I will survey recent work with Alexander Fribergh (Montreal) and Manuel Cabezas (Santiago) on the problem of “the ant in the labyrinth”, as coined by De Gennes in 1976, i.e random walks on critical structures. We consider here the case of random walks on critical high dimensional critical clusters. The first question is to compute the dynamical scaling exponent a.k.a the spectral dimension. This value was conjectured in the 80s by Alexander and Orbach and proven in 2008 by Kozma and Nachmias.
We go further and want to understand really the scaling limit of this process. We conjecture that this scaling limit is a process we call the BISE, i.e the Brownian motion on the Integrated SuperBrownian Excursion.
I will begin by explaining how this limit process is a finite dimensional embedding of the Brownian Motion on Aldous’ CRT (continuous random tree). I will also show how this process can be related to the class of Spatially Subordinated Brownian Motions (SSBMs) as introduced as limit of trap randon walks in the work with Cabezas, Cerny, and Royfman (2015). I will then briefly cover our results about convergence given in a series of 4 recent papers.
Professor Ben Arous works on probability theory (stochastic analysis, large deviations, random media and random matrices) and its connections with other domains of mathematics (partial differential equations, dynamical systems), physics (statistical mechanics of disordered media), or industrial applications. He is mainly interested in the time evolution of complex systems, and the universal aspects of their long time behavior and of their slow relaxation to equilibrium, in particular how complexity and disorder imply aging. He is a Fellow of the Institute of Mathematical Statistics (as of August 2011) and an elected member of the International Statistical Institute.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai