Nonperturbative renormalization group for the stationary Kardar-Parisi-Zhang equation: scaling functions and amplitude ratios in 1+1, 2+1 and 3+1 dimensions

TL;DR

This paper applies a nonperturbative renormalization group method to study the stationary KPZ equation across dimensions 1 to 3, deriving critical exponents, scaling functions, and amplitude ratios with high accuracy and computational efficiency.

Contribution

It introduces a simplified second-order NPRG approximation that accurately computes scaling functions and amplitude ratios for the KPZ equation in multiple dimensions.

Findings

High-accuracy scaling functions in 1+1 dimensions

Scaling functions and ratios in 2+1 and 3+1 dimensions

Reliable approximation up to dimension 3.5

Abstract

We investigate the strong-coupling regime of the stationary Kardar-Parisi-Zhang equation for interfaces growing on a substrate of dimension d=1, 2, and 3 using a nonperturbative renormalization group (NPRG) approach. We compute critical exponents, correlation and response functions, extract the related scaling functions and calculate universal amplitude ratios. We work with a simplified implementation of the second-order (in the response field) approximation proposed in a previous work [PRE 84, 061128 (2011) and Erratum 86, 019904 (2012)], which greatly simplifies the frequency sector of the NPRG flow equations, while keeping a nontrivial frequency dependence for the 2-point functions. The one-dimensional scaling function obtained within this approach compares very accurately with the scaling function obtained from the full second-order NPRG equations and with the exact scaling function. Furthermore, the approach is easily applicable to higher dimensions and we provide scaling functions and amplitude ratios in d=2 and d=3. We argue that our ansatz is reliable up to d \simeq 3.5.