KPZ equation limit of random walks in random environments
Abstract
We show that under a certain moderate deviation scaling, the multiplicativenoise 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 spacetime. 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2023
 DOI:
 10.48550/arXiv.2311.09151
 arXiv:
 arXiv:2311.09151
 Bibcode:
 2023arXiv231109151D
 Keywords:

 Mathematics  Probability;
 Mathematical Physics
 EPrint:
 62 pages, 1 figure