Universität Duisburg-Essen
Fachbereich Mathematik, Campus Essen
Anton Klimovsky and Anita Winter

04th April 2015

Special Topic Course (2SWS)
Sommer Semester 2015
Tuesday, 10.15-11.45; WSC-S-U-3.03

Gaussian Free Fields

A realization of discrete GFF on a square grid $60 \times 60$

Brownian motion is a centered Gaussian Markov process $W = \{ W(t) \}_{t \in I}$ indexed by the half-line $I := [0,+\infty)$ and having stationary independent increments. We think of $I$ as of the time line along which Brownian motion evolves. Alternatively, to define Brownian motion, we could have just postulated that $W$ has the covariance $\mathbb{E} [ W(t) W(s)] = t \wedge s$ for all $t, s \in I$. (We require that $W(\cdot)$ is a.s. continuous.)

Often, complementary to processes evolving in time, we want to model spatial processes, where to each point in space, say $x \in \mathbb{R}^2$, we attach a real random variable $h_x$. (As an example, one can think, e.g., of the temperature distribution on a planar surface). Such spatially indexed collections of random variables $\{ h(x) \}_{x \in I}$ are called random fields. In the case $I = \mathbb{R}^2$, we can visualize the random field as a random surface in $\mathbb{R}^3$, where $h(x)$ is the height (or depth, if negative) of the surface at point $x \in I$ above (below) the reference level $0$. (See the above picture.)

Gaussian free field (GFF) is a generalization of Brownian motion to such spatially indexed settings. Similarly to Brownian motion, GFF is a centered Gaussian process which satisfies a spatial generalization of the Markov property -- the Gibbs property (sometimes also called Markov random field property). Apart from that, GFF is tightly linked to the behavior of random walks on graphs: the covariance of GFF is given by the inverse graph Laplacian. Given this, it is not surprising that GFF is an important probabilistic object which has recently received much attention in the mathematics literature.

In this course, we shall present the main results currently available on the behavior of GFF together with the tools that are used to study these objects. Specifically, we will learn about

• Gaussian processes indexed by general index spaces;
• Spatial behavior of random walks on graphs: local times, potential theory, Green's function, Dynkin's isomorphism;
• Time-permitting: Extreme-value theory: point processes, extremal distributions, extremal processes.

Literature

What should I know if I want to participate?
This special topic course (Vertiefungsvorlesung) is offered as part of the Master Program in Mathematics. Nevertheless, if you are interested in probability theory you are very welcome to participte. Preknowledge in probability theory, stochastic processes and Brownian motion are of great advantage. Don't hesitate to send me an e-mail for more information:

anita(dot)winter `at' due(dot)de

• The first session will be on Tuesday, 07th of April.