Open ASEP in the Weakly Asymmetric Regime

Abstract

We consider ASEP on a bounded interval and on a half-line with sources and sinks. On the full line, Bertini and Giacomin in 1997 proved convergence under weakly asymmetric scaling of the height function to the solution of the KPZ equation. We prove here that under similar weakly asymmetric scaling of the sources and sinks as well, the bounded interval ASEP height function converges to the KPZ equation on the unit interval with Neumann boundary conditions on both sides (different parameter for each side), and likewise for the half-line ASEP to KPZ on a half-line. This result can be interpreted as showing that the KPZ equation arises at the triple critical point (maximal current / high density / low density) of the open ASEP.

Singular SPDEs in domains with boundaries

“…This is the point of view that was taken in [4] where the authors showed that the height function associated to a large but finite discrete system of particles performing a weakly asymmetric simple exclusion process converges to the solutions to (1.4) with boundary conditions c ± that are related to the boundary behaviour of the discrete system in a straightforward way. In particular, if the 'net flow' of particles at each boundary is 0, then c ± = 0.…”

“…Remark 1. 4 We have chosen to include the arbitrary constant 1 2 in front of the term ∂ 2 x u so that the corresponding semigroup at time t is given by the Gaussian with variance t.…”

“…It is however possible to define a notion of "KPZ equation with Neumann boundary conditions" via the Hopf-Cole transform. Indeed, it suffices to realise that, at least formally, if u solves 4) then the process Z = exp(2u) solves…”

The KPZ equation on the real line

“…However, in many situations (including for the KPZ equation) the higher order terms can only be constructed with the help of a suitable renormalisation, and this means that the solution we eventually find does not solve the original equation, but a renormalised version of it [BCCH17]. But while we now have a good understanding in what sense the Cole-Hopf solution solves the KPZ equation and how to interpret its renormalisation, all this is restricted to the equation on the torus or in a finite volume with boundary conditions [GH18a, CS16,GPS17]. Since one of the main interests in the KPZ equation comes from its large scale behavior it would be more natural to solve it on R, a space that can be arbitrarily rescaled.…”

KPZ Equation Limit of Stochastic Higher Spin Six Vertex Model

“…We will focus on discussing the method for proving this self-averaging result in the rest of the section. We remark that other recent KPZ equation convergence results using the Hopf-Cole transform include ASEP-(q, j) [CST18], Hall-Littlewood PushTASEP [Gho17], weakly asymmetric bridges [Lab17], open ASEP [CS18,Par19].…”