Abstract
We study the one-dimensional branching Brownian motion starting at the origin and investigate the correlation between the rightmost and leftmost visited sites up to time . At each time step the existing particles in the system either diffuse (with diffusion constant ), die (with rate ), or split into two particles (with rate ). We focus on the regime where these two extreme values and are strongly correlated. We show that at large time , the joint probability distribution function (PDF) of the two extreme points becomes stationary . Our exact results for demonstrate that the correlation between and is nonzero, even in the stationary state. From this joint PDF, we compute exactly the stationary PDF of the (dimensionless) span , which is the distance between the rightmost and leftmost visited sites. This span distribution is characterized by a linear behavior for small spans, with . In the critical case this distribution has a nontrivial power law tail for large spans. On the other hand, in the subcritical case , we show that the span distribution decays exponentially as for large spans, where is a nontrivial function of , which we compute exactly. We show that these asymptotic behaviors carry the signatures of the correlation between and . Finally we verify our results via direct Monte Carlo simulations.
4 More- Received 2 February 2015
DOI:https://doi.org/10.1103/PhysRevE.91.042131
©2015 American Physical Society