Kardar-Parisi-Zhang universality class in ($d+1$)-dimensions

TL;DR

This paper proposes an analytical formula for the KPZ growth exponents in any dimension, confirming it through extensive simulations and RG calculations up to 15 dimensions, advancing understanding of the universality class.

Contribution

It introduces a new analytical expression for KPZ exponents in arbitrary dimensions, supported by numerical and RG evidence, addressing a key open problem in statistical physics.

Findings

Analytical formula for KPZ exponents: rac{7}{8d+13}

Excellent agreement with literature estimates

Validation through Monte Carlo simulations and RG calculations up to 15 dimensions

Abstract

The determination of the exact exponents of the KPZ class in any substrate dimension $d$ is one of the most important open issues in Statistical Physics. Based on the behavior of the dimensional variation of some exact exponent differences for other growth equations, I find here that the KPZ growth exponents (related to the temporal scaling of the fluctuations) are given by $\beta_d = \frac{7}{8d+13}$. These exponents present an excellent agreement with the most accurate estimates for them in the literature. Moreover, they are confirmed here through extensive Monte Carlo simulations of discrete growth models and real space renormalization group (RG) calculations for directed polymers in random media (DPRM), up to $d=15$. The left-tail exponents of the probability density functions for the DPRM energy provide another striking verification of the analytical result above.