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

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 $β\in [0,β_c)$. 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.

Figures (4)

• Figure 1.1: Graph of the functions $\psi_{\text{typ}}$ and $\psi_{\text{mean}}$ such that, for any $a\in(0,1)$ and up to polynomial factors, $\nu_{\beta, t}([a, 1])$ decays like $\mathrm{e}^{-\psi_{\text{typ}}(\beta) at}$ and $\mathbb{E}\mathopen{}\mathclose{\left[\nu_{\beta, t}([a, 1])\right]$ decays like $\mathrm{e}^{-\psi_{\text{mean}}}(\beta) at}$.
• Figure 1.2: Behaviors of the particles that contribute typically to the overlap distribution.
• Figure 1.3: Graph of the function $v$ such that particles that contribute to $\mathbb{E}\mathopen{}\mathclose{\left[\nu_{\beta, t}([a, 1])\right]$ are "near" $v(\beta)at$ at time $at$.
• Figure 1.4: Behaviors of the particles that contribute to the mean overlap.

Theorems & Definitions (10)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Remark 2.1
• Remark 2.2
• Remark 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• Lemma 2.6