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 [J. Phys. A 46 (2013) 465205, 25] and further studied in [Int. Math. Res. Not. IMRN 14 (2015) 5577–5603], 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 [Phys. Rev. E 58 (1998) 4181]. 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.