Seminars & Discussions
Week beginning 14 September 2014
Wednesday 17 Sep 14 - 11:30am
Correlated Extreme Values in Branching Brownian Motion
Kabir Ramola (LPTMS, France)
We investigate one dimensional branching Brownian motion (BBM) 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 which individually behave diffusively. 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 particles. 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~(D/|b - d|)^(1/2). 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. We also extend our study to the spatial extent of this process (the distance between the rightmost and leftmost visited sites). We derive exact results for the PDF of this spatial extent for the cases b <= d where the two extreme points are strongly correlated. Once again we find an emergent power law at the critical point with a correlation length ~(D//|b - d|)^(1/2) away from criticality. Direct Monte Carlo simulations confirm our predictions.
Monday 22 Sep 14 - 1:00pm - Room 2511
Li Tao (University of Edinbugh)