The ASEP and determinantal point processes

TL;DR

This paper introduces a new family of discrete determinantal point processes linked to orthogonal polynomials, and applies them to analyze the asymptotic behavior of the ASEP and stochastic six vertex models.

Contribution

It establishes a connection between ASEP height functions and a new class of determinantal point processes, enabling asymptotic analysis in various regimes.

Findings

Derived large time asymptotics for ASEP height function

Established GUE Tracy-Widom distribution for ASEP

Obtained KPZ universality class results for ASEP

Abstract

We introduce a family of discrete determinantal point processes related to orthogonal polynomials on the real line, with correlation kernels defined via spectral projections for the associated Jacobi matrices. For classical weights, we show how such ensembles arise as limits of various hypergeometric orthogonal polynomials ensembles. We then prove that the q-Laplace transform of the height function of the ASEP with step initial condition is equal to the expectation of a simple multiplicative functional on a discrete Laguerre ensemble --- a member of the new family. This allows us to obtain the large time asymptotics of the ASEP in three limit regimes: (a) for finitely many rightmost particles; (b) GUE Tracy-Widom asymptotics of the height function; (c) KPZ asymptotics of the height function for the ASEP with weak asymmetry. We also give similar results for two instances of the stochastic six vertex model in a quadrant. The proofs are based on limit transitions for the corresponding determinantal point processes.