Large scale limit of interface fluctuation models

Abstract

† and NYU Shanghai ‡ We extend the weak universality of KPZ in Hairer and Quastel [Forum Math. Pi 6 (2018) e3] to weakly asymmetric interface models with general growth mechanisms beyond polynomials. A key new ingredient is a pointwise bound on correlations of trigonometric functions of Gaussians in terms of their polynomial counterparts. This enables us to reduce the problem of a general nonlinearity with sufficient regularity to that of a polynomial.

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

“…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. Similar results have also been obtained in [GP16] for models at stationarity.…”

“…Instead, in [HX18b], the authors expanded F ( √ εΨ ε ) in terms of Fourier transform, developed a procedure in obtaining pointwise correlation bounds on trigonometric functions of Gaussians, and deduced the desired convergence from those bounds.…”

“…In the recent work [FG17], the authors developed different methods based on Malliavin calculus to control similar objects. The methods developed in [FG17] and [HX18b] to treat general nonlinearities are both robust enough to cover both KPZ and Φ 4 3 equations as well as other similar situations.…”

Stationary directed polymers and energy solutions of the Burgers equation

“…Note that complicated non-linearities leading to Burgers equation where considered in the works [29,30] and later in [21] in the context of energy solutions. The works [30] and [21] deal with general examples of weakly asymmetric scaling. Our work is an example of intermediate disorder scaling.…”

Some recent progress in singular stochastic partial differential equations

“…Proving universality theorems in this continuum setting remains an active direction. More recently, [HX18b] generalized the result of [HQ18] by allowing more general function F (not necessarily polynomial); they essentially only need to assume certain decay of the local distributional norm of the Fourier transform of F . In particular, it includes the case F (u) = √ 1 + u 2 as originally considered in [KPZ86].…”