Dyck Paths, Motzkin Paths and Traffic Jams

TL;DR

This paper explores the exact solutions of the parallel-update ASEP model, linking its phase transitions to lattice path models like Dyck and Motzkin paths, and demonstrates the robustness of its phase diagram through thermodynamic equivalence.

Contribution

It extends the application of Lee-Yang theory to the parallel-update ASEP and introduces thermodynamic equivalence among lattice path models, explaining phase diagram robustness.

Findings

ASEP normalization relates to Dyck and Motzkin path models.

Lee-Yang theory applies to the parallel-update ASEP.

Phase diagram robustness is due to thermodynamic equivalence.

Abstract

It has recently been observed that the normalization of a one-dimensional out-of-equilibrium model, the Asymmetric Exclusion Process (ASEP) with random sequential dynamics, is exactly equivalent to the partition function of a two-dimensional lattice path model of one-transit walks, or equivalently Dyck paths. This explains the applicability of the Lee-Yang theory of partition function zeros to the ASEP normalization. In this paper we consider the exact solution of the parallel-update ASEP, a special case of the Nagel-Schreckenberg model for traffic flow, in which the ASEP phase transitions can be intepreted as jamming transitions, and find that Lee-Yang theory still applies. We show that the parallel-update ASEP normalization can be expressed as one of several equivalent two-dimensional lattice path problems involving weighted Dyck or Motzkin paths. We introduce the notion of thermodynamic equivalence for such paths and show that the robustness of the general form of the ASEP phase diagram under various update dynamics is a consequence of this thermodynamic equivalence.