School of Mathematical Sciences

Department Colloquium

Schreiber 006, 12:15

Weizmann Institute

The Fisher-Kolmogorov-Petrovsky-Piscounov equation describes the propagation of a self interacting

Abstract:

wave such as in flame propagation. Probabilistically, for appropriate initial conditions it describes the evolution of

the location of the maximal particle in a system of branching random walks (where the total population size increases

exponentially). That particle system was analyzed by Bramson using a mixture of probabilistic and analytic methods

in the early 80's. In particular, the ``front'' of the propagating wave is delayed (by a logarithmic factor) behind the

stationary, linear speed propagation.

The Gaussian free field is a random Gaussian field that is indexed by points in $R^d$; in the critical dimension d=2,

it represents a random distribution. A discrete analogue of the GFF can be defined on any finite graph. Of special

interest are properties of the maxima of the field, and in particular the fluctuations of the maximum.

After describing a general point of view that allows one to analyze branching particle systems, I will describe newly

observed phase transitions that occur in the case of time-inhomogeneous media. I will then describe recent work

that links the two objects, branching random walks and Gaussian free fields, in the critical dimension $d=2$. In

particular, I will explain how Bramson's work can be adapted in order to show that the maximum of the two

dimensional (discrete) GFF has fluctuations which are of order $1$. The talk is based on joint works with Maury

Bramson, with Jian Ding and with Ming Fang.

Coffee will be served at 12:00 before the lecture

at Schreiber building 006