Shock Fluctuations in Flat TASEP Under Critical Scaling

Abstract

We consider TASEP with two types of particles starting at every second site. Particles to the left of the origin have jump rate $1$, while particles to the right have jump rate $\alpha$. When $\alpha <1$ there is a formation of a shock where the density jumps to $(1-\alpha )/2$. For $\alpha <1$ fixed, the statistics of the associated height functions around the shock is asymptotically (as time $t\rightarrow \infty$) a maximum of two independent random variables as shown in Ferrari and Nejjar (Probab Theory Rel Fields 161:61–109, 2015). In this paper we consider the critical scaling when $1-\alpha =a t^{-1/3}$, where $t\gg 1$ is the observation time. In that case the decoupling does not occur anymore. We determine the limiting distributions of the shock and numerically study its convergence as a function of $a$. We see that the convergence to $F_\mathrm{GOE}^2$ occurs quite rapidly as $a$ increases. The critical scaling is analogue to the one used in the last passage percolation to obtain the BBP transition processes (Baik et al. in Ann Probab 33:1643–1697, 2006).

Appendix: Kernel \(K_a\) in Terms of Airy Functions

Here we give the explicit form of $K_a$ that we used for the numerical evaluation of $G_a$ and its statistics.

Lemma 3.6

Denote $u_{i,a}=u_i+a$ we have (with the conjugation transferred to the diffusion part)

$$\begin{aligned}&K_a (u_1,\xi _1;u_2,\xi _2)\nonumber \\&\quad \mathop {=}\limits ^\mathrm{conj}-\frac{e^{\frac{2}{3}u_{1,a}^{3}+u_{1,a}\xi _1}}{e^{\frac{2}{3}u_{2,a}^{3}+u_{2,a}\xi _2}} \frac{e^{-(\xi _2-\xi _1)^{2}/(4(u_2-u_1))}}{\sqrt{4\pi (u_2-u_1)}}\mathbb {1}(u_2>u_1)\end{aligned}$$

(4.1)

$$\begin{aligned}&\qquad \quad +\int _{0}^{\infty }\mathrm {d}\lambda \mathrm {Ai}(\xi _1+u_{1,a}^{2}+\lambda )\mathrm {Ai}(\xi _2+u_{2,a}^2+\lambda )e^{\lambda (u_2-u_1)}\end{aligned}$$

(4.2)

$$\begin{aligned}&\qquad \quad +\int _{0}^{\infty }\mathrm {d}\lambda \mathrm {Ai}(\xi _1+u_{1,a}^{2}-\lambda )\mathrm {Ai}(\xi _2+u_{2,a}^2+\lambda )e^{\lambda (2a+u_1+u_2)}\end{aligned}$$

(4.3)

$$\begin{aligned}&\qquad \quad -\int _{0}^{\infty }\mathrm {d}\lambda \mathrm {Ai}(\xi _1+u_{1,a}^{2}+\lambda )\mathrm {Ai}(\xi _2+u_{2,a}^2+\lambda )e^{\lambda (4a+u_2-u_1)} \end{aligned}$$

(4.4)

$$\begin{aligned}&\qquad \quad +\int _{0}^{\infty }\mathrm {d}\lambda \mathrm {Ai}(\xi _1+u_{1,a}^{2}+\lambda )\mathrm {Ai}(\xi _2+u_{2,a}^2-\lambda )e^{\lambda (2a-u_1-u_2)}. \end{aligned}$$

(4.5)

Proof

The result is an easy computation that uses the identities

$$\begin{aligned}&\frac{-1}{2\pi \mathrm {i}}\int _{\delta +\mathrm {i}\mathbb {R}}\mathrm {d}v e^{v^{3}/3+xv^{2}+yv}=\mathrm {Ai}(x^{2}-y)e^{\frac{2}{3}x^{3}-xy},\nonumber \\&\quad \frac{1}{z}=\int _{0}^{\infty }\mathrm {d}\lambda e^{-\lambda z} \qquad (z \in \mathbb {C},\,{\text {Re}}(z)>0), \end{aligned}$$

(4.6)

for any $\delta >\max \{0,x\}$. $\square$

Remark 3.7

Alternatively, via the identity (A.6) of [9], one has

$$\begin{aligned} (A.3)=&\ -\int _{-\infty }^{0}\mathrm {d}\lambda e^{\lambda (u_{2,a}+u_{1,a})}\mathrm {Ai}(\xi _1+u_{1,a}^{2}-\lambda )\mathrm {Ai}(\xi _2+u_{2,a}^{2}+\lambda )\nonumber \\&+2^{-1/3}\mathrm {Ai}\left( 2^{-1/3}(\xi _1+\xi _2)+2^{-4/3}(u_1-u_2)^{2}\right) e^{-\frac{1}{2}(u_{1,a}+u_{2,a})(\xi _2+u_{2,a}^{2}-\xi _1-u_{1,a}^{2})}, \end{aligned}$$

(4.7)

with an analogous formula for (4.5).