We study the one-dimensional branching Brownian motion starting at the origin and investigate the correlation between the rightmost $\left(X_{max}\ge 0\right)$ and leftmost $\left(X_{min}\le 0\right)$ 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 $p\left(\zeta \right)$ 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 $p\left(\zeta \right)\sim \frac{1}{2}(1+\Delta )\zeta$ for small spans, with $\Delta =(\frac{a}{b}-1)$. In the critical case $\left(\Delta =0\right)$ this distribution has a nontrivial power law tail $p\left(\zeta \right)\sim 8\pi \sqrt{3}/{\zeta }^3$ for large spans. On the other hand, in the subcritical case $\left(\Delta >0\right)$, we show that the span distribution decays exponentially as $p\left(\zeta \right)\sim (A^2/2)\zeta exp(-\sqrt{\Delta }\zeta )$ for large spans, where $A$ is a nontrivial 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.

• Received 2 February 2015

DOI:https://doi.org/10.1103/PhysRevE.91.042131