Stationary probability of the identity for the TASEP on a ring

TL;DR

This paper analyzes the circular TASEP Markov chain on permutations, calculating the stationary probability of the identity permutation, thereby resolving a conjecture by Thomas Lam.

Contribution

It provides an explicit computation of the stationary probability of the identity permutation in the circular TASEP, confirming a conjecture and extending understanding of the chain's long-term behavior.

Findings

Exact stationary probability of the identity permutation computed.

Results confirm and resolve Thomas Lam's conjecture.

Analysis extends to sorted words beyond the identity.

Abstract

Consider the following Markov chain on permutations of length $n$. At each time step we choose a random position. If the letter at that position is smaller than the letter immediately to the left (cyclically) then these letters swap positions. Otherwise nothing happens, corresponding to a loop in the Markov chain. This is the circular TASEP. We compute the average proportion of time the chain spends at the identity permutation (and, in greater generality, at sorted words). This answers a conjecture by Thomas Lam.