School of Mathematical Sciences
Monday, January 14, 2013
Schreiber 006, 12:15
From KPP equations to the maxima of Gaussian free fields
Abstract: The Fisher-Kolmogorov-Petrovsky-Piscounov equation describes the propagation of a self interacting
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