The $q$-Hahn asymmetric exclusion process

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.

40 pages,11 figures. v3: Presentation improved in Introduction and Section 4. to appear in Ann. Appl. Probab

References in corpus (4)

• Asymptotics in ASEP with Step Initial Condition
• Facilitated Asymmetric Exclusion
• A phase transition for $q$-TASEP with a few slower particles
• A deformation of affine Hecke algebra and integrable stochastic particle system

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