2015/01/31 by Kabir Ramola, Satya N. Majumdar, Grégory Schehr +1 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Branching (polymer chemistry) #Brownian motion #Combinatorics #Diffusion and Search Dynamics #Dimensionless quantity #Distribution (mathematics) #Mathematical analysis #Mathematics #Physics #Power law #Quantum mechanics #Statistical physics #Statistics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.stat-mech #math-ph #math.MP #math.PR
paper · pdf · doi:10.1103/physreve.91.042131
published as Phys. Rev. E 91, 042131 (2015) · 15 pages (2 columns), 11 figures, slightly revised version
arxiv created 2015/03/13 · openalex publication_date 2015/04/23 · arxiv updated 2015/04/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study the one-dimensional branching Brownian motion starting at the origin and investigate the correlation between the rightmost (Xmax\ensuremath≥0) and leftmost (Xmin\ensuremath≤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\ensuremath≤a where these two extreme values Xmax and Xmin 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\ensuremath→\ensuremath∞)\ensuremath→p(X,Y). Our exact results for p(X,Y) demonstrate that the correlation between Xmax and Xmin is nonzero, even in the stationary state. From this joint PDF, we compute exactly the stationary PDF p(\ensuremathζ) of the (dimensionless) span \ensuremathζ=(Xmax\ensuremath-Xmin)/√(D/b), which is the distance between the rightmost and leftmost visited sites. This span distribution is characterized by a linear behavior p(\ensuremathζ)\ensuremath∼(1)/(2)(1+\ensuremathΔ)\ensuremathζ for small spans, with \ensuremathΔ=((a)/(b)\ensuremath-1). In the critical case (\ensuremathΔ=0) this distribution has a nontrivial power law tail p(\ensuremathζ)\ensuremath∼8\ensuremathπ√(3)/\ensuremathζ3 for large spans. On the other hand, in the subcritical case (\ensuremathΔ>0), we show that the span distribution decays exponentially as p(\ensuremathζ)\ensuremath∼(A2/2)\ensuremathζexp(\ensuremath-√\ensuremathΔ\ensuremathζ) for large spans, where A is a nontrivial function of \ensuremathΔ, which we compute exactly. We show that these asymptotic behaviors carry the signatures of the correlation between Xmax and Xmin. Finally we verify our results via direct Monte Carlo simulations.