| Date: | January 29, 2018 |
| Time: | 2:00pm |
| Location: | BLOC 628 |
| Speaker: | paul jung, Korea Advanced Institute of Science and Technology |
| Title: | Infinite-volume Gibbs measures for the 1D-Coulomb jellium |
| Abstract: | 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 (non-ergodicity of the Gibbs measures) at low densities of electrons. Using tools from probability theory including the Feyman-Kac formula and Markov chains, we demonstrate crystallization for the quantum one-dimensional jellium at all densities. |
|
| Date: | February 26, 2018 |
| Time: | 2:00pm |
| Location: | BLOC 220 |
| Speaker: | Erik Lundberg, Florida Atlantic University |
| Title: | Random matrices arising in the study of random fields |
| Abstract: | 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. |
|
| Date: | March 19, 2018 |
| Time: | 2:00pm |
| Location: | BLOC 220 |
| Speaker: | Johan Tykesson, chalmers university of technology |
| Title: | Generalized divide and color models |
| Abstract: | 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 1-p, 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 well-studied 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.
|
|
| Date: | March 26, 2018 |
| Time: | 2:00pm |
| Location: | BLOC 220 |
| Speaker: | Pierre Tarrago, CIMAT |
| Title: | Subordination methods for free deconvolution |
| Abstract: | 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 non-commutative 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). |
|
| Date: | April 2, 2018 |
| Time: | 2:00pm |
| Location: | BLOC 220 |
| Speaker: | Eliran Subag, Courant Institute NYU |
| Title: | The geometry of pure states in spherical spin glasses |
| Abstract: | Following Parisi's celebrated replica symmetry breaking solution for mean-field spin glasses (1980), physicists invested considerable efforts to interpret it in terms of `physical' properties of the system. One of the central ideas in their theory was that the system decomposes into `pure states', organized in an ultrametric structure. In his seminal work Talagrand (2010) proved for a wide class of models the existence of such a decomposition -- a sequence of subsets on which the Gibbs measure asymptotically concentrates. Panchenko (2013) established the famous ultrametricity conjecture, which implies, in particular, that those subsets are organized in a certain hierarchical structure.
In the context of the spherical models, I will describe a new geometric picture for the above, in which the hierarchy is expressed through a tree of nested spherical bands. In particular, the pure states concentrate on bands corresponding to the leaves of this tree. |
|
| Date: | April 30, 2018 |
| Time: | 2:00pm |
| Location: | BLOC 220 |
| Speaker: | Jeffery Kuan, Columbia |
| Title: | Universality questions for the (2+1)--dimensional AKPZ class |
| Abstract: | We consider the (2+1)--dimensional Anisotropic Kardar-Parisi-Zhang (KPZ) universality class, which is a variant of the (1+1)--dimensional KPZ class. Several models, arising from random matrices, representation theory, and non--commutative probability, lead to conjectures for the scaling exponents and limiting distribution.
|
|
| Date: | May 21, 2018 |
| Time: | 2:00pm |
| Location: | BLOC 628 |
| Speaker: | Pascal Maillard, Université Paris-Sud |
| Title: | Fluctuations of the Gibbs measure of branching Brownian motion at critical temperature |
| Abstract: | Branching Brownian motion is a prototype of a disordered system and a toy model for spin glasses and log-correlated fields. It also has an exact duality relation with the FKPP equation, a well-known reaction diffusion equation. In this talk, I will present recent results (obtained with Michel Pain) on the fluctuations of the Gibbs measure at the critical temperature. By Gibbs measure I mean here the measure whose atoms are the positions of the particles, weighted by their Gibbs weight. It is known that this Gibbs measure, after a suitable scaling, converges to a deterministic measure. We prove a non-standard central limit theorem for the integral of a function against the Gibbs measure, for a large class of functions. The possible limits are 1-stable laws with arbitrary asymmetry parameter depending on the function. In particular, the derivative martingale and the usual additive martingale satisfy such a central limit theorem with, respectively, a totally asymmetric and a Cauchy distribution. |
|
| Date: | May 22, 2018 |
| Time: | 2:00pm |
| Location: | BLOC 628 |
| Speaker: | Xinyi Li, University of Chicago |
| Title: | Minkowski Content for Brownian cut points |
| Abstract: | Consider 2 or 3-dimensional Brownian motion and the set of its cut points. In this talk, we will discuss about its relationship with the intersection exponent of Brownian motion and prove the existence of its Minkowski content as a random Borel measure. As an application, we are able to identify this measure as the scaling limit of the counting measure of pivotal points for percolation on the triangular lattice in the 2-dimensional case. This is a joint project with Nina Holden, Greg Lawler and Xin Sun.
|
|
Menu