Cutoff profile of ASEP on a segment

Abstract

This paper studies the mixing behavior of the Asymmetric Simple Exclusion Process (ASEP) on a segment of length N. Our main result is that for particle densities in (0, 1), the total-variation cutoff window of ASEP is $$N^{1/3}$$ N 1 / 3 and the cutoff profile is $$1-F_{\mathrm {GUE}},$$ 1 - F GUE , where $$F_{\mathrm {GUE}}$$ F GUE is the Tracy-Widom distribution function. This also gives a new proof of the cutoff itself, shown earlier by Labbé and Lacoin. Our proof combines coupling arguments, the result of Tracy–Widom about fluctuations of ASEP started from the step initial condition, and exact algebraic identities coming from interpreting the multi-species ASEP as a random walk on a Hecke algebra.

Riemann surfaces for KPZ with periodic boundaries

“…That distribution was initially introduced in [269] in the context of random matrix theory for the fluctuations of the largest eigenvalue of a large complex valued Hermitian random matrix distributed according to the Gaussian unitary ensemble (GUE). In the context of driven particles, the GUE Tracy-Widom distribution thus describes the fluctuations of the current [267] for some initial conditions, but interestingly also controls the convergence of the time-dependent probabilities of microstates to their stationary values in a finite system [270,271]. For flat initial condition, one finds instead [272][273][274] that −h(x ), whose statistics is translation invariant in x , has the same GOE Tracy-Widom distribution 7 [275] as the largest eigenvalue of large real valued symmetric random matrices distributed according to the Gaussian orthogonal ensemble (GOE).…”

Stationary measure for the open KPZ equation

“…Let us also mention related work of Refs. [5,27,28] for the stochastic Burgers equation, and work on the mixing time of open ASEP [19,24,31,37,49].…”

“…Depending on the value of 𝑣, we see that the parameters in Equations (6.16) and (6.17) either satisfy P or N1 in Definition 6.5 and either way we arrive at Equation (6.15) and verify that the right-hand side is a probability measure. Depending on the value of 𝑢, the parameters in Equations (6 19…”

Double coset Markov chains

“…The interpretation of multiplication on the Hecke algebra as various ‘systematic scan’ Markov chains is developed in [18], [10]. It works for other types in several variations.…”