Intermediate disorder directed polymers and the multi-layer extension of the stochastic heat equation

TL;DR

This paper extends the understanding of directed polymers in disordered environments by proving that multiple non-intersecting random walks under intermediate disorder scaling converge to a multi-layer stochastic heat equation, generalizing prior single-walk results.

Contribution

It establishes the convergence of multi-layer directed polymer models to the multi-layer stochastic heat equation, broadening the scope of intermediate disorder universality.

Findings

Convergence of multi-layer polymer partition functions to the multi-layer SHE.

Extension of single-walk results to multiple non-intersecting walks.

Validation of the multi-layer SHE as a universal limit in this setting.

Abstract

We consider directed polymer models involving multiple non-intersecting random walks moving through a space-time disordered environment in one spatial dimension. For a single random walk, Alberts, Khanin and Quastel proved that under intermediate disorder scaling (in which time and space are scaled diffusively, and the strength of the environment is scaled to zero in a critical manner) the polymer partition function converges to the solution to the stochastic heat equation with multiplicative white noise. In this paper we prove the analogous result for multiple non-intersecting random walks started and ended grouped together. The limiting object now is the multi-layer extension of the stochastic heat equation introduced by O'Connell and Warren.