Multivariate Eulerian polynomials and exclusion processes

TL;DR

This paper provides a new combinatorial interpretation of the stationary distribution of exclusion processes using decorated trees and permutations, linking it to multivariate Eulerian polynomials and analyzing their stability properties.

Contribution

It introduces a novel combinatorial framework for exclusion processes and connects it to multivariate Eulerian polynomials, extending recent mathematical developments.

Findings

Stationary distribution expressed via decorated trees and permutations

Partition functions related to multivariate Eulerian polynomials

Partition functions exhibit stability and negative dependence properties

Abstract

We give a new combinatorial interpretation of the stationary distribution of the (partially) asymmetric exclusion process on a finite number of sites in terms of decorated alternative trees and colored permutations. The corresponding expressions of the multivariate partition functions are then related to multivariate generalizations of Eulerian polynomials for colored permutations considered recently by N. Williams and the third author, and others. We also discuss stability-- and negative dependence properties satisfied by the partition functions.