Upper critical dimension of the KPZ equation

TL;DR

This paper investigates the upper critical dimension of the KPZ equation through numerical analysis of the Directed Polymer model in 1+4 dimensions, revealing that 4+1 is not the upper critical dimension in weak disorder conditions.

Contribution

The study provides new numerical evidence that challenges the assumption that 4+1 is the upper critical dimension for the KPZ equation.

Findings

Confirmed the meandering exponent zeta ≈ 0.57 in weak disorder

Observed crossover behavior from Min-Max to weak disorder regimes

Established that 1+4 is not the upper critical dimension for KPZ

Abstract

Numerical results for the Directed Polymer model in 1+4 dimensions in various types of disorder are presented. The results are obtained for system size considerably larger than that considered previously. For the extreme strong disorder case (Min-Max system), associated with the Directed Percolation model, the expected value of the meandering exponent, zeta = 0.5 is clearly revealed, with very week finite size effects. For the week disorder case, associated with the KPZ equation, finite size effects are stronger, but the value of seta is clearly seen in the vicinity of 0.57. In systems with "strong disorder" it is expected that the system will cross over sharply from Min-Max behavior at short chains to weak disorder behavior at long chains. This is indeed what we find. These results indicate that 1+4 is not the Upper Critical Dimension (UCD) in the week disorder case, and thus 4+1 does not seem to be the upper critical dimension for the KPZ equation.