Analys- och sannolikhetsseminarium

Abstrakt finns enbart på engelska: The Gaussian Free Field (GFF) was introduced by physicists and is a central object in probability theory which has been studied extensively and which arises in many different contexts. It is essentially a Gaussian random function on a domain D in the plane with covariance structure given by the Green's function and as such is a generalization of Brownian motion where time becomes 2-dimensional. We will obtain it from random walk by (1) replacing 1-d time by 2-d time obtaining the discrete Gaussian free field and then (2) taking a "scaling limit" like random walk going to Brownian motion. However, things are more subtle than for Brownian motion since the discrete GFFs do not converge in the space of functions due to their rapid fluctuations. Instead, one needs to embed the set of functions into the larger space of so-called Schwartz distributions to obtain convergence. In particular, the GFF, in the end, can be defined as a random (Gaussian) Schwartz distribution whose covariance structure is determined by the Green's function of D (or as a random element in what is called a negative Sobolov space).

This seminar will also serve as a pre-seminar for next week's seminar by Lukas Schoug.