Probability Seminar

Date Time 
Location  Speaker 
Title – click for abstract 

01/29 2:00pm 
BLOC 628 
paul jung Korea Advanced Institute of Science and Technology 
Infinitevolume Gibbs measures for the 1DCoulomb jellium
The jellium is a model, introduced by Wigner, for a gas of electrons moving in a uniform neutralizing background of positive charge. In two dimensions, the model is related to the Gaussian free field while in one dimension the model is used to study dimerization and crystallization. For the quantum 1D jellium, Brascamp and Lieb (1975) proved crystallization (nonergodicity of the Gibbs measures) at low densities of electrons. Using tools from probability theory including the FeymanKac formula and Markov chains, we demonstrate crystallization for the quantum onedimensional jellium at all densities. 

02/26 2:00pm 
BLOC 220 
Erik Lundberg Florida Atlantic University 
Random matrices arising in the study of random fields
Certain problems in random fields, such as studying the solutions to a random system of equations (e.g., the critical points of a random potential energy landscape) have made important use of random matrix theory. After surveying some applications related to classical random matrix ensembles, we present a new direction in random fields concerning the solutions to problems in enumerative geometry (e.g., the number of lines on a random cubic surface). The resulting random matrices are of a special structured type. We conclude with some open problems that are simple to state. This is joint work with Saugata Basu, Antonio Lerario, and Chris Peterson. 

03/19 2:00pm 
BLOC 220 
Johan Tykesson chalmers university of technology 
Generalized divide and color models
In this talk, we consider the following model: one starts with a finite or countable set V, a random partition of V and a parameter p in [0,1]. The corresponding Generalized Divide and Color Model is the {0,1}valued process indexed by V obtained by independently, for each partition element in the random partition chosen, with probability p, assigning all the elements of the partition element the value 1, and with probability 1p, assigning all the elements of the partition element the value 0.
A very special interesting case of this is the ``Divide and Color Model'' (which motivates the name we use) introduced and studied by Olle Häggström.
Some of the questions which we study here are the following. Under what situations can different random partitions give rise to the same color process? What can one say concerning exchangeable random partitions? What is the set of product measures that a color process stochastically dominates? For random partitions which are translation invariant, what ergodic properties do the resulting color processes have?
The motivation for studying these processes is twofold; on the one hand, we believe that this is a very natural and interesting class of processes that deserves investigation and on the other hand, a number of quite varied wellstudied processes actually fall into this class such as the Ising model, the stationary distributions for the Voter Model, random walk in random scenery and of course the original Divide and Color Model.


03/26 2:00pm 
BLOC 220 
Pierre Tarrago CIMAT 
Subordination methods for free deconvolution
The classical deconvolution of measures is an important problem which consists in recovering the distribution of a random variable from the knowledge of the random variable modified by an independent noise with known distribution.
In this talk, I will discuss the free version of this problem: how can we recover
the distribution of a noncommutative random variable from the knowledge of
the distribution of the random variable modified by the addition (or multiplication) of a free independent noise? Since large independent random matrices
in general positions are approximately free, an answer to the former question is
a first step in the extraction of the spectral distribution of a large matrix from
the knowledge of the matrix with an additive or multiplicative noise.
Contrary to the classical case, the free convolution is not described by an
integral kernel like the Fourier transform. This problem has been circumvented
by Biane, Voiculescu, Belinschi and Bercovici which developed a fixed point
method called subordination. I will explain how this method can be used to
reduce the free deconvolution problem to a classical one. This is a joint work
with Octavio Arizmendi (CIMAT) and Carlos Vargas (CIMAT). 
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