Stability of weak disorder phase for directed polymer with applications to limit theorems

TL;DR

This paper investigates the stability of the weak disorder phase in directed polymer models, proving its persistence under small biases, and provides new proofs and improvements for limit theorems and large deviation results.

Contribution

It establishes the stability of weak disorder under perturbations and offers new proofs for the CLT and large deviation principles in directed polymer models.

Findings

Weak disorder persists under small bias perturbations.

New proof of the CLT in the weak disorder phase.

Large deviation rate function matches that of the underlying random walk.

Abstract

We study the directed polymer model in a bounded environment with bond disorder and show that, in the interior of the weak disorder phase, weak disorder continues to hold upon perturbation by a small bias. Using this stability result, we give a new proof for the central limit theorem (CLT) in probability for the directed polymer model in the interior of the weak disorder phase. We also show that the large deviation rate function agrees with that of the underlying random walk. For the Brownian polymer model, we improve the convergence in the CLT to almost sure convergence in the whole weak disorder phase. The main technical tools are a new moment bound from \cite{J21_1} and a quantitative comparison between the associated martingales at different inverse temperatures.