TASEP and generalizations: Method for exact solution

TL;DR

This paper extends an explicit biorthogonalization method to solve a broad class of determinantal measures in particle systems within the KPZ universality class, providing exact formulas for multipoint distributions.

Contribution

The paper generalizes the biorthogonalization method to various discrete and continuous time TASEP variants and related models, enabling exact solutions for their distribution functions.

Findings

Derived explicit Fredholm determinant formulas for multipoint distributions.

Extended the method to systems of interacting caterpillars.

Unified treatment of multiple TASEP variants with exact solutions.

Abstract

The explicit biorthogonalization method, developed in [arXiv:1701.00018] for continuous time TASEP, is generalized to a broad class of determinantal measures which describe the evolution of several interacting particle systems in the KPZ universality class. The method is applied to sequential and parallel update versions of each of the four variants of discrete time TASEP (with Bernoulli and geometric jumps, and with block and push dynamics) which have determinantal transition probabilities; to continuous time PushASEP; and to a version of TASEP with generalized update. In all cases, multipoint distribution functions are expressed in terms of a Fredholm determinant with an explicit kernel involving hitting times of certain random walks to a curve defined by the initial data of the system. The method is further applied to systems of interacting caterpillars, an extension of the discrete time TASEP models which generalizes sequential and parallel updates.