A phase transition for $q$-TASEP with a few slower particles

TL;DR

This paper studies a variant of the $q$-TASEP model with a few slower particles, revealing a BBP phase transition in particle positions, extending previous results that assumed identical rates.

Contribution

It extends the analysis of $q$-TASEP to include particles with different speeds, demonstrating the BBP transition without restrictions on parameters or positions.

Findings

Observation of BBP phase transition in $q$-TASEP with slower particles

Refinement of Ferrari-Vet"o's proof technique

No restrictions on $q$ or particle positions

Abstract

We consider a $q$-TASEP model started from step initial condition where all but finitely many particles have speed $1$ and a few particles are slower. It is shown in [9] that the rescaled particles position of $q$-TASEP with identical hopping rates obeys a central limit theorem \`a la Tracy-Widom. We adapt this work to the case of different hopping rates and show that one observes the so-called BBP transition. Our proof is a refinement of Ferrari-Vet\"o's and does not require any condition on the parameter $q$ nor the macroscopic position of particles.