# zbMATH — the first resource for mathematics

Microscopic structure at the shock in the asymmetric simple exclusion. (English) Zbl 0679.60094
Let $$P_ t$$ be the semigroup of the asymmetric simple exclusion process on $${\mathbb{Z}}$$ evolving in the following way: $\eta \to \eta^{x,x+1},\quad at\quad rate\quad p\eta (x)(1-\eta (z));$ $\eta \to \eta^{x,x+1},\quad at\quad rate\quad q\eta (x)(1-\eta (x)),$ where $$q<p<1$$ and $$p+q=1$$. Let $$\nu_{\rho}$$ be the Bernoulli measure of parameter $$\rho$$ and take as initial distribution the product measure $${\bar \nu}{}_{\rho}$$ whose marginals are $${\bar \nu}{}_{\rho}\{\eta (x)=1\}=0$$ for $$x<0$$, $$=1$$ for $$x=0$$, $$=\nu_{\rho}\{\eta (x)=1\}$$ for $$x>0$$. Set $$u_{\rho}=(p-q)(1-\rho)$$ and denote by $$r_ x$$ the space shift by x to the left as an operator acting on the probability measure on $$\{0,1\}^ Z$$. Finally, let $$\lambda (r,t)=P[B_ t>r],$$ where $$B_ t$$ is the Brownian motion with diffusion coefficient $$u_{\rho}$$. The authors prove that for all $$r\in R$$, $\lim_{\epsilon \to 0}r_{u_{\rho}\epsilon^{-1}t+r\epsilon^{-1/2}}{\bar \nu}_{\rho}P_{\epsilon^{-1}t}=\lambda (r,t)\nu_ 0+(1-\lambda (r,t))\nu_{\rho}.$ Moreover, let $$X_ t$$ be the position of the leftmost particle at time t, then the following central limit theorem holds: $$(X_ t-u_{\rho}t)/\sqrt{t}$$ converges to the distribution of $$B_ 1$$.
Reviewer: Mufa Chen

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory
Full Text: