Spatial Extent of Branching Brownian Motion

Abstract

We study the one dimensional branching Brownian motion starting at the origin and investigate the correlation between the rightmost ($X_{\max }\ge 0$) and leftmost ($X_{\min }\le 0$) 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 $b\le a$ 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.

15 pages (2 columns), 11 figures, slightly revised version

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