Some recent progress on the periodic KPZ equation

TL;DR

This paper reviews recent advances in understanding the periodic KPZ equation, focusing on limit theorems for interface height and winding number, with novel explicit formulas and probabilistic laws derived.

Contribution

It introduces new explicit expressions for the homogenization corrector and establishes the law of the iterated logarithm for the height function.

Findings

Central limit theorems for height and winding number

Effective variances expressed via independent Brownian bridges

Law of the iterated logarithm for the height function

Abstract

We review recent progress on the study of the Kardar-Parisi-Zhang (KPZ) equation in a periodic setting, which describes the random growth of an interface in a cylindrical geometry. The main results include central limit theorems for the height of the interface and the winding number of the directed polymer in a periodic random environment. We present two different approaches for each result, utilizing either a homogenization argument or tools from Malliavin calculus. A surprising finding in the case of a $1+1$ spacetime white noise is that the effective variances for both the height and the winding number can be expressed in terms of independent Brownian bridges. Additionally, we present two new results: (i) the explicit expression of the corrector used in the homogenization argument, and (ii) the law of the iterated logarithm for the height function.