Paris Diderot University
Ever since the works of the founding fathers of statistical mechanics, the derivation of the laws of macroscopic transport as the result of the motion of the microscopic components has been a major challenge which remains largely unsolved to this day. I will present a new model that can be seen as a random lattice Lorentz gas and for which a macroscopic diffusion equation can be rigorously derived from the microscopic dynamics. The proof is based on the fact that in high dimension, random walks have a small probability of making loops or intersecting each other when starting sufficiently far apart.
Université Paris VI
Consider a random walk. What can be said about its most visited sites? The study was initiated by Erdos and Revesz in 1984 for random walk on the line. I am going to mention a few conjectures by Erdos and Revesz, and make some elementary discussions when the random walk is on a tree. Joint work with Yueyun Hu.
Beijing Jiaotong University
In this talk, we want to introduce and discuss a new class of processes, web Markov skeleton processes (WMSP), arising from information retrieval on the Web. The framework of WMSP covers various known classes of processes, such as Markov chains and semi-Markov processes; it contains also important new classes of processes, such as mirror semi-Markov processes. We mainly state some applications of WMSPs in computing page importance on the Web.
Consider the speed v(λ) of the λ-biased random walk on Galton-Watson trees. It was proved by Lyons, Pemantle & Peres that the speed v(λ) exists. Recently E. Aidekon gave a nice formula for v(λ) by computing the invariant measure for the walk. It was conjectured that v(λ) is monotone on λ for 0 <λ< m, where m is the mean of offspring. The conjecture is verified for λ very close to m, and very close to 0 respectively, by G. Ben Arous, Y. Hu, S. Olla and O. Zeitouni, G. Ben Arous, A. Fribergh and V. Sidoravicius. The monotonicity problem received many recent interests and has many alternatives. In the same spirit we consider the simple random walk on the infinite cluster of the Bernoulli bond percolation of trees, and investigate the relation between the speed of the simple random walk and the retaining probability p by studying three classes of trees. A sufficient condition is established for Galton-Watson trees.