Fluctuations of partition functions of directed polymers in weak disorder beyond the $L^2$-phase

TL;DR

This paper investigates the behavior of directed polymers in a weak disorder regime beyond the $L^2$-phase, focusing on the rate of homogenization of the associated martingale field in a bounded environment.

Contribution

It establishes the convergence rate of spatial averages of the polymer martingale field in a regime where traditional $L^2$ methods do not apply.

Findings

Spatial averages over sets of diameter $n^{1/2}$ converge to zero.

The convergence rate is explicitly characterized as a function of inverse temperature $eta$.

The results extend understanding of polymer fluctuations beyond the $L^2$-phase.

Abstract

We study the directed polymer model in a bounded environment in weak disorder but without $L^2$-boundedness, specifically the speed of homogenization for the field $(W_n^{0,x})_{x\in\mathbb Z^d}$, where $W_n^{0,x}$ denotes the associated martingale for the polymer starting from $x$. We show that a suitably re-centered spatial average over a set of diameter $n^{1/2}$ convergence to zero at rate $n^{-\xi+o(1)}$, where the exponent is an explicit function of the inverse temperature $\beta$.