Overlap, Disorder and Directed Polymers: A Renormalization Group Approach -

TL;DR

This paper employs a Renormalization Group approach to analyze the overlap behavior of directed polymers in a random medium across different dimensions, revealing critical scaling laws and confirming previous numerical results.

Contribution

It introduces a novel RG-based framework to derive the scaling behavior of polymer overlap and mutual repulsion in disordered media, extending understanding across dimensions.

Findings

Overlap vanishes at critical temperature with a specific power law in high dimensions.

The derived formula matches numerical simulations in one dimension.

Determined the scaling exponent for mutual repulsion of two chains.

Abstract

The overlap of a $d+1$ dimensional directed polymer of length $t$ in a random medium is studied using a Renormalization Group approach. In $d>2$ it vanishes at $T_c$ for $t\rightarrow \infty$ as $t^{\Sigma}$ where $\Sigma=[\frac{d-1}{3-2d}]\frac{d}{z}$ and $z$ is the transverse spatial rescaling exponent. The same formula holds in $d=1$ for any finite temperature and it agrees with previous numerical simulations at $d=1$. Among other results we also determine the scaling exponent for mutual repulsion of two chains in the random medium.