Asymptotic fluctuations of geometric q-TASEP, geometric q-PushTASEP and q-PushASEP

TL;DR

This paper studies the long-term fluctuation behavior of three related particle systems, showing they converge to well-known distributions like GUE Tracy-Widom and Baik-Ben Arous-Peche under certain conditions.

Contribution

It establishes the asymptotic fluctuation limits of geometric q-TASEP, q-PushTASEP, and q-PushASEP, including effects of perturbations in jump rates.

Findings

Rescaled particle positions converge to GUE Tracy-Widom distribution in homogeneous case.

Perturbations in jump rates lead to Baik-Ben Arous-Peche distribution.

Top eigenvalue fluctuations of finite GUE matrices describe the perturbed cases.

Abstract

We investigate the asymptotic fluctuation of three interacting particle systems: the geometric q-TASEP, the geometric q-PushTASEP and the q-PushASEP. We prove that the rescaled particle position converges to the GUE Tracy-Widom distribution in the homogeneous case. If the jump rates of the first finitely many particles are perturbed in the first two models, we obtain that the limiting fluctuations are governed by the Baik-Ben Arous-Peche distribution and that of the top eigenvalue of finite GUE matrices.