A Class of Growth Models Rescaling To kpz

Abstract

We consider a large class of 1 + 1-dimensional continuous interface growth models and we show that, in both the weakly asymmetric and the intermediate disorder regimes, these models converge to Hopf-Cole solutions to the KPZ equation.

Open ASEP in the Weakly Asymmetric Regime

“…For ASEP on the whole line, the seminal work of Bertini and Giacomin [7] demonstrated convergence of ASEP under a similar scaling to the KPZ equation on R via a discrete Cole-Hopf transformation introduced by Gärtner [36]. There has since been great progress in expanding this weak universality of the KPZ equation on R using methods similar to those of Bertini and Giacomin [15,16,19,57], as well as using energy solutions (introduced by Assing [2] as well as Jara and Gonçalves [38] and proved to be unique by Gubinelli and Perkowski [46]) [26, 38-40, 45, 47, 48], and Hairer's regularity structures (introduced in [50]) [51,52]. There has since been great progress in expanding this weak universality of the KPZ equation on R using methods similar to those of Bertini and Giacomin [15,16,19,57], as well as using energy solutions (introduced by Assing [2] as well as Jara and Gonçalves [38] and proved to be unique by Gubinelli and Perkowski [46]) [26, 38-40, 45, 47, 48], and Hairer's regularity structures (introduced in [50]) [51,52].…”

“…They assumed near-equilibrium initial data, and narrow wedge initial data was treated in [1]. There has since been great progress in expanding this weak universality of the KPZ equation on R using methods similar to those of Bertini and Giacomin [15,16,19,57], as well as using energy solutions (introduced by Assing [2] as well as Jara and Gonçalves [38] and proved to be unique by Gubinelli and Perkowski [46]) [26, 38-40, 45, 47, 48], and Hairer's regularity structures (introduced in [50]) [51,52]. These works have dealt entirely with the KPZ equation on R or the torus (OE0; 1 with periodic boundary conditions).…”

Local Solution to the Multi-layer KPZ Equation

“…In terms of the algebraic theory of renormalization of regularity structures the situation here is essentially 2 the same as those found in previous treatments of the KPZ equation [HQ18,HS17]. A full specification of the BPHZ model is unnecessary for our purposes, we will only need to understand its action on certain elements of the regularity structure present so that we can derive the renormalized equation.…”

Sharp Convergence of Nonlinear Functionals of a Class of Gaussian Random Fields

“…The KPZ equation is expected to be the universal model for weakly asymmetric interface growth at large scales. In [HQ15], the authors considered continuous microscopic models of the type…”

“…for any even polynomial F and smooth stationary Gaussian random fieldξ. The main result in [HQ15] is that there exists C ε → +∞ such that the rescaled and re-centered whereΨ = ∂ x P * ξ and P is the heat kernel. [HX18b] extended the result to arbitrary even functions F with sufficient regularity and polynomial growth.…”