Multiplicative SHE limit of random walks in space-time random environments

TL;DR

This paper demonstrates that under certain scaling, the fluctuations of a 1D random walk in space-time random environments converge to a multiplicative stochastic heat equation, revealing a nontrivial noise coefficient and independent limiting noise.

Contribution

It establishes a novel limit theorem for random walk densities in random environments, showing convergence to SHE without chaos expansion and identifying the nature of the limiting noise.

Findings

The quenched density solves a discrete SPDE resembling SHE.

The limit involves a nontrivial noise coefficient.

Independent noise emerges in the limit, not converging to the original driving noise.

Abstract

We show that under a certain moderate deviation scaling, the multiplicative-noise stochastic heat equation (SHE) arises as the fluctuations of the quenched density of a 1D random walk whose transition probabilities are iid [0,1]-valued random variables. In contrast to the case of directed polymers in the intermediate disorder regime, the variance of our weights is fixed rather than vanishing under the diffusive rescaling of space-time. Consequently, taking a naive limit of the chaos expansion fails for this model, and a nontrivial noise coefficient is observed in the limit. Rather than using chaos techniques, our proof instead uses the fact that in this regime the quenched density solves a discrete SPDE which resembles the SHE. As a byproduct of our techniques, it is shown that independent noise is generated in the limit, in the sense that the prelimiting noise field does not converge to the driving noise of the limiting SPDE.