Generalizations of TASEP in discrete and continuous inhomogeneous space

TL;DR

This paper introduces a new class of exactly solvable particle systems generalizing TASEP with spatial inhomogeneity, revealing phase transitions, limit shapes, and fluctuation behaviors within the KPZ universality class.

Contribution

It develops a novel framework for inhomogeneous TASEP-like systems, providing explicit limit shapes, fluctuation results, and connections to Schur measures, expanding understanding of phase transitions in these models.

Findings

Explicit limit shapes for special initial data

Asymptotic fluctuation results in KPZ class

Deformations of Tracy-Widom distribution at traffic jams

Abstract

We investigate a rich new class of exactly solvable particle systems generalizing the Totally Asymmetric Simple Exclusion Process (TASEP). Our particle systems can be thought of as new exactly solvable examples of tandem queues, directed first- or last-passage percolation models, or Robinson-Schensted-Knuth type systems with random input. One of the novel features of the particle systems is the presence of spatial inhomogeneity which can lead to the formation of traffic jams. For systems with special step-like initial data, we find explicit limit shapes, describe hydrodynamic evolution, and obtain asymptotic fluctuation results which put the systems into the Kardar-Parisi-Zhang universality class. At a critical scaling around a traffic jam in the continuous space TASEP, we observe deformations of the Tracy-Widom distribution and the extended Airy kernel, revealing the finer structure of this novel type of phase transitions. A homogeneous version of a discrete space system we consider is a one-parameter deformation of the geometric last-passage percolation, and we obtain extensions of the limit shape parabola and the corresponding asymptotic fluctuation results. The exact solvability and asymptotic behavior results are powered by a new nontrivial connection to Schur measures and processes.