Center for Mathematical Sciences Lectures

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The Lewiner Institute for Theoretical Physics

Distinguished Lecture Series at LITP and CMS Professor Thomas Spencer Institute for Advanced Study, Princeton will give a series of lectures on "Statistical Mechanics, Phase Transitions and Random Matrices"

Program of Lectures:

lectures	Date and Time	Place	Title of Lectures
Physics Colloquium	Thursday 11-Mar-2010 at 16:30	Rosen Auditorium (323)	Classical models of Quantum localization
Mathematical Physics Seminar	Sunday 14-Mar-2010 at 14:30	Lewiner Institute Seminar Room	A Phase transition for a supersymmetric hyperbolic sigma model.
Mathematics Colloquium	Monday 15-Mar-2010 at 15:30	Amado Building, 232	Central Limit theorems for Statistical Mechanics
Probability Seminar	Tuesday 16-Mar-2010 at 11:30	Electrical Engineering Building, 861	Central Limit theorems for Random Surfaces

Physics Colloquium (11.3):

Classical models of Quantum localization

After reviewing some mathematical aspects of quantum localization, I shall discuss

3 classical models which are closely related to localization: The first is the motion of a classical particle on the Manhattan lattice with random obstructions. This model (due to Cardy et al.) is equivalent to a quantum network model with random unitary evolution. The second model is a history dependent walk called edge reinforced walk. This walk favors edges it has visited more frequently in the past. The third model is a SUSY sigma model due to Zirnbauer. This model is equivalent to random walk in a correlated random environment and has an Anderson like transition in 3D. It seems to be closely related to the edge reinforced random walk where the strength of the reinforcement is proportional to the disorder.

Physics Seminar (14.3):

A Phase transition for a supersymmetric hyperbolic sigma model.

Spectral properties of random band matrices and other disordered quantum systems can be expressed in terms of SUSY statistical mechanics models. This talk will discuss a simplified version of these models due to Zirnbauer. The advantage of this model is that after integrating out the fermions, the action is real so that probabilistic methods can be applied. Correlations can be expressed as a random walk in a random environment. In 3D this model is shown to have an Anderson like transition from diffusive to localized states. The proof of these results relies heavily on Ward identities coming from the internal supersymmetry. This is joint work with M. Disertori and M. Zirnbauer.

Mathematics Colloquium (15.3):

Central Limit theorems for Statistical Mechanics

This talk will review some results and conjectures about the role of central limit theorems in statistical mechanics. In one dimension, fluctuations in statistical mechanics are described via the usual central limit theorem. In two dimensions, central limit theorems which arise in the study of anharmonic membranes, coulomb gases, and dimers are more difficult to establish due to strong correlations. We shall describe a strong version of the central limit theorem which is valid in two dimensions but which is false for sums of independent random variables. Central limit theorems for the self-avoiding walk and the Ising model in 4 or more dimensions will also be discussed.

Probability Seminar (16.3):

Central Limit theorems for Random Surfaces

This talk will review principles of optimal transportation and its relation to the integration by parts formula of Helffer, Sjöstrand and Witten. These ideas are combined with the theory of singular integral operators to obtain a strong central limit theorem for a class of random surfaces governed by Gibbs weight with a convex action. Upper and lower bounds on correlations are also established. This is joint work with J. Conlon.