Graduate Summer School Program

Course Descriptions:

The Renormalisation Group and Equilibrium Statistical Mechanics; David Brydges, University of British Columbia

In the theory of critical phenomena in statistical mechanics, the idea of a scaling limit is exemplified by observing a very long self-avoiding walk from far away so that individual steps become invisible and one sees (the occupation density of) a path in the continuum. The scaling limit is the probability law for this random continuum path.
The Renormalisation Group (RG) is a nascent program to construct and classify scaling limits in d dimensions based on the Nobel prize work of Ken Wilson on the renormalisation group (RG). RG is a map acting on a space of statistical mechanical models. Models are probability measures on random fields and RG acts on a model by integrating out the short distance fluctuations giving rise to a new model whose typical random field has the same long distance fluctuations but suppressed short distance fluctuations. Finding scaling limits corresponds to determining the fixed points of RG.
I will introduce some statistical mechanical models, in particular dipoles on Zd, formulate the RG precisely, and show that the fixed point for RG in the case of the dipole gas is the massless Gaussian field.

Background reading for lectures by David Brydges: (download .pdf)

(updated 5/30/07)

Lectures on Random matrices; Alice Guionnet, Ecole Normale Supérieure de Lyon

The theory of random matrices has developed rapidly during the last 15 years in connection with fields as diverse as statistics, theoretical physics, number theory and combinatorics. This course will be an introduction to part of this theory. After a general overview of random matrices, we will focus on the so-called Wigner matrices which are self-adjoint matrices with independent entries (modulo the symmetry constraint). We will study the empirical measure of the eigenvalues of such matrices, or more generally traces of words in such matrices. We will show that it converges (Wigner's theorem), study its fluctuations and prove that concentration of measure holds. We will then restrict ourselves to Gaussian entries and discuss large deviations and matrix models in connection with the enumeration of certain graphs called maps. The prerequisites are basic probability theory.

Background reading: see Lecture Notes on St. Flour at www.umpa.ens-lyon.fr/~aguionne/

Dimers and random surfaces; Richard Kenyon, University of British Columbia

The goal of the lectures will be to introduce the dimer model and discuss the role it plays in recent results about limit shapes for crystal surfaces. The dimer model can be viewed as a model of random surfaces, and we intend to show how in the scaling limit (when the lattice spacing tends to zero) the random surfaces can have non-random limit shapes which arise from energy minimization considerations.
Time permitting we will discuss connections with SLE and the Gaussian free field.

Recommended course reading: Chapters 1-4, An introduction to the dimer model at http://arxiv.org/PS_cache/math/pdf/0310/0310326v1.pdf

An Introduction to the Schramm-Loewner Evolution; Gregory Lawler, University of Chicago
An introduction to the mathematics of SLE (Schramm-Loewner evolution):

Topics include:
--- Basics of univalent functions
--- Loewner differential equation
--- Definition of Schramm-Loewner evolution
--- Phases and dimension of the path
--- Conformal transformations of SLE
--- Restriction, locality, and the fundamental
martingales
--- Relation with Brownian loops

I will assume stochastic calculus through Ito's formula and complex variables through the Riemann mapping theorem.

Much of the material will come from my book Conformally Invariant Processes in the Plane (2005). You may wish to read some of that, especially the initial section entitled “Some discrete processes.”

Zeros of Gaussian Analytic Functions, determinantal processes and gravitational allocation; Yuval Peres, Microsoft Research and University of California, Berkeley

Lecture 1: Point Processes and Repulsion.
Point processes (random scatters of points in space) have applications in many areas, including statistics and cosmology. Recently, there has been increasing interest in processes that exhibit "repulsion". We will see why zeros of random polynomials have this property, and describe the effect of repulsion on matching and allocation problems.

Lecture 2: Zeros of Gaussian Analytic Functions.
Zeros of Gaussian analytic functions have a remarkable rigidity property, discovered by M. Sodin: The first order intensity determines the whole process. For each of the classical geometries, planar, spherical and hyperbolic, there is a one-parameter family of Gaussian analytic functions with isometry-invariant zeros.

Lecture 3: Determinantal Processes.
Discrete and continuous point processes where the joint intensities are determinants arise in Combinatorics (Random spanning trees) and Physics (Fermions, eigenvalues of Random matrices). For these processes the number of points in a region can be represented as a sum of independent, zero-one valued variables, one for each eigenvalue of the relevant operator.

Lecture 4: Zeros of the I.I.D. Gaussian Power Series.
The power series with i.i.d. complex Gaussian coefficients has zeros that form an isometry-invariant determinantal process in the disk model of the hyperbolic plane. (Joint work with B. Virag). This allows an exact calculation of the law of the number of zeros in a subdisk. We also analyze the dynamic version where the coefficients perform Brownian motion.

Lecture 5: The Translation-Invariant Planar Gaussian Zeros.
Sodin-Tsirelson analyzed the zeros of the Gaussian power series with Euclidean symmetry. Their results reveal a surprising analogy with a four-dimensional Poisson process. In particular, the probability of a large disk of radius R to be free of zeros decays like exp(-cR^4). A remarkable "gravitational allocation" that allots a unit of area to each zero in a translation invariant way was discovered by Sodin and Tsirelson. Nazarov, Sodin and Volberg ahowed that the diameters of the domains of attraction have exponential tails.

Lecture 6: Gravitational Allocation for Poisson Points.
While the method of gravitational allocation is not applicable to the planar Poisson process, it does apply to the Poisson process in dimensions 3 and higher; (This is Joint work with S. Chatterjee, R. Peled, D. Romik). The argument starts with the classical calculation by Chandrasekar of the total gravity force acting on a point, which has a stable law. Here also the domains of attraction have an exponential tail, and the proof uses ideas of dependent percolation.
See http://pcmi.ias.edu/current/Peresimages.htm for images related to Peres’ course.

Gaussian Analytic Functions Book - PDF Download (3.2MB)

Conformal invariant models; Wendelin Werner, Université Paris-Sud

We will first discuss some two-dimensional discrete models from statistical physics (critical percolation, uniform spanning trees, etc.) and study their large-scale properties and in particular their conformal invariance. We shall see how and why the SLE (Schramm-Loewner Evolutions) that will have been studied in Greg Lawler's course can then be used to derive results concerning the discrete models. Finally, we plan to describe continuous two-dimensional systems in which SLE loops are naturally embedded.

Background/preparation reading:

G.R. Grimmett, Percolation, Springer
Basic material on conformal invariance (Riemann's mapping theorem &
Morrera's theorem) -- for example in Ahlfors' Complex Analysis.

The course will be otherwise self-contained.

Material related (I will probably follow another structure) to the lectures:

My lecture notes from Saint-Flour http://arxiv.org/abs/math/0303354 and Les
Houches http://front.math.ucdavis.edu/math.PR/0511268

The Graduate Summer School receives major support from award #DMS-0437137 from the National Science Foundation. The Graduate Summer School is also supported in part by Mathematical Sciences Research Institute (MSRI).