The spectral gap for random regular graphs

Speaker: 

Tobias Johnson

Institution: 

USC

Time: 

Tuesday, January 12, 2016 - 11:00am to 12:00pm

Host: 

Location: 

RH 306

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.

Diffusive limits for stochastic kinetic equtions

Speaker: 

Arnaud Debussche

Institution: 

Univ. Rennes

Time: 

Tuesday, November 10, 2015 - 11:00am to 12:00pm

Host: 

Location: 

RH 306

In this talk, we consider kinetic equations containing random

terms. The kinetic models contain a small parameter and it is well

known that, after scaling, when this parameter goes to zero the limit

problem is a diffusion equation in the PDE sense, ie a parabolic equation

of second order. A smooth noise is added, accounting for external perturbation.

It scales also with the small parameter. It is expected that the limit

equation is then a stochastic parabolic equation where the noise is in

Stratonovitch form.

Our aim is to justify in this way several SPDEs commonly used.

We first treat linear equations with multiplicative noise. Then show how

to extend the methods to nonlinear equations or to the more physical

case of a random forcing term.

The results have been obtained jointly with S. De Moor and J. Vovelle.

Low Correlation Noise Stability of Euclidean Sets

Speaker: 

Steve Heilman

Institution: 

UCLA

Time: 

Tuesday, November 24, 2015 - 11:00am to 12:00pm

Host: 

Location: 

RH 306

The noise stability of a Euclidean set is a well-studied quantity.  This quantity uses the Ornstein-Uhlenbeck semigroup to generalize the Gaussian perimeter of a set.  The noise stability of a set is large if two correlated Gaussian random vectors have a large probability of both being in the set.  We will first survey old and new results for maximizing the noise stability of a set of fixed Gaussian measure.  We will then discuss some recent results for maximizing the low-correlation noise stability of three sets of fixed Gaussian measures which partition Euclidean space.  Finally, we discuss more recent results for maximizing the low-correlation noise stability of symmetric subsets of Euclidean space of fixed Gaussian measure.  All of these problems are motivated by applications to theoretical computer science.

Geometric properties of eigenfunctions for the fractional Laplacian

Speaker: 

Rodrigo Banuelos

Institution: 

Purdue University

Time: 

Tuesday, October 13, 2015 - 1:00pm to 2:00pm

Location: 

RH 306

 

Abstract

A classical result of H.J Brascamp and E.H. Lieb says that the ground state eigenfunction for the Laplacian in convex regions (and of Schr ̈odinger operators with convex potentials on Rn) is log-concave. A proof can be given (interpreted) in terms of the finite dimensional distributions of Brownian motion. Some years ago the speaker raised similar questions (and made some con- jectures) when the Brownian motion is replaced by other stochastic processes and in particular those with transition probabilities given by the heat kernel of the fractional Laplacian–the rota- tionally symmetric stable processes. These problems (for the most part) remain open even for the unit interval in one dimension. In this talk we elaborate on this topic and outline a proof of a result of M. Kaßmann and L. Silvestre concerning superharmonicity of eigenfunctions for certain fractional powers of the Laplacian. Our proof is joint work with D. DeBlassie. 

Synchronization by Noise

Speaker: 

Michael Scheutzow

Institution: 

Technische Universitat, Berlin

Time: 

Tuesday, September 22, 2015 - 11:00am to 11:50am

Host: 

Location: 

RH 306

 

Whenever a  deterministic system like an ODE or PDE does not possess an

asymptotically stable constant solution but if noise is added then there

exists a random  attractor which consists of a single (random) point,

then we call this phenomenon "synchronization by noise".

 

We first provide some specific examples and then present sufficient

conditions for synchronization to occur. Our results can be applied to

a large class of SDEs and some SPDEs with additive noise and to rather

general order-preserving random dynamical systems.

 

This is joint work with Franco Flandoli (Pisa) and Benjamin Gess (Leipzig).

Limit shapes of restricted integer partitions under non—multiplicative conditions

Speaker: 

Stephen DeSalvo

Institution: 

UCLA

Time: 

Tuesday, February 24, 2015 - 11:00am to 12:00pm

Host: 

Location: 

R 306

Abstract: Limit shapes are an increasingly popular way to understand
the large—scale characteristics of a random ensemble.  The limit shape
of unrestricted integer partitions has been studied by many authors
primarily under the uniform measure and Plancherel measure.  In
addition, asymptotic properties of integer partitions subject to
restrictions has also been studied, but mostly with respect to
\emph{independent} conditions of the form ``parts of size $i$ can
occur at most $a_i$ times.”  While there has been some progress on
asymptotic properties of integer partitions under other types of
restrictions, the techniques are mostly ad hoc.  In this talk, we will
present an approach to finding limit shapes of restricted integer
partitions which extends the types of restrictions currently
available, using a class of asymptotically stable bijections.  This is
joint work with Igor Pak.

Rare events for point process limits of random matrices.

Speaker: 

Diane Holcomb

Institution: 

University of Arizona

Time: 

Tuesday, January 27, 2015 - 11:00am to 12:00pm

Host: 

Location: 

RH 306

The Gaussian Unitary and Orthogonal Ensembles (GUE, GOE) are some of the most studied Hermitian random matrix models. When appropriately rescaled the eigenvalues in the interior of the spectrum converge to a translation invariant limiting point process called the Sine process. On large intervals one expects the Sine process to have a number of points that is roughly the length of the interval times a fixed constant (the density of the process). We solve the large deviation problem which asks about the asymptotic probability of seeing a different density in a large interval as the size of the interval tends to infinity. Our proof works for a one-parameter family of models called beta-ensembles which contain the Gaussian orthogonal, unitary and symplectic ensembles as special cases.

Brownian Motion in Three Dimensions Conditioned to have the Origin as a Recurrent Point

Speaker: 

Patrick Fitzsimmons

Institution: 

UCSD

Time: 

Tuesday, January 20, 2015 - 11:00am to 12:00pm

Host: 

Location: 

RH 306

I will discuss aspects of a polymer model based on three-dimensional Brownian motion conditioned to hit (and keep returning to) the origin 

introduced by Mike Cranson and co-authors.   The construction and certain properties of this conditioned Brownian motion will be approached from two points of view (i) Dirichlet forms, and (ii) excursion theory. The latter gives a nice interpretation of the Johnson-Helms example from martingale theory. It turns out that this diffusion process is not a semimartingale, even though its radial part is just a one-dimensional Brownian motion reflected at the origin. 

 

(This talk is based on joint work with Liping Li of Fudan University.)

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