On limiting operators related to circular ensembles.

Speaker: 

Joseph Najnudel

Institution: 

Univ. Paul Sabatier, Toulouse

Time: 

Wednesday, June 19, 2013 - 11:00am to 12:00pm

It is a classical result, by Dyson, that the behavior of the eigenvalues of a random unitary matrix following uniform measure tend, when the dimension goes to infinity, after a suitable scaling, to a random set of points, called adeterminantal sine-kernel process. By defining the model in all dimensions on a single probability space, we are able to show that the convergence stated above can occur almost surely. Moreover, in an article with K. Maples and A. Nikeghbali, we interpret the limiting point process as the spectrum of a random operator.

Metastable Densities for Contact Processes on Random Graphs

Speaker: 

Daniel Valesin

Institution: 

University of British Columbia

Time: 

Wednesday, June 12, 2013 - 10:00am to 11:00am

Host: 

Location: 

RH 340P

We consider the contact process on a random graph chosen with a fixed degree, power law distribution, according to a model proposed by Newman, Strogatz and Watts (2001). We follow the work of Chatterjee and Durrett (2009) who showed that for arbitrarily small infection parameter λ

Changes of the filtration and the default event risk premium

Speaker: 

Delia Coculescu

Institution: 

Univsitat Zurich

Time: 

Wednesday, May 29, 2013 - 2:00pm to 3:00pm

Host: 

Location: 

RH 440R

In this talk we aim at emphasizing the role of information in financial markets (public information versus insider information). In particular, if the information about a particular event (as for instance the default event of a company) is incorporated into a pricing model, then by a change of the underlying filtration, one can compute risk premiums attached to particular events. We also show that modeling of the information leads eventually to modeling of dependencies.

A new rearrangement inequality around infinity and applications to Lévy processes.

Speaker: 

Alexander Drewitz

Institution: 

Columbia University

Time: 

Tuesday, April 16, 2013 - 11:00am to 12:00pm

Location: 

Rowland Hall 306

We start with showing how rearrangement inequalities may be used in probabilistic contexts such as e.g. for obtaining bounds on survival probabilities in trapping models. This naturally motivates the need for a new rearrangement inequality which can be interpreted as involving symmetric rearrangements around infinity. After outlining the proof of this inequality we proceed to give some further applications to the volume of Lévy sausages as well as to capacities for Lévy processes.
(Joint work with P. Sousi and R. Sun)

"Dimension Spectrum of SLE Boundary Collisions"

Speaker: 

Tom Alberts

Institution: 

Cal Tech

Time: 

Tuesday, March 12, 2013 - 11:00am to 12:00pm

Location: 

306 RH

In the range 4 < \kappa < 8, the intersection of the Schramm-Loewner Curve (one of the central objects in the theory of 2-D Conformally Invariant Systems) with the boundary of its domain is a random fractal set. After reviewing some previous results on the dimension and measure of this set, I will describe recent joint work with Ilia Binder and Fredrik Viklund that partitions this set of points according to the generalized "angle" at which the curve hits the boundary, and computes the Hausdorff dimension of each partition set. The Hausdorff dimension as a function of the angle is what we call the dimension spectrum.

Modeling the genealogy of populations using coalescents with multiple mergers

Speaker: 

Jason Schweinsberg

Institution: 

UCSD

Time: 

Tuesday, February 19, 2013 - 11:00am to 12:00pm

Host: 

Location: 

RH 306

Suppose we take a sample of size n from a population and follow the ancestral lines backwards in time.  Under standard assumptions, this process can be modeled by Kingman's coalescent, in which each pair of lineages merges at rate one.  However, if some individuals have large numbers of offspring or if the population is affected by selection, then many ancestral lineages may merge at one time.  In this talk, we will introduce the family of coalescent processes with multiple mergers and discuss some circumstances under which populations can be modeled by these coalescent processes.  We will also describe how genetic data, such as the number of segregating sites and the site frequency spectrum, would be affected by multiple mergers of ancestral lines, and we will discuss the implications for statistical inference.

Path properties of the random polymer in the delocalized regime

Speaker: 

Ken Alexander

Institution: 

USC

Time: 

Tuesday, February 5, 2013 - 11:00am to 12:00pm

Location: 

RH 306

 We study the path properties of the random polymer attracted to a defect line by a potential with disorder, and we prove that in the delocalized regime, at any temperature, the number of contacts with the defect line remains in a certain sense ``tight in probability'' as the polymer length varies. On the other hand we show that at sufficiently low temperature, there exists a.s. a subsequence where the number of contacts grows like the log of the length of the polymer.

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