The Intermediate Disorder Regime for Brownian Directed Polymers in Poisson Environment

TL;DR

This paper studies the behavior of Brownian directed polymers in a Poisson environment under intermediate disorder, showing convergence of the partition function to the stochastic heat equation with Gaussian noise.

Contribution

It establishes a convergence result for the partition function in the intermediate disorder regime using Wiener-Ito chaos expansion, bridging strong and weak disorder phases.

Findings

Partition function converges in law to the stochastic heat equation solution.

Uses Wiener-Ito chaos expansion as a key tool.

Provides a natural Poissonian setting for the model.

Abstract

We consider the Brownian directed polymer in Poissonian environment in dimension 1+1, under the so-called intermediate disorder regime, which is a crossover regime between the strong and weak disorder regions. We show that, under a diffusive scaling involving different parameters of the system, the renormalized point-to-point partition function of the polymer converges in law to the solution of the stochastic heat equation with Gaussian multiplicative noise. The Poissonian environment provides a natural setting and strong tools, such as the Wiener-Ito chaos expansion, which, applied to the partition function, is the basic ingredient of the proof.