On the genealogy of branching random walks and of directed polymers

TL;DR

This paper investigates the finite size corrections to the overlap distribution in directed polymers, revealing universal properties and connecting these findings to genealogical structures in branching processes.

Contribution

It provides an explicit computation of the universal finite size correction to the overlap distribution in directed polymers, linking it to genealogical properties of branching processes.

Findings

Finite size correction to overlap distribution is universal.

Explicit formula for the correction is derived.

Results connect directed polymers to genealogical properties of branching processes.

Abstract

It is well known that the mean field theory of directed polymers in a random medium exhibits replica symmetry breaking with a distribution of overlaps which consists of two delta functions. Here we show that the leading finite size correction to this distribution of overlaps has a universal character which can be computed explicitly. Our results can also be interpreted as genealogical properties of branching Brownian motion or of branching random walks.