Maxima and entropic repulsion of Gaussian free field: on and beyond Z^d

Joe P. Chen (University of Connecticut)

Abstract :

The Gaussian free field on an (infinite) graph is a centered Gaussian process whose covariance is the Green's function for simple random walk on the graph. As a model random surface which favors minimization of "gradient", one would like to study its stochastic geometry, such as its maximal height. A related problem, known as "entropic repulsion," concerns the height of the field conditioned upon it being positive everywhere, as if a "hard wall" were imposed at zero height.

In this talk, I will present quantitative results on the maxima and the entropic repulsion in the Z^d and non-Z^d settings (a Sierpinski carpet graph being one of the latter examples), and discuss similarities (and differences) between these two problems. A distinction will be made between graphs which support strongly recurrent random walk and those which support transient random walk.