COMPACTNESS AND LARGE DEVIATIONS

In a reasonable topological space, large deviation estimates essentially deal
with probabilities of events that are asymptotically (exponentially) small,
and in a certain sense, quantify the rate of these decaying probabilities.
In such estimates, upper bounds for such small probabilities often require 
compactness of the ambient space, which is often absent in problems arising in
statistical mechanics (for example, distributions of local times of
Brownian motion in the full space R^d). Motivated by such a problem, we
present a robust theory of “translation-invariant compactification”
of probability measures in R^d. Thanks to an inherent shift-invariance of
the underlying problem, we are able to apply this abstract theory painlessly
and solve a long standing problem in statistical mechanics, the mean-field
polaron problem.

This talk is based on joint works with S. R. S. Varadhan (New York), as
well as with Erwin Bolthausen(Zurich)and Wolfgang Koenig (Berlin).

We consider a drift-diffusion process on a smooth potential landscape

with small noise. We give a new proof of the Eyring-Kramers formula

which asymptotically characterizes the spectral gap of the generator of

the diffusion. The proof is based on a refinement of the two-scale

approach introduced by Grunewald, Otto, Villani, and Westdickenberg and

of the mean-difference estimate introduced by Chafai and Malrieu. The

new proof exploits the idea that the process has two natural

time-scales: a fast time-scale resulting from the fast convergence to a

metastable state, and a slow time-scale resulting from exponentially

long waiting times of jumps between metastable states. A nice feature

of the argument is that it can be used to deduce an asymptotic formula

for the log-Sobolev constant, which was previously unknown.

Abstract: We will review the techniques used to prove that a positive proportion of the zeros of the Riemann zeta-Function lie on the critical line Re(s)=1/2. The famous Riemann hypothesis states that all the zeros lie there. We will then discuss the mollifiers that allow us to show that > 41% of zeros are critical. This is joint work with A. Roy and A. Zaharescu.

I will review some (actually quite old) results by C. Borgs, J.T. Chayes, R. Kotecky and myself concerning the partition function zeros of the Ising model. The focus will be on the fact that, for specific boundary conditions, the zeros lie (in a suitable representation) on the unit circle. I will explain (1) the classic proof of the Lee-Yang circle theorem and (2) how one can nail the positions of the zeros up to exponentially small errors in the system size for the periodic boundary conditions. I may find time to explain how one uses this result to prove the so called Griffiths singularities in site-diluted Ising model.

Expander graphs are useful across mathematics, all the way from number theory to applied computer science. The smaller the second eigenvalue of a regular graph, the better expander it is. Since this connection was discovered in the 1980s, researchers have tried to pinpoint the second eigenvalue of random regular graphs. The most prominent work in this direction was Joel Friedman's proof of Noga Alon's conjecture from 1985 that for a random d-regular graph on n vertices, the second eigenvalue is almost as small as possible, with high probability as n tends to infinity with d held fixed.

We consider the case of denser graphs, where d and n are both growing. Here, the best result (Broder, Frieze, Suen, Upfal 1999) holds only if d = o(n^(1/2)). We extend this to d = O(n^(2/3)). Our result relies on new concentration inequalities for statistics of random regular graphs based on the theory of size biased couplings, an offshoot of Stein's method. The theory we develop should be useful for proving concentration inequalities in a broad range of settings. This is joint with Nicholas Cook and Larry Goldstein.