Spatial Extent of Branching Brownian Motion

TL;DR

This paper analyzes the joint distribution of the maximum and minimum positions in a one-dimensional branching Brownian motion, revealing their strong correlation and deriving the span distribution's exact form in different regimes.

Contribution

The study provides the first exact characterization of the joint PDF of extremal positions and the span distribution in branching Brownian motion, highlighting the correlation effects.

Findings

Joint PDF of extremal positions becomes stationary at large time

Span distribution exhibits linear, power-law, or exponential decay depending on parameters

Correlation between maximum and minimum persists in the stationary state

Abstract

We study the one dimensional branching Brownian motion starting at the origin and investigate the correlation between the rightmost ($X_{\max}\geq 0$) and leftmost ($X_{\min} \leq 0$) visited sites up to time $t$. At each time step the existing particles in the system either diffuse (with diffusion constant $D$), die (with rate $a$) or split into two particles (with rate $b$). We focus on the regime $b \leq a$ where these two extreme values $X_{\max}$ and $X_{\min}$ are strongly correlated. We show that at large time $t$, the joint probability distribution function (PDF) of the two extreme points becomes stationary $P(X,Y,t \to \infty) \to p(X,Y)$. Our exact results for $p(X,Y)$ demonstrate that the correlation between $X_{\max}$ and $X_{\min}$ is nonzero, even in the stationary state. From this joint PDF, we compute exactly the stationary PDF $p(\zeta)$ of the (dimensionless) span $\zeta = {(X_{\max} - X_{\min})}/{\sqrt{D/b}}$, which is the distance between the rightmost and leftmost visited sites. This span distribution is characterized by a linear behavior ${p}(\zeta) \sim \frac{1}{2} \left(1 + \Delta \right) \zeta$ for small spans, with $\Delta = \left(\frac{a}{b} -1\right)$. In the critical case ($\Delta = 0$) this distribution has a non-trivial power law tail ${p}(\zeta) \sim 8 \pi \sqrt{3} /\zeta^3$ for large spans. On the other hand, in the subcritical case ($\Delta > 0$), we show that the span distribution decays exponentially as ${p}(\zeta) \sim (A^2/2) \zeta \exp \left(- \sqrt{\Delta}~\zeta\right)$ for large spans, where $A$ is a non-trivial function of $\Delta$ which we compute exactly. We show that these asymptotic behaviors carry the signatures of the correlation between $X_{\max}$ and $X_{\min}$. Finally we verify our results via direct Monte Carlo simulations.