Pascal Maillard

 Near the volcano Paricutín, Mexico

Postal address Département de Mathématiques
Bâtiment 425, Bureau 032
Faculté des Sciences d'Orsay
Université Paris-Sud
91405 Orsay Cedex
Email pascal.maillard at u-psud.fr
Telephone +33 1 69 15 57 37 (France)
Visit my Mendeley profile

I am Maître de Conférence (Assistant Professor) at Université Paris-Sud. Previously, I was a post-doctoral fellow at the Weizmann Institute of Science in Israel, my host was Ofer Zeitouni. I did my PhD at the Université Pierre et Marie Curie (UPMC) in Paris, France, under the supervision of Zhan Shi.

Curriculum Vitae (French, version of Jan 4, 2014)

Curriculum Vitae (version of December 8, 2012)

A picture taken during a talk given at the European parliament

Research interests

Probability theory.

I work or have worked on the following topics, which are in fact often interrelated:

  • branching processes, in particular branching random walks and branching Brownian motion
  • self-similar or fractal structures/processes (in particular Gaussian multiplicative chaos or Liouville quantum gravity), heavy tails
  • limiting behavior of complex stochastic processes (particle systems, stochastic evolution equations..)
  • random walk in random environment

Publications and preprints

Note: list might be incomplete. Please check on the arXiv for a complete list.

  • with Rémi Rhodes, Vincent Vargas and Ofer Zeitouni, Liouville heat kernel: regularity and bounds arXiv:1406.0491, submitted
  • with Olivier Hénard, On trees invariant under edge contraction arXiv:1403.5491
  • with Elliot Paquette, Choices and intervals arXiv:1402.3931, submitted
  • with Jean Bérard, The limiting process of N-particle branching random walk with polynomial tails, Electronic Journal of Probability (19), no. 22, 1-17 (2014) Journal Preprint
  • with Itai Benjamini, Point-to-point distance in first passage percolation on (Tree) x Z, Geometric Aspects of Functional Analysis, Israel Seminar (GAFA) 2011-2013, Klartag, Bo'az, Milman, Emanuel (Eds.), Lecture Notes in Mathematics, Vol. 2116, Springer (2014) SpringerLink Preprint
  • with Ofer Zeitouni, Slowdown in branching Brownian motion with inhomogeneous variance arXiv:1307.3583, submitted.
  • Speed and fluctuations of N-particle branching Brownian motion with spatial selection, arXiv:1304.0562, submitted.
  • with Ofer Zeitouni, Performance of the Metropolis algorithm on a disordered tree: the Einstein relation, Annals of Applied Probability (24), no. 5, 2070-2090 (2014) Journal Preprint
  • A note on stable point processes occurring in branching Brownian motion, Electronic Communications in Probability, no. 5, 1-9 (2013) Journal
  • The number of absorbed individuals in branching Brownian motion with a barrier, Ann. I.H.P. Prob. Stat., vol. 49, no. 2, 428-455 (2013) Journal, Preprint
  • with Robert Görke, Andrea Schumm, Christian Staudt and Dorothea Wagner, Dynamic graph clustering combining modularity and smoothness, J. Exp. Algorithmics, vol. 18, no. 1.5 (2013) Journal Preprint
  • with Robert Görke, Christian Staudt and Dorothea Wagner, Modularity-driven clustering of dynamic graphs, Experimental Algorithms, vol. 6049, 436-448, P. Festa (Ed.), Springer Berlin / Heidelberg (2010) Preprint

PhD thesis

  • Branching Brownian motion with selection, Ph.D. thesis, Université Pierre et Marie Curie (2012) arXiv:1210.3500.

Unpublished notes

  • Branching Brownian motion with selection of the N right-most particles: An approximate model, arXiv:1112.0266 (new version published as “Speed and fluctuations of N-particle branching Brownian motion with spatial selection”)
  • A characterisation of superposable random measures, arXiv:1102.1888v1 (new version published as “A note on stable point processes occurring in branching Brownian motion”)

Pretty pictures

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.

 
users/maillard/index.txt · Last modified: 2014/12/18 11:31 by maillard
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