A product formula for the TASEP on a ring

TL;DR

This paper investigates the TASEP on a ring, revealing a conditional independence property of entries in the stationary distribution, with proofs using multi-line queues and a combinatorial interpretation, and proposes a conjecture for a more general case.

Contribution

It introduces a new conditional independence result for the TASEP on a ring and connects it to combinatorial structures, extending understanding of its stationary distribution.

Findings

Conditional independence of first and last entries in the TASEP on a ring

Enumeration of configurations using multi-line queues

Conjecture for the case without separation of entries

Abstract

For a random permutation sampled from the stationary distribution of the TASEP on a ring, we show that, conditioned on the event that the first entries are strictly larger than the last entries, the order of the first entries is independent of the order of the last entries. The proof uses multi-line queues as defined by Ferrari and Martin, and the theorem has an enumerative combinatorial interpretation in that setting. Finally, we present a conjecture for the case where the small and large entries are not separated.