The $q$-Hahn asymmetric exclusion process

TL;DR

This paper introduces a new integrable asymmetric exclusion process generalizing the $q$-Hahn models, providing exact formulas for particle locations, and confirming KPZ universality and Tracy-Widom fluctuations through asymptotic analysis.

Contribution

It extends the $q$-Hahn models to include bidirectional jumps, derives explicit moment and distribution formulas, and analyzes asymptotic behaviors confirming universality predictions.

Findings

Derived Fredholm determinant formulas for particle positions.

Confirmed KPZ scaling limit constants via steepest descent analysis.

Proved Tracy-Widom GUE fluctuations for the first particle in the partially asymmetric case.

Abstract

We introduce new integrable exclusion and zero-range processes on the one-dimensional lattice that generalize the $q$-Hahn TASEP and the $q$-Hahn Boson (zero-range) process introduced in [Pov13] and further studied in [Cor14], by allowing jumps in both directions. Owing to a Markov duality, we prove moment formulas for the locations of particles in the exclusion process. This leads to a Fredholm determinant formula that characterizes the distribution of the location of any particle. We show that the model-dependent constants that arise in the limit theorems predicted by the KPZ scaling theory are recovered by a steepest descent analysis of the Fredholm determinant. For some choice of the parameters, our model specializes to the multi-particle-asymmetric diffusion model introduced in [SW98]. In that case, we make a precise asymptotic analysis that confirms KPZ universality predictions. Surprisingly, we also prove that in the partially asymmetric case, the location of the first particle also enjoys cube-root fluctuations which follow Tracy-Widom GUE statistics.