Kardar-Parisi-Zhang Equation And Its Critical Exponents Through Local Slope-Like Fluctuations

TL;DR

This paper develops a method to determine the roughness exponent and critical exponents of interface growth models, including the KPZ equation, by analyzing local slope fluctuations on discrete lattices across various dimensions.

Contribution

It introduces a novel approach to calculate exact roughness exponents for linear and nonlinear stochastic growth equations using slope-slope couplings and symmetries.

Findings

Exact roughness exponents for KPZ and EW equations in all dimensions.

Method applicable to linear and nonlinear stochastic equations with non-conserved noise.

Derived precise asymptotic exponents for higher-order growth equations.

Abstract

Growth of interfaces during vapor deposition are analyzed on a discrete lattice. Foe a rough surface, relation between the roughness exponent alpha, and corresponding step-step (slope-slope) couplings is obtained in (1+1) and (2+1) dimensions. From the discrete form and the symmetries of the growth problem, the step -step couplings can be determined. Thus alpha can be obtained. The method is applied to Edward-Wilkinson type and Kardar- Parisi -Zhang equations in all the dimensions to obtain exact values of alpha. It is further applied to the fourth order linear and non linear terms. Exact values of roughness coefficients in these cases are obtained. The method is thus applicable to any linear or nonlinear stochastic equation with non conserved noise for obtaining the exact asymptoic exponents.