Localization length of the $1+1$ continuum directed random polymer

TL;DR

This paper investigates the localization length of the 1+1 continuum directed polymer, proving its distribution converges and confirming a physics-predicted power law decay using recent mathematical results.

Contribution

It provides the first rigorous proof of the distribution and decay behavior of the localization length in the continuum directed polymer model.

Findings

Localization length converges in distribution in the thermodynamic limit.

Explicit density formula for the limiting distribution is derived.

Density exhibits a 3/2-power law decay, confirming physics predictions.

Abstract

In this paper, we study the localization length of the $1+1$ continuum directed polymer, defined as the distance between the endpoints of two paths sampled independently from the quenched polymer measure. We show that the localization length converges in distribution in the thermodynamic limit, and derive an explicit density formula of the limiting distribution. As a consequence, we prove the $\tfrac32$-power law decay of the density, confirming the physics prediction of Hwa-Fisher \cite{fisher}. Our proof uses the recent result of Das-Zhu \cite{daszhu}.