Convergence of the law of the Environment Seen by the Particle for Directed Polymers in Random Media in the $L^2$ region

TL;DR

This paper proves the convergence of the environment's law viewed by a directed polymer in an i.i.d. environment within the $L^2$ region, using concentration inequalities and local limit theorems.

Contribution

It establishes the $L^1$ convergence of the environment law for directed polymers, improving existing concentration bounds for bounded environments.

Findings

Proves convergence of the environment law in the $L^2$ region.

Establishes a lower tail concentration inequality for the partition function.

Demonstrates $L^1$ convergence of the environment density.

Abstract

We consider the model of Directed Polymers in an i.i.d. gaussian or bounded Environment in the $L^2$ region. We prove the convergence of the law of the environment seen by the particle. As a main technical step, we establish a lower tail concentration inequality for the partition function for bounded environments. Our proof is based on arguments developed by Talagrand in the context of the Hopfield Model. This improves in some sense a concentration inequality obtained by Carmona and Hu for gaussian environments. We use this and a Local Limit Theorem to prove the $L^1$ convergence of the density of the law of the environment seen by the particle with respect to the product measure.