Distribution of a Second-Class Particle’s Position in the Two-Species ASEP with a Special Initial Configuration

Abstract

In this paper, we consider the two-species asymmetric simple exclusion process (ASEP) consisting of $N-1$ first-class particles and one second-class particle. We assume that all particles are located at arbitrary positions but the second-class particle is the rightmost particle at time $t=0$. We find the exact formula of the distribution of the second-class particle’s position at time t by directly using the transition probabilities of the two-species ASEP, which is a different approach from the coupling method Tracy and Widom used in [J Phys A 42:425002, 2009].

Appendix A: Formulas of \([\textbf{A}_{\sigma }]_{\nu _n,\nu _N}\) in (1)

In this appendix, we briefly summarize how to find the formulas of $[\textbf{A}_{\sigma }]_{\nu _n,\nu _N}$ in (1), which were developed in [15, 16]. For a permutation $\sigma$ in the symmetric group ${\mathcal {S}}_N$, $\textbf{A}_{\sigma }$ is an $N^N \times N^N$ matrix which is obtained as follows. Let $\textrm{T}_i$ be the simple transposition which interchanges the $i^{th}$ element and the $(i+1)^{st}$ element and leaves everything else fixed. Then, any permutation $\sigma \in {\mathcal {S}}_N$ can be written as a product of simple transpositions, that is,

$$\begin{aligned} \sigma = \textrm{T}_{i_j}\cdots \textrm{T}_{i_1} \end{aligned}$$

(63)

for some $i_1,\dots , i_j \in \{1,\dots , N-1\}$. Of course, the form of (63) is not unique. For example, a permutation $(321) \in {\mathcal {S}}_3$ is written as

$$\begin{aligned} (321) =\textrm{T}_1\textrm{T}_2\textrm{T}_1 = \textrm{T}_2\textrm{T}_1\textrm{T}_2. \end{aligned}$$

If $\textrm{T}_i$ interchanges $\alpha$ at the $i^{th}$ slot and $\beta$ at the $(i+1)^{st}$ slot in a permutation, that is,

$$\begin{aligned} \textrm{T}_i(~\cdots ~ \alpha \beta ~\cdots ) =(~\cdots ~ \beta \alpha ~\cdots ), \end{aligned}$$

we denote this $\textrm{T}_i$ by $\textrm{T}_i(\beta ,\alpha )$ to show explicitly which numbers are interchanged. For example,

$$\begin{aligned} (321) = \textrm{T}_1(3,2)\textrm{T}_2(3,1)\textrm{T}_1(2,1). \end{aligned}$$

Hence, we may have an expression

$$\begin{aligned} \sigma = \textrm{T}_{i_j}(\beta _j,\alpha _j) \cdots \textrm{T}_{i_1}(\beta _1,\alpha _1) \end{aligned}$$

(64)

for any permutation $\sigma$. Given $\textrm{T}_{i}(\beta ,\alpha )$, we define $N^N \times N^N$ matrix $\textbf{T}_{i}(\beta ,\alpha )$ by

$$\begin{aligned} \textbf{T}_{i}(\beta ,\alpha ) =\underbrace{\textbf{I}_N ~\otimes ~ \cdots ~ \otimes ~ \textbf{I}_N}_{(i-1) ~\text {times}}~ \otimes ~ \textbf{R}_{\beta \alpha } ~\otimes ~ \underbrace{\textbf{I}_N ~\otimes ~ \cdots ~\otimes ~ \textbf{I}_N}_{(N-i-1)~\text {times}} \end{aligned}$$

where $\textbf{I}_N$ is the $N \times N$ identity matrix and $\textbf{R}_{\beta \alpha }$ is an $N^2 \times N^2$ matrix, where $\textbf{R}_{\beta \alpha }$ is defined as follows. Let us label all columns and rows of $N^2 \times N^2$ matrices by ij where $i,j =1,\dots , N$ in the lexicographical order. The (ij, kl) entry of the matrix $\textbf{R}_{\beta \alpha }$ is defined to be

$$\begin{aligned} \big [\textbf{R}_{\beta \alpha }\big ]_{ij,kl} = {\left\{ \begin{array}{ll} S_{\beta \alpha }&{} ~\textrm{if}~~ij=kl~\textrm{with}~i=j;\\ P_{\beta \alpha }&{} ~\textrm{if}~~ij=kl~\textrm{with}~i<j;\\ Q_{\beta \alpha }&{} ~\textrm{if}~~ij=kl~\textrm{with}~i>j;\\ pT_{\beta \alpha }&{}~\textrm{if}~~ij=lk~\textrm{with}~i<j;\\ qT_{\beta \alpha }&{}~\textrm{if}~~ij=lk~\textrm{with}~i>j;\\ 0 &{}~\text {for all other cases} \end{array}\right. } \end{aligned}$$

where

$$\begin{aligned} S_{\beta \alpha }&= -\frac{p+q\xi _{\alpha }\xi _{\beta } -\xi _{\beta }}{p+q\xi _{\alpha }\xi _{\beta } - \xi _{\alpha }}, \quad P_{\beta \alpha } = \frac{(p-q\xi _{\alpha }) (\xi _{\beta }-1)}{p+q\xi _{\alpha }\xi _{\beta } - \xi _{\alpha }}\\ T_{\beta \alpha }&= \frac{\xi _{\beta }-\xi _{\alpha }}{p+q\xi _{\alpha } \xi _{\beta } - \xi _{\alpha }}, \quad Q_{\beta \alpha } =\frac{(p-q\xi _{\beta })(\xi _{\alpha }-1)}{p+q\xi _{\alpha } \xi _{\beta } - \xi _{\alpha }}. \end{aligned}$$

Now, for a given expression (64), define

$$\begin{aligned} \textbf{A}_{\sigma } := \textbf{T}_{i_j}(\beta _j,\alpha _j) ~\cdots ~ \textbf{T}_{i_1}(\beta _1,\alpha _1). \end{aligned}$$

(65)

It is a known fact that $\textbf{A}_{\sigma }$ is well-defined in the sense that (65) represents the same matrix for any expression (64). We have just constructed an $N^N \times N^N$ matrix $\textbf{A}_{\sigma }$ for a given permutation $\sigma$. Now, let us label all columns and rows of $\textbf{A}_{\sigma }$ by $\nu =i_1\cdots i_N$ where $i_j \in \{1,\dots , N\},\, j=1,\dots , N$ in the lexicographical order. In this paper, we denote $2\cdots 212\cdots 2$ by $\nu _n$ if 1 is the $n^{\textrm{th}}$ leftmost number.

Since $\textbf{A}_{\sigma }$ is a product of matrices, in general, the matrix elements $[\textbf{A}_{\sigma }]_{\pi ,\nu }$ are possibly written as a sum of the products of the matrix elements of $\textbf{T}_{i_j}(\beta _j,\alpha _j),\dots , \textbf{T}_{i_1}(\beta _1,\alpha _1)$ in (65). However, it was shown in [16] that, for some special cases of initial order of particles $\nu$, $[\textbf{A}_{\sigma }]_{\pi ,\nu }$ are expressed as a product of the matrix elements of $\textbf{T}_{i_j}(\beta _j,\alpha _j),\dots , \textbf{T}_{i_1}(\beta _1,\alpha _1)$, not a sum of the products. In particular, for the initial order of particles $\nu _N$, that is, $2\cdots 21$, the formulas $[\textbf{A}_{\sigma }]_{\nu _n,\nu _N}$ are given in Theorem 1.2 (a), Theorem 1.4, Theorem 1.6, Theorem 1.7, and Proposition 1.8 in [16].