Abstract:
The continuum Gaussian free field (GFF) is a natural model of generalized Gaussian field satisfying a domain Markov property. In dimension 2 it is conformally invariant in law. Since the works of Kurt Symanzik and Evgeniy Dynkin, functionals of the GFF were known to be related to Brownian sample paths. In dimension 2 we are able to push this connection further. Despite the GFF not being pointwise defined, we are able to construct certain level sets of the GFF, which have zero Lebesgue measure, but on which the field is non-trivial. We further show that these level sets have same law as clusters in a Poisson point process of Brownian loops and Brownian boundary to boundary excursions.
Biography:
Titus Lupu is now a CNRS researcher at Sorbonne University, Paris. He defended his Ph.D. in 2015 under the direction of Professor Yves Le Jan, and afterwards was Post-doc at ETH Zurich, in Wendelin Werner's group. His domains of interest are the Gaussian free field, isomorphism theorems, conformally invariant processes, determinantal point processes, self-interacting random walks and diffusions.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai