|Postal address|| Faculty of Mathematics and Computer Science
The Weizmann Institute of Science
Rehovot 76100, ISRAEL
|pascal.maillard at weizmann.ac.il|
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I am currently a post-doctoral fellow at the Weizmann Institute of Science in Israel, my tutor there is Ofer Zeitouni. I did my PhD at the Université Pierre et Marie Curie (UPMC) in Paris, France, under the supervision of Zhan Shi. My research focuses on branching random walks and branching Brownian motion, in particular on the behaviour of these systems under selection and the distribution of their extremal particles.
Here are some pictures and videos of simulations of a branching random walk in two dimensions. At every step a particle branches into two particles with probability p and every particle jumps to one of the directions NE,NW,SW,SE with equal probability. The program uses a numerical trick described in Brunet and Derrida (1999), Microscopic models of traveling wave equations, Computer Physics Communications, 1999 121-122, 376-381 to simulate huge numbers of particles (10^300 and more): The number of particles on a specific site that branch or that jump to another given site is simply a binomial variable. When the number of particles is relatively small (say < 10^9), we use the binomial number generator from the Boost C++ libraries based on the BTRD algorithm, when the number of particles is bigger, we simply approximate this variable by a Gaussian.
p = 0.05, 3000 steps. See the video here (2 steps per image).
Another video, with p = 0.05, 3000 steps, 2 steps per image. Here you see nice irregularities of the border, that eventually stabilize.
Particles in the videos are represented by black points, the more particles, the darker the point, the saturation depending linearly on the logarithm of the number of particles. The green curve is the linear speed of the process, i.e. the first-order approximation, as described in [Biggins, J. D. (1978). The Asymptotic Shape of the Branching Random Walk. Advances in Applied Probability, 10(1), 62. doi: 10.2307/1426719] while the red curve is the second-order approximation, i.e. with the logarithmic correction term, whose existence was first proven in [Bramson, M. D. (1978). Maximal displacement of branching brownian motion. Communications on Pure and Applied Mathematics, 31(5), 531-581. doi: 10.1002/cpa.3160310502] for branching Brownian motion, and in [Hu, Y., & Shi, Z. (2009). Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. The Annals of Probability, 37(2), 742-789. doi: 10.1214/08-AOP419] for the branching random walk.
p = 1, 1000 steps. See the video here (2 steps per image).
While with small p, the shape of the branching random walk looks like a circle, with bigger p one clearly sees that it actually isn't a circle. Indeed, the rate function of the walk is the function
I(x,y) = H(x) + H(y) - log(1+p) , with H(t) = -(t+1/2) log(t+1/2) - (1/2-t) log(1/2-t)
and the asymptotic shape is the (convex) set of points (x,y) with I(x,y)<0. For p >= 3, this shape is actually a rectangle.