# Kardar-Parisi-Zhang Equation with temporally correlated noise: a non-perturbative renormalization group approach

## TL;DR

This paper examines how temporal correlations in noise affect the universality class of the KPZ equation, showing that short-range correlations restore the original KPZ behavior, while long-range correlations lead to new fixed points and anomalous scaling.

## Contribution

The study applies non-perturbative renormalization group methods to analyze the KPZ equation with temporally correlated noise, revealing the stability of the SR-KPZ fixed point and identifying a new long-range fixed point for power-law correlations.

## Key findings

- Short-range noise restores KPZ universality at large scales.
- Long-range noise with exponent above a critical value leads to a new fixed point.
- Anomalous scaling emerges in the long-range phase in 1+1 dimensions.

## Abstract

We investigate the universal behavior of the Kardar-Parisi-Zhang (KPZ) equation with temporally correlated noise. The presence of time correlations in the microscopic noise breaks the statistical tilt symmetry, or Galilean invariance, of the original KPZ equation with delta-correlated noise (denoted SR-KPZ). Thus it is not clear whether the KPZ universality class is preserved in this case. Conflicting results exist in the literature, some advocating that it is destroyed even in the limit of infinitesimal temporal correlations, while others find that it persists up to a critical range of such correlations. Using non-perturbative and functional renormalization group techniques, we study the influence of two types of temporal correlators of the noise: a short-range one with a typical time-scale $\tau$, and a power-law one with a varying exponent $\theta$. We show that for the short-range noise with any finite $\tau$, the symmetries (the Galilean symmetry, and the time-reversal one in $1+1$ dimension) are dynamically restored at large scales, such that the long-distance and long-time properties are governed by the SR-KPZ fixed point. In the presence of a power-law noise, we find that the SR-KPZ fixed point is still stable for $\theta$ below a critical value $\theta_{\textrm{th}}$, in accordance with previous renormalization group results, while a long-range fixed point controls the critical scaling for $\theta>\theta_{\textrm{th}}$, and we evaluate the $\theta$-dependent critical exponents at this long-range fixed point, in both $1+1$ and $2+1$ dimensions. While the results in $1+1$ dimension can be compared with previous studies, no other prediction was available in $2+1$ dimension. We finally report in $1+1$ dimension the emergence of anomalous scaling in the long-range phase.