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
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,
- A casual first meeting to get to know each other and
decide on a topic for the semester.
- 26 January, Adam Quinn Jaffe,
- 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
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
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
- 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
- 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
- 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!