Correlated Extreme Values in Branching Brownian Motion
Kabir Ramola, LPTMS
We investigate one dimensional branching Brownian motion in which at each time step particles either diffuse (with diffusion constant D), die (with rate d), or split into two particles (with rate b). When the birth rate exceeds the death rate (b > d), there is an exponential proliferation of particles and the process is explosive. When b < d, the process eventually dies. At the critical point (b = d) this system is characterized by a fluctuating number of particles with a constant average. Quite remarkably, although the individual positions of these particles have a non-trivial finite time behaviour, the average distances between successive particles (the gaps) become stationary at large times, implying strong correlations between them. We compute the probability distribution functions (PDFs) of these gaps, by conditioning the system to have a fixed number of particles at a given time t. At large times we show that these PDFs are characterized by a power law tail ~1/g^3 (for large gaps g) at the critical point and ~exp(- g/c) otherwise, with a correlation length c~\sqrt(D/|b – d|). We discuss the emergence of these two length scales, the dominant overall length scale of the individual positions, and the sub-dominant gap length scale in this system. Direct Monte Carlo simulations confirm our predictions.