# Probability Seminar

## Thursday, September 15, Jun Yin, University of Wisconsin - Madison

Some recent results on random matrices with almost independent entries.

In this talk, we are going to introduce some recent work on a large class of random matrices, whose entries are (almost) independent. For example, the Wigner matrix, generalized Wigner matrix, Band random matrix, Covariance matrix and Sparse random matrix. We mainly focus on the local statistics of the eigenvalues and eigenvectors of these random matrix ensembles. We will also introduce some applications of these results and some long-standing open questions.

## Thursday, September 22, Philip Matchett Wood, University of Wisconsin - Madison

Survey of the Circular Law

What do the eigenvalues of a random matrix look like? This talk will focus on large square matrices where the entries are independent, identically distributed random variables. In the most basic case, the distribution of the eigenvalues in the complex plane (suitably scaled) approaches the uniform distribution on the unit disk, which is called the circular law. We will discuss some of the methods that have been used to prove the circular law, including recent work that has extended the circular law to the most general situation, and we will also discuss generalizations to situations where the eigenvalue distributions are stable, but non-circular.

## Thursday, September 29, Antonio Auffinger, University of Chicago

A simplified proof of the relation between scaling exponents in first passage percolation

In first passage percolation, we place i.i.d. non-negative weights on the nearest-neighbor edges of Z^d and study the induced random metric. A long-standing conjecture gives a relation between two "scaling exponents": one describes the variance of the distance between two points and the other describes the transversal fluctuations of optimizing paths between the same points. In a recent breakthrough work, Sourav Chatterjee proved a version of this conjecture using a strong definition of the exponents. I will discuss work I just completed with Michael Damron, in which we introduce a new and intuitive idea that replaces Chatterjee's main argument and gives an alternative proof of the scaling relation. One advantage of our argument is that it does not require a non-trivial technical assumption of Chatterjee on the weight distribution.

## Tuesday, October 4, 2:30 PM, VV901, Gregorio Moreno Flores, University of Wisconsin - Madison

UNUSUAL TIME

Airy process and the polymer end point distribution

We give an explicit formula for the joint density of the max and argmax of the Airy process minus a parabola. The argmax has a universal distribution which governs the rescaled endpoint of directed polymers in 1+1 dimensions.

## Thursday, October 20, Kay Kirkpatrick, University of Illinois at Urbana-Champaign

Bose-Einstein condensation and a phase transition for the nonlinear Schrodinger equation

Near absolute zero, a gas of quantum particles can condense into an unusual state of matter, called Bose-Einstein condensation (BEC), that behaves like a giant quantum particle. The rigorous connection has recently been made between the physics of the microscopic dynamics and the mathematics of the macroscopic model, the cubic nonlinear Schrodinger equation (NLS). I'll discuss work with Sourav Chatterjee about a phase transition for invariant measures of the discrete focusing NLS. Using techniques from probability theory, we show that the thermodynamics of the NLS are exactly solvable in dimensions three and higher. There are a number of consequences of this result, including a prediction for experimentalists to look for a new spatially localized phase of BEC.

## Monday, October 31, 2:30pm, VV B341, Ankit Gupta, Ecole Polytechnique, Centre de Mathematiques Appliqees

UNUSUAL TIME AND PLACE

Modeling adaptive dynamics for structured populations with functional traits

We develop the framework of adaptive dynamics for populations that are structured by age and functional traits. The functional trait of an individual may express itself differently during the life of an individual according to her age and a random parameter that is chosen at birth to capture the environmental stochasticity. The population evolves through birth, death and selection mechanisms. At each birth, the new individual may be a clone of its parent or a mutant. Starting from an individual based model we use averaging techniques to take the large population and rare mutation limit under a well-chosen time-scale separation. This gives us the Trait Substitution Sequence process that describes the adaptive dynamics in our setting. Assuming small mutation steps we also derive the Canonical Equation which expresses the evolution of advantageous traits as a function-valued ordinary differential equation.

This is joint work with J.A.J Metz (Leiden University) and V.C. Tran (University of Lille).

## Friday, November 4, 2:30pm, VV B341, Michael Cranston, University of California, Irvine

UNUSUAL TIME AND PLACE

Overlaps and Pathwise Localization in the Anderson Polymer Model

We consider large time behavior of typical paths under the Anderson poly- mer measure. We establish that the polymer measure gives a macroscopic mass to a typical path and this mass grows to 1 in the limit as a particular parameter tends to infinity. This gives a rigourous approach to the polymer localization. The localization is measured by considering the overlap between two independent samples drawn from the polymer measure.

## Thursday, November 10, David Anderson, University of Wisconsin - Madison

Computational methods for stochastic models in biology

I will focus on computational methods for continuous time Markov chains, which includes the large class of stochastically modeled biochemical reaction networks and population processes. I will show how different computational methods can be understood and analyzed by using different representations for the processes. Topics discussed will be a subset of: approximation techniques, variance reduction methods, parameter sensitivities.

## Wednesday, November 30, 2:30pm, B325, Shamgar Gurevich, University of Wisconsin - Madison

UNUSUAL TIME AND PLACE

Fast Matched Filter and Group Representation: What? Why? How?