Asymmetric simple exclusion process

The asymmetric simple exclusion process (ASEP) is an interacting particles system introduced in 1970 by Frank Spitzer in Interaction of Markov Processes and thousands articles have been published on it in the physics and mathematics litterature since then.

The process with parameters $p,\,q\geqslant 0,\,p+q=1,\,p=q$ is a continuous time Markov process on $S=\lbrace 0,1\rbrace ^{\mathbb {Z} }$ , the 1’s being thought of as particles and the 0’s as holes. Each particle waits a random exponent mean one amount of time and then attempts a jump, one site to the right with probability $p$ and one site to the left with probability $q$. However, the jump is only performed if there is no particle at the target site. Otherwise, nothing happens and the particle waits another exponential time. All particles are doing this independently of each other.

The model is related to the Kardar–Parisi–Zhang equation in the weakly asymmetric limit, i.e. when $p-q$ tends to zero. Recently, progress has been made to understand the statistics of the current of particles and it appears that the Tracy–Widom distribution plays a key role.