The Kardar-Parisi-Zhang exponents for the $2+1$ dimensions

TL;DR

This paper introduces new analytical methods to determine the growth exponents of the KPZ equation in 2+1 dimensions, aligning with experimental and simulation data, and explores the possibility of no upper critical dimension.

Contribution

It presents a novel geometric analytical approach to calculate KPZ exponents in 2+1 dimensions, extending understanding beyond the known 1+1 case.

Findings

KPZ exponents in 2+1 dimensions match experimental and simulation results.

Proposes a geometric method linking exponents to fractal dimensions.

Suggests no upper critical dimension exists for KPZ universality.

Abstract

The Kardar-Parisi-Zhang (KPZ) equation has been connected to a large number of important stochastic processes in physics, chemistry and growth phenomena, ranging from classical to quantum physics. The central quest in this field is the search for ever more precise universal growth exponents. Notably, exact growth exponents are only known for $1+1$ dimensions. In this work, we present physical and geometric analytical methods that directly associate these exponents to the fractal dimension of the rough interface. Based on this, we determine the growth exponents for the $2+1$ dimensions, which are in agreement with the results of thin films experiments and precise simulations. We also make a first step towards a solution in $d+1$ dimensions, where our results suggest the inexistence of an upper critical dimension.