We present a general framework for approximating combinatorial assemblies when both the size $n$ and the number of components $k$ is specified.  The approach is an extension of the usual saddle point approximation, and we demonstrate near-universal behavior when the rank $r := n-k$ is small relative to $n$ (hence the name `low rank’).  

In particular, for $\ell = 1, 2, \ldots$, when $r \asymp n^\alpha$, for $\alpha \in \left(\frac{\ell}{\ell+1}, \frac{\ell+1}{\ell+2}\right)$, the size~$L_1$ of the largest component converges in probability to $\ell+2$.  When $r \sim t\, n^{\ell/(\ell+1)}$ for any $t>0$ and any positive integer $\ell$, $\P(L_1 \in \{\ell+1, \ell+2\}) \to 1$.  We also obtain as a corollary bounds on the number of such combinatorial assemblies, which in the special case of set partitions fills in a countable number of gaps in the asymptotic analysis of Louchard for Stirling numbers of the second kind. 

This is joint work with Richard Arratia.

The seminal invariance principle of Mossel-O'Donnell-Oleszkiewicz implies the following. Suppose we have a multilinear polynomial Q, all of whose partial derivatives are small. Then the distribution of Q on i.i.d. uniform {-1,1} inputs is close to the distribution of Q on i.i.d. standard Gaussian inputs. The case that Q is a linear function recovers the Berry-Esseen Central Limit Theorem. In this way, the invariance principle is a nonlinear version of the Central Limit Theorem. We prove the following version of one of the two inequalities of the invariance principle, which we call a majorization principle. Suppose we have a multilinear polynomial Q with matrix coefficients, all of whose partial derivatives are small. Then, for any even K>1, the Kth moment of Q on i.i.d. uniform {-1,1} inputs is larger than the Kth moment of Q on (carefully chosen) random matrix inputs, minus a small number. The exact statement must be phrased carefully in order to avoid being false. Time permitting, we discuss applications of this result to anti-concentration, and to computational hardness for the noncommutative Grothendieck inequality. (joint with Thomas Vidick) (

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.