Dyck paths, Motzkin paths and traffic jams

Abstract

It has recently been observed that the normalization of a one-dimensional out-of-equilibriummodel, the asymmetric exclusion process (ASEP) with random sequential dynamics, isexactly equivalent to the partition function of a two-dimensional lattice path model ofone-transit walks, or equivalently Dyck paths. This explains the applicabilityof 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 ofthe Nagel–Schreckenberg model for traffic flow, in which the ASEP phase transitions can beinterpreted as jamming transitions, and find that Lee–Yang theory still applies. We showthat the parallel-update ASEP normalization can be expressed as one of severalequivalent two-dimensional lattice path problems involving weighted Dyck or Motzkinpaths. We introduce the notion of thermodynamic equivalence for such paths andshow that the robustness of the general form of the ASEP phase diagram undervarious update dynamics is a consequence of this thermodynamic equivalence.