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Eyal Lubetzky
We will survey recent developments and several of the many problems that remain open on the rate of convergence to equilibrium of Glauber dynamics for the Potts and FK model in Z^2 at criticality.
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Peter Winkler
The “displacement ratio” R is the ratio of two natural measures of how close a permutation is to the identity, and is known always to be between 1 and 2. We want to know: what is R for a random permutation in S_n with a given number m(n) of inversions? The behavior of R as m(n) changes is not entirely proven but seems rather surprising.
Joint work with David Bevan, U. of Strathclyde.
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Sandra Cerrai
I will discuss the validity of the so-called Smoluchowski-Kramers approximation for systems with an infinite number of degrees of freedom in a finite time. Then, I will investigate the validity of such approximation for large time. In particular, I will address the problem of the convergence, in the small mass limit, of statistically invariant states for a class of semi-linear damped wave equations, perturbed by an additive Gaussian noise, with quite general nonlinearities. More precisely, I will show how the first marginals of any sequence of invariant measures for the stochastic wave equation converge in a suitable Wasserstein metric to the unique invariant measure of the limiting stochastic semi-linear parabolic equation obtained in the Smoluchowski-Kramers approximation.
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Joe Neeman
The Gaussian isoperimetric inequality states that if we want to partition R^n into two sets with prescribed Gaussian measure while minimizing the Gaussian surface area of the interface between the sets, then the optimal partition is obtained by cutting R^n with a hyperplane. We prove an extension to more than two parts. For example, the optimal way to partition R^3 into three parts involves cutting along three rays that meet at 120-degree angles at a common point. This is the Gaussian analogue of the famous Double Bubble Theorem in Euclidean space, which describes the shape that results from blowing two soap bubbles and letting them stick together.
Joint work with Emanuel Milman.
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Vadim Gorin
I will talk about a special class of large-dimensional stochastic systems with strong correlations. The main examples will be random tilings, non-colliding random walks, eigenvalues of random matrices, and measures governing decompositions of group representations into irreducible components.
It is believed that macroscopic fluctuations in such systems are universally described by log-correlated Gaussian fields. I will present an approach to handle this question based on the notion of the Schur generating function of a probability distribution, and explain how it leads to a rigorous confirmation of this belief in a variety of situations.