Spring 2022 Seminar

This is the homepage for the UC Berkeley Student Probability Seminar, a venue for graduate students in the departments of mathematics, statistics, and others to study aspects of modern probability theory. We meet on Wednesdays in Evans 732 from 2:00 PM - 3:00 PM

Organizers: Ella Hiesmayr and Adam Quinn Jaffe.

The topic for this semester's seminar is the Gaussian free field. We will primarily follow the lecture notes by Werner and Powell [WP] and by Berestycki [B], but we will also consult some other sources along the way. The list of talks and abstracts can be found below.

• 19 January, Initial Meeting.
• A casual first meeting to get to know each other and decide on a topic for the semester.
• 26 January, Adam Quinn Jaffe, Introductory Talk.
• First, we give a non-rigorous construction of the continuum Gaussian free field (GFF). Roughly speaking, the GFF is a canonical sort of a random harmonic function; in one dimension it is just the Brownian bridge, and in two or more dimensions it is not a function at all but rather a generalized function. We also outline some of its probabilistic properties. Second, we sketch a handful of the GFF's many connections to some importants elements of modern probability theory. For one example, we describe how its interfaces follow a Schramm-Loewner evolution (SLE) and its level lines follow a conformal loop ensemble (CLE). For another example, we describe how the GFF gives rise to a random measure called the Gaussian multiplicative chaos (GMC); better yet, we describe how the GMC gives rise to a canonical random compact Riemmann surface called the Liouville quantum gravity (LQG).
• 2 Februrary, Ella Hiesmayr, Green's function and the construction of the GFF, WP Chapter 3.1 and 3.2
• We will start by motivating the definition of the continuum Green’s function using the discrete analogue and then define it and derive its properties. We will then show how this allows us to prove the existence of the continuum GFF.
• 9 Februrary, Daniel Raban, Concrete construction of the GFF, WP Chapter 3.2 and 3.3
• We will use spectral theory of the Laplacian to prove a diagonalization formula for the Green's function. This will lead to a concrete construction and characterization of Gaussian Free Fields.
• 16 Februrary, Alexander Tsigler, Construction of GFF via eigenfunctions of the Laplacian, WP Chapter 3.3
• We will discuss the spectral decomposition of the Laplacian and see how it helps connect L^2 structure with differentiability and define Sobolev spaces. Then we will explicitly construct GFF on measures whose densities belong to those Sobolev spaces.
• 23 Februrary, Yang Chu, Green's functions and GFF in dimension 2, WP Chapter 3.2 and 3.3
• Today we will discuss GFF in dimension 2 where it has special properties such as conformal invariance. We will show that the Green's function in dimension 2 is scale invariant and give examples such that the domain is upper-half plane and unit disk. Combined with the Riemann mapping theorem, we will see it is enough to find the Green's function on any particular simply-connected domain, and then one can extend this to all other simply-connected domains.
• 2 March, Yassine El Maazouz, The Markov property for GFF, WP Chapter 4.1
• Given a compact subset A of a bounded domain D satisfying some regularity conditions, we construct a decomposition of a GFF Gamma on D into two independently Gaussian processes. One part of this decomposition is again a GFF on D\A and the second part is obtained as a restriction of the original GFF to A and on D\A as a suitable harmonic extension.
• 9 March, Zack McNulty, A spatial strong Markov property for the continuum GFF, WP Chapter 4.2
• We extend the spatial Markov property of the continuum GFF from deterministic sets to local sets, a special class of random sets which in some sense act as the spatial analog of stopping times. We start by analyzing these sets in the discrete GFF setting, and move to the continuum setting through dyadic discretization.
• 16 March, No talk.
• 23 March, No talk.
• 30 March, Nick Liskij, Circle Averages, WP Chapter 3.3 and B Section 1.7
• First, we will define the circle averages for the GFF. We will show that the circle averages around a point are a time-changed Brownian motion. Moreover, we can construct a jointly continuous modification. If time allows, we will briefly discuss "thick points" of the GFF.
• 6 April, Meredith Shea, On the size of the set of thick points of the GFF, Hu, Miller, Peres
• Previously we have seen that the circle average process of the GFF behaves (almost surely) as a Brownian motion (as r goes to 0), nevertheless the GFF still admits thick points. In this talk we will consider the size of the set of thick points. In particular, we will prove an upper bound on the Hausdorff dimension on the size of the set of a-thick points.
• 13 April, Adam Quinn Jaffe, More on the size of the set of thick points of the GFF, Hu, Miller, Peres
• When constructing the circle average process of the GFF, we saw that, while any particular is almost surely typical, there is in fact some random set of atypical (thick) points. In the last talk we established an upper bound on the Hausdorff dimension of this random set, and in this talk we prove the corresponding lower bound.
• 20 April, Karissa Huang, The L2 phase of Liouville Quantum Gravity, B Section 2.2
• In this talk we will study the L2 phase of Liouville Quantum Gravity. In particular, we will show that in the L2 phase, the measure that is expressed as the exponential of a Gaussian Free Field is integrable and converges in probability to a limit.
• 27 April, Ella Hiesmayr, Convergence of the Gaussian multiplicative chaos for the entire subcritical phase, B Section 2.4
• As we saw last week, for small parameters we can prove that the exponential of the Gaussian Free Field has a limiting measure using convergence in L^2. This week we will prove the same result for all meaningful parameters, by showing that we can ignore all points that are "too thick".

Thanks for the great semester, everyone!