On the scaling limits of weakly asymmetric bridges

Abstract

We consider a discrete bridge from (0, 0) to (2N, 0) evolving according to the corner growth dynamics, where the jump rates are subject to an upward asymmetry of order N −α with α ∈ (0, ∞). We provide a classification of the asymptotic behaviours -invariant measure, hydrodynamic limit and fluctuations -of this model according to the value of the parameter α.MSC 2010 subject classifications: Primary 60K35; secondary 60H15; 82C24.

$\operatorname{ASEP}(q,j)$ converges to the KPZ equation

“…Besides the work of [BG97,ACQ11] (and previous to the present work), the only other result of this type via Gärtner / Cole-Hopf transform is due to [DT16], wherein they show KPZ equation convergence for a class of weakly asymmetric non-simple exclusion processes with hopping range at most 3. Another work which was posted slightly after our present article is by Labbé [Lab16b,Lab16a] who showed that in particular INTRODUCTION range of scaling regimes the fluctuations of the weakly asymmetric bridges converge to the KPZ equation, also via the method of Gärtner transform.…”

Weakly asymmetric bridges and the KPZ equation

“…An important feature of this hyperbolic PDE is that, for any initial condition, it reaches the macroscopic stationary state x → x ∧ (1 − x) in finite time. Notice that for α ≥ 1, the hydrodynamic limit is given by a heat equation so that it takes infinite time to reach the stationarity state, see [Lab16]. This feature of the PDE has a non-trivial consequence at the level of the KPZ fluctuations, that we now investigate.…”

“…Theorem 1.1 ( [Lab16]) Take α ∈ (0, 1). The law of the process u N under µ N converges to the law of the centred Gaussian process (B α (x), x ∈ R), with covariance…”

“…Theorem 1.2 ( [Lab16]) The sequence u N , starting from the equilibrium measure µ N , converges in distribution in the Skorohod space D([0, ∞), C(R)) towards the solution u of the following stochastic PDE…”

“…This choice of asymmetry, combined with our boundary conditions, gives rise to scaling behaviours which, to the best of our knowledge, have not been observed previously in the literature, see below. A complete description of the full range α ∈ (0, ∞) of scaling limits of this model is presented in a companion paper [Lab16]. In the present paper, we will always assume α ∈ (0, 1).…”