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} and the cutoff profile is 1-F_{mathrm {GUE}}, where F_{mathrm {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.
Highlights
We consider Asymmetric Simple Exclusion Process (ASEP) on the segment [1; N ] := {1, . . . , N } with k ≤ N particles
The dynamics of ASEP can be described as follows: Each particle waits an exponential time with parameter 1, after which with probability p > 1/2 it attempts to make a unit step to the right, and with probability q = 1 − p < 1/2 it attempts to make a unit step to the left
Theorem 1 gives the cutoff window and the cutoff profile of ASEP, we refer to Chapter 18 of the textbook [23] by Levin-Peres for definitions and examples in the general context of Markov chains
Summary
We consider ASEP on the segment [1; N ] := {1, . . . , N } with k ≤ N particles. This is a continuous time Markov chain with state space. We define the maximal total-variation distance between the distribution at given time and the stationary distribution as d N,k (t) := max ||Ptξ − πN,k ||TV, ξ ∈ N ,k and for c ∈ R, we define the time point. Where f (α) = (√(αα+(1√−1α−))α1/)64/3 , and FGUE is the GUE Tracy-Widom distribution defined in (10) Proof This is an immediate consequence of Theorem 3 (which gives an upper bound for the limit on the lefthand side), proven, and Theorem 6 (which gives a lower bound), proven in Sect. 4. Theorem 1 gives the cutoff window and the cutoff profile (or shape) of ASEP, we refer to Chapter 18 of the textbook [23] by Levin-Peres (with contributions by Wilmer) for definitions and examples in the general context of Markov chains. Our Theorem 1 confirms (and gives a precise meaning to) this conjecture
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