Asymptotics of the overlap distribution of branching Brownian motion at high temperature

TL;DR

This paper analyzes the decay rate of overlap probabilities in branching Brownian motion at high temperature, revealing different thresholds in conditioned and unconditioned cases.

Contribution

It provides a detailed study of the asymptotic decay rates of overlap probabilities across the entire subcritical temperature phase, highlighting distinct thresholds.

Findings

Overlap probability decays to zero at high temperature.

Different thresholds for overlap probability in conditioned vs. unconditioned cases.

Identification of two sub-phases within the inverse temperature range.

Abstract

At high temperature, the overlap of two particles chosen independently according to the Gibbs measure of the branching Brownian motion converges to zero as time goes to infinity. We investigate the precise decay rate of the probability to obtain an overlap greater than $a$, for some $a>0$, in the whole subcritical phase of inverse temperatures $\beta \in [0,\beta_c)$. Moreover, we study this probability both conditionally on the branching Brownian motion and non-conditionally. Two sub-phases of inverse temperatures appear, but surprisingly the threshold is not the same in both cases.