Guillaume Barraquand (ENS, Paris)

Diffusion in random environment

Consider a particle hopping on a one dimensional lattice. At each time step, the particle moves by +1 or -1 with probability p or 1-p. It is well known that the trajectory of such particle at large scale is a Brownian motion. What if the probabilities p=p_{x,t} depend on the position x and time t ? Say that the p_{x,t} are independent and uniformly distributed in [0,1]. Then, at large scale, the behaviour is still Brownian. However, the extreme behaviour of such particles radically differ from classical diffusion. The large deviations rather follow scales and statistics characteristic to the Kardar-Parisi-Zhang universality class. I will give a panorama of several results obtained in the last 10 years in this direction, in the mathematics and physics literature.