Duality relations for the ASEP conditioned on a low current

TL;DR

This paper establishes duality relations for the ASEP conditioned on low current, revealing a microscopic traveling-wave property and shock dynamics using quantum algebra symmetries.

Contribution

It introduces new duality relations for the ASEP under low current conditioning, derived from quantum algebra symmetry, and demonstrates a microscopic traveling-wave property of shock measures.

Findings

Duality relations derived from $U_q[ ext{gl}(2)]$ symmetry.

Microscopic traveling-wave property of the conditioned process.

Shock-antishock measure dynamics expressed via $K$-particle transition probabilities.

Abstract

We consider the asymmetric simple exclusion process (ASEP) on a finite lattice with periodic boundary conditions, conditioned to carry an atypically low current. For an infinite discrete set of currents, parametrized by the driving strength $s_K$, $K \geq 1$, we prove duality relations which arise from the quantum algebra $U_q[\mathfrak{gl}(2)]$ symmetry of the generator of the process with reflecting boundary conditions. Using these duality relations we prove on microscopic level a travelling-wave property of the conditioned process for a family of shock-antishock measures for $N>K$ particles: If the initial measure is a member of this family with $K$ microscopic shocks at positions $(x_1,\dots,x_K)$, then the measure at any time $t>0$ of the process with driving strength $s_K$ is a convex combination of such measures with shocks at positions $(y_1,\dots,y_K)$. which can be expressed in terms of $K$-particle transition probabilities of the conditioned ASEP with driving strength $s_N$.