Statistics of TASEP with Three Merging Characteristics

Abstract

In this paper we consider the totally asymmetric simple exclusion process, with non-random initial condition having three regions of constant densities of particles. From left to right, the densities of the three regions are increasing. Consequently, there are three characteristics which meet, i.e. two shocks merge. We study the particle fluctuations at this merging point and show that they are given by a product of three (properly scaled) GOE Tracy–Widom distribution functions. We work directly in TASEP without relying on the connection to last passage percolation.

Appendix: Some Bounds

Lemma A.1

Let $\nu \in (0,1)$. There exists a $t_0\in (0,\infty )$ such that for all $t\ge t_0$,

$$\begin{aligned} \begin{aligned}&{\mathbb {P}}(x^{\mathrm{step}}_{\nu t}(t)\ge (1-2\sqrt{\nu })t-s t^{1/3})\le C_1\, e^{-c_1 (-s)^{3/2}},\quad s\le 0,\\&{\mathbb {P}}(x^{\mathrm{step}}_{\nu t}(t)\le (1-2\sqrt{\nu })t -s t^{1/3})\le C_2 \,e^{-c_2 s},\quad s\ge 0, \end{aligned} \end{aligned}$$

(A.1)

where the constants $C_i,c_i$ are positive and independent of s. Further, for any given $\varepsilon >0$, the constants in the bounds for step initial conditions can be chosen independent of $\nu \in [\varepsilon ,1-\varepsilon ]$.

The first estimate in (A.1) was obtained in [2] in terms of TASEP height function. The idea is to bound the Fredholm determinant which gives the distribution function of $x^{\mathrm{step}}_{t/4}(t)$ by the exponential of the trace of the kernel, see Sect. 4 of [2]. The method was used before by Widom in [31]. The other two estimates in (A.1) follow directly from the exponential estimates on the correlation kernel for step initial condition. Although we do not need it here, let us mention that the estimates can be improved to optimal decay power [22].