A CLT for dependent variables and some homogenization problems.

Speaker: 

Professor Nikos Zygouras

Institution: 

USC

Time: 

Tuesday, April 15, 2008 - 11:00am

Location: 

MSTB 254

I will present a criterion for the validity of the central limit theorem
for a class of dependent random variables and then I will discuss some applications of
it on random, boundary homogenization problems of nonlinear PDEs such nonlinear
parabolic ones and Navier walls.

Overlap distribution in the multiple spherical SK models (joint work Michel Talagrand).

Speaker: 

Professor Dmitry Panchenko

Institution: 

Texas A&M

Time: 

Tuesday, April 29, 2008 - 11:00am

Location: 

MSTB 254

One possible approach to the study of the geometry of the Gibbs measure in the Sherrington-Kirkpatrick
type models (for example, the chaos and ultrametricity problems) is based on the analysis of the free energy
on several replicas of the system under some constraints on the distances between replicas. In general, this
approach runs into serious technical difficulties, but we were able to make some progress in the setting of the
spherical p-spin SK models where many computations become more explicit.

Negotiations in the network bargaining problem

Speaker: 

Professor Alan Hammond

Institution: 

Courant Institute

Time: 

Thursday, May 29, 2008 - 11:00am

Location: 

MSTB 254

Abstract: consider a finite graph, with an actor sitting at each node, and a
dollar on each edge. Negotiations will be conducted between pairs of
adjacent actors over splitting the dollar on the intervening edge.
At the end of negotiations, each actor may sign at most one contract with a
neighbour, agreeing on some possibly uneven split of the dollar.
How much money is each actor likely to receive? And which matchings of the
graph are likely to arise?
Kleinberg and Tardos analysed the limiting answer - a balanced solution -
that arises from assuming that actors iteratively revise current deals using
Nash bargaining, taking the best alternative deal currently available as a
backup.

Most of the talk will be expository, I'll explain the concepts of Nash bargaining and balanced solution. If there is time, I will discuss
the rate of
convergence to the balanced solution of this type of negotiation.

Two results about large random matrices.

Speaker: 

Professor Amir Dembo

Institution: 

Stanford University

Time: 

Tuesday, April 22, 2008 - 1:00pm

Location: 

MSTB 254

We show that the properly scaled spectral measures
of symmetric Hankel and Toeplitz matrices of size N by N generated by
i.i.d. random variables of zero mean and unit variance converge weakly
in N to universal, non-random, symmetric
distributions of unbounded support, whose moments are
given by the sum of volumes of solids related to Eulerian numbers.
The universal limiting spectral distribution for
large symmetric Markov matrices
generated by off-diagonal i.i.d. random variables
of zero mean and unit variance, is more explicit, having
a bounded smooth density given by the free convolution of the
semi-circle and normal densities.

Time permitting, I will also explain the formula for the
large deviations rate function
for the number of open path of length k in random graphs
on N>>1 vertices with
each edge chosen independently with probability 0

Branching random walks and diffusions on hyperbolic spaces: recurrence, transience and Hausdorff dimension of limiting sets

Speaker: 

Professor Yuri Suhov

Institution: 

Cambridge University

Time: 

Tuesday, April 8, 2008 - 1:00pm

Location: 

MSTB 254

This talk focuses on asymptotic properties
of geometric branching processes on hyperbolic spaces
and manifolds. (In certain aspects, processes on hyperbolic spaces are
simpler than on Euclidean spaces.)
The first paper in this direction was
by Lalley and Sellke (1997) and dealt with a homogenous branching diffusion on a hyperbolic (Lobachevsky) plane).
Afterwards, Karpelevich, Pechersky and Suhov (1998) extended
it to general homogeneous branching processes on
hyperbolic spaces of any dimension. Later on, kelbert and Suhov
(2006, 2007) proceeded to include non-homogeneous branching
processes. One of the main questions here is to calculate
the Hausdorff dimension of the limiting set on the absolute.
I will not assume any preliminary knowledge of hyperbolic
geometry.

Spin glasses and Stein's method

Speaker: 

Professor Sourav Chatterjee

Institution: 

UC Berkeley

Time: 

Tuesday, March 4, 2008 - 11:00am

Location: 

MSTB 254

The high temperature phase of the Sherrington-Kirkpatrick model of spin glasses is solved by the famous Thouless-Anderson-Palmer (TAP) system of equations. The only rigorous proof of the TAP equations, based on the cavity method, is due to Michel Talagrand. The basic premise of the cavity argument is that in the high temperature regime, certain objects known as `local fields' are approximately gaussian in the presence of a `cavity'. In this talk, I will show how to use the classical Stein's method from probability theory to discover that under the usual Gibbs measure with no cavity, the local fields are asymptotically distributed as asymmetric mixtures of pairs of gaussian random variables. An alternative (and seemingly more transparent) proof of the TAP equations automatically drops out of this new result, bypassing the cavity method.

Equilibrium Fluctuations for Coagulating-Fragmenting Brownian Particles

Speaker: 

Professor Fraydoun Rezakhanlou

Institution: 

UC Berkeley

Time: 

Tuesday, February 5, 2008 - 11:00am

Location: 

MSTB 254

Smoluchowski's equation is used to describe the coagulation-fragmentation
process macroscopically. Microscopically clusters of various sizes coalesce to
form larger clusters and fragment into smaller clusters. I formulate a conjecture about the nature of the fluctuations of the microscopic
clusters about the solutions to the Smoluchowski's equation. I also sketch the proof of the conjecture when the model is in equilibrium.

The quantum Ising model via stochastic geometry.

Speaker: 

Professor Geoffrey Grimmett

Institution: 

Cambridge University

Time: 

Tuesday, January 29, 2008 - 11:00am

Location: 

MSTB 254

Abstract: (Thanks to work of Abel Klein and others) it is understood how
to represent the quantum Ising model in terms of a certain classical model
of stochastic geometry called the `continuum random-cluster model'. In the
regime of large external field, this geometrical model is subcritical. By
developing bounds for its `ratio weak' mixing rate, one obtains estimates
involving the reduced density matrix of the quantum Ising model. The
implications of these estimates for entanglement do not appear to be best
possible, but they are at least robust for disordered systems. [Joint work
with Tobias Osborne and Petra Scudo.]

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