Duality relations for the ASEP conditioned on a low current

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 $s_K$, $K\ge 1$, 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 $s_N$.

26 pages

References in corpus (6)

• Fluctuation theorems for stochastic dynamics
• Duality and hidden symmetries in interacting particle systems
• Matrix representation of the stationary measure for the multispecies TASEP
• Self-Duality for the Two-Component Asymmetric Simple Exclusion Process
• Stochastic duality of ASEP with two particle types via symmetry of quantum groups of rank two
• Quantum algebra symmetry of the ASEP with second-class particles

Cited by in corpus (3)

• Asymmetric stochastic transport models with ${\mathcal{U}}_q(\mathfrak{su}(1,1))$ symmetry
• A Multi-species ASEP(q,j) and q-TAZRP with Stochastic Duality
• On dualities for SSEP and ASEP with open boundary conditions