Weak convergence of directed polymers to deterministic KPZ at high temperature

TL;DR

This paper proves that high-temperature directed polymers in dimensions three and above, when smoothed and scaled appropriately, converge to a deterministic KPZ equation, revealing a transition from randomness to determinism.

Contribution

It establishes the convergence of high-temperature directed polymer surfaces to the deterministic KPZ equation in dimensions three and higher, under suitable smoothing and scaling.

Findings

Directed polymer surfaces converge to deterministic KPZ at high temperature.

Convergence occurs after smoothing and in a specific scaling limit.

Results extend understanding of the KPZ universality class in higher dimensions.

Abstract

It is shown that when $d\ge 3$, the growing random surface generated by the $(d+1)$-dimensional directed polymer model at sufficiently high temperature, after being smoothed by taking microscopic local averages, converges to a solution of the deterministic KPZ equation in a suitable scaling limit.