrm{R}_{z_\ell };\mathrm{L}_{z_\ell })}\right]\\ &\qquad \times \left[\prod _{\ell =2}^m\frac{z_{\ell -1}^L}{z_{\ell -1}^L-z_\ell ^L}\right]\left[\prod _{\ell =2}^m\frac{\Delta (\mathrm{R}_{z_\ell };\mathrm{L}_{z_{\ell -1}})}{\prod _{u\in \mathrm{L}_{z_{\ell -1}}}(-u)^N\prod _{v\in \mathrm{R}_{z_\ell }}(v+1)^{L-N}}\right], \end{split} {\tag{4.8}} \end{equation}$$

where

$$\begin{equation} \mathrm{E}_i(z):=\prod _{u\in \mathrm{L}_{z}}(-u)^{k_i-N-1}\prod _{v\in \mathrm{R}_{z}}(v+1)^{-a_i+k_i-N}e^{t_iv} {\tag{4.9}} \end{equation}$$

for $i=1,\cdots ,m$, and $\mathrm{E}_0(z):=1$.

It is easy to see that all terms in $C(\mathbf{z})$ other than $\prod _{\ell =2}^m\frac{z_{\ell -1}^L}{z_{\ell -1}^L-z_\ell ^L}$ are analytic for $z_1,\cdots ,z_m$ within the disk $\{z;|z|<\mathbb{r}_0\}$. Hence $C(\mathbf{z})$ is analytic in the disk except for the simple poles when $z_{\ell -1}^L=z_\ell ^{L}$, $\ell =2, \cdots ,m$.

We now define $D(\mathbf{z})$. It is given by a Fredholm determinant. Set

$$\begin{equation} \begin{split} F_{i}(w) &:= w^{-k_i+N+1} (w+1)^{-a_i+k_i-N} e^{t_iw}\quad \text{for $i=1,\cdots ,m$,}\\ F_{0}(w) &:=1. \end{split} {\tag{4.10}} \end{equation}$$

Define

$$\begin{equation} f_i(w)=\begin{dcases} \frac{F_i(w)}{F_{i-1}(w)} \quad &\text{for $\Re (w)<-\rho $,} \\ \frac{F_{i-1}(w)}{F_i(w)} \quad & \text{for $\Re (w)>-\rho $.} \end{dcases} {\tag{4.11}} \end{equation}$$

Also, set

$$\begin{equation} \mathrm{J}(w)= \frac{w(w+1)}{L(w+\rho )}. {\tag{4.12}} \end{equation}$$

Define, for $0<|z|<\mathbb{r}_0$,

$$\begin{equation} \mathrm{l}_z(w) = \frac{1}{(w+1)^{L-N}}\prod _{u\in \mathrm{L}_{z}}(w-u), \qquad \mathrm{r}_z(w) = \frac{1}{w^N} \prod _{u\in \mathrm{R}_{z}}(w-u). {\tag{4.13}} \end{equation}$$

Note that $\mathrm{l}_z(w) \mathrm{r}_z(w) = \frac{q_z(w) }{(w+1)^{L-N}w^N} = \frac{w^N(w+1)^{L-N}-z^L}{w^N(w+1)^{L-N}}$. Set

$$\begin{equation} H_z(w) := \begin{dcases} \mathrm{l}_z(w) \qquad &\text{for $\Re (w)>-\rho $,} \\ \mathrm{r}_z(w) \qquad &\text{for $\Re (w)<-\rho $.} \end{dcases} {\tag{4.14}} \end{equation}$$

When $z=0$, we define $\mathrm{l}_z(w)=\mathrm{r}_z(w)=1$ and hence $H_z(w)=1$.

Define two sets

$$\begin{equation} \mathrm{S_1}:=\mathrm{L}_{z_1}\cup \mathrm{R}_{z_2} \cup \mathrm{L}_{z_3}\cup \cdots \cup \begin{cases} \mathrm{L}_{z_m}& \text{if $m$ is odd,} \\ \mathrm{R}_{z_m}& \text{if $m$ is even,} \\ \end{cases} \tag{4.15} \end{equation}$$

and

$$\begin{equation} \mathrm{S_2}:=\mathrm{R}_{z_1}\cup \mathrm{L}_{z_2} \cup \mathrm{R}_{z_3}\cup \cdots \cup \begin{cases} \mathrm{R}_{z_m}& \text{if $m$ is odd,} \\ \mathrm{L}_{z_m}& \text{if $m$ is even.} \end{cases} \tag{4.16} \end{equation}$$

See Figure 7. We define two operators

$$\begin{equation} K_1:\ell ^2(\mathrm{S_2})\to \ell ^2(\mathrm{S_1}), \qquad K_2:\ell ^2(\mathrm{S_1})\to \ell ^2(\mathrm{S_2}) \tag{4.17} \end{equation}$$

by kernels. If $w\in \mathcal{R}_{z_i}\cap \mathrm{S_1}$ and $w'\in \mathcal{R}_{z_j}\cap \mathrm{S_2}$ for some $i,j\in \{1, \cdots , m\}$, we set

$$\begin{equation} \begin{split} K_1(w,w') &= (\delta _i(j)+\delta _i (j+(-1)^i) ) \frac{\mathrm{J}(w)f_i(w) (H_{z_{i}}(w))^2 }{H_{z_{i-(-1)^{i}}}(w) H_{z_{j-(-1)^{j}}}(w')(w-w')} \mathrm{Q}_1(j) . \end{split} {\tag{4.18}} \end{equation}$$

Similarly, if $w\in \mathcal{R}_{z_i}\cap \mathrm{S_2}$ and $w'\in \mathcal{R}_{z_j}\cap \mathrm{S_1}$ for some $i,j\in \{1, \cdots , m\}$, we set