**UCSD Probability Seminar, 2017-2018**

*The probability seminar
meets at 10am on Thursdays in AP&M 6402, unless specifically indicated
otherwise. Please send any inquiries to the organizer *

*T. Zheng (
tzheng2 at math dot ucsd dot edu.)*** **

**Winter 2018**

á Thursday
February 8, 2018, 10am

**Yumeng**** Zhang, Stanford
University**

**Title: **Rapid
mixing of Glauber dynamics on hypergraph independent
set

**Abstract: **Independent sets in hypergraphs can be encoded as 0-1
configurations on the set of vertices such that each hyperedge
is adjacent to at least one 0. This model has been studied in the CS community
for its large gap between efficient MCMC algorithms (previously d <(k-1)/2)
and the conjectured onset of computational hardness (d > O(2^{k/2}) ), where d is the largest degree of vertices and k is the
minimum size of hyperedges. In
this talk we use a percolation
approach to show that the Glauber dynamics is
rapid mixing for d < O(2^{k/2} ),
closing the gap up to a multiplicative constant.

This is joint work with Jonathan Hermon and Allan Sly.

á Thursday
February 15, 2018, 10am

**Leonid Petrov, University of
Virginia**

**Title:** Nonequilibrium particle systems in
inhomogeneous space

**Abstract:** I will discuss stochastic interacting particle systems in the KPZ
universality class evolving in one-dimensional inhomogeneous space. The
inhomogeneity means that the speed of a particle depends on its location. I
will focus on integrable examples of such systems,
i.e., for which certain observables can be written in exact form suitable for
asymptotic analysis. Examples include a continuous-space version of TASEP
(totally asymmetric simple exclusion process), and the pushTASEP
(=long-range TASEP). For integrable systems, density
limit shapes can be described in an explicit way. We also obtain asymptotics of fluctuations, in particular, around slow
bonds and infinite traffic jams caused by slowdowns.

á Thursday March
8, 2018, 10am

**Georg Menz, UCLA**

**Title: ** A quantitative theory of the hydrodynamic limit

**Abstract: **The hydrodynamic limit of the Kawasaki dynamics states that a
certain stochastic evolution of a lattice system converges macroscopically to a
deterministic non-linear heat equation. We will discuss how the statement of
the hydrodynamic limit can be made quantitative. The key step is to introduce
an additional evolution on a mesoscopic scale that emerges from projecting the
microscopic observables onto splines. The hydrodynamic limit is then deduced in
two steps. In the first step one shows the convergence of the microscopic to
the mesoscopic evolution and in the second step one deduces the convergence of
the mesoscopic to the macroscopic evolution.

The talk is about a
joint work with Deniz Dizdar,
Felix Otto and Tianqi Wu.

á Thursday March
15, 2018, 10am

**Karl Liechty, De Paul University**

**Title: **Tacnode
processes, winding numbers, and Painleve II

**Abstract: **Abstract: I will discuss a model of nonintersecting Brownian
bridges on the unit circle, which produces quite a few universal determinantal processes as scaling limits. I will focus on
the tacnode process, in which two groups of particles
meet at a single point in space-time before separating, and introduce a new
version of the tacnode process in which a finite
number of particles "switch sides" before the two groups separate. We
call this new process the k-tacnode process, and it
is defined by a kernel expressed in terms of a system of tau-functions for the Painleve II equation. Technically, our model of
nonintersecting Brownian bridges on the unit circle is studied using a system
of discrete orthogonal polynomials with a complex (non-Hermitian) weight, so
I'll also discuss some of the analytical obstacles to that analysis.

This is joint work with Dong Wang and Robert Buckingham

**Spring 2018**

á Thursday April
19, 2018, 10am

**Nick Cook, UCLA**

**Title:**

**Abstract: **

á Thursday May 3,
2018, 10am

**Lucien Beznea, IMAR**

**Title:**

**Abstract: **

á Thursday May 10,
2018, 10am

**Pascal Maillard, UniversitŽ Paris-Sud**

**Title:**

**Abstract: **

á Thursday June
7, 2018, 10am

**Dan Romik, UC Davis**

**Title:**

**Abstract: **

**Fall 2017**

á Thursday, Oct 5, 2017, 10am

**Jean-Dominique Deuschel, TU Berlin**

**Title:** Random walks in dynamical balanced environment

**Abstract:** We prove a quenched invariance principle and local limit theorem for
a random walk in an ergodic balanced time dependent environment on the
lattice. Our proof relies on the parabolic Harnack
inequality for the adjoint operator. This is
joint work with X. Guo.

á Thursday, Oct 12, 2017, 10am

**Pierre-Olivier Goffard, UC Santa
Barbara**

**Title:** Boundary
Crossing Problems with Applications to Risk Management.

**Abstract:** Many
problems in stochastic modeling come down to study the crossing time of a
certain stochastic process through a given boundary, lower or upper. Typical
fields of application are in risk theory, epidemic modeling, queueing,
reliability and sequential analysis. The purpose of this talk is to present a
method to determine boundary crossing probabilities linked to stochastic point
processes having the order statistic property. A very well-known boundary
crossing result is revisited, a detailed proof is given. the same arguments may
be used to derive results in trickier situations. We further discuss the
practical implications of this classical.

á Thursday, Oct 19, 2017, 10am

**Omer Tamuz, Caltech**

**Title: **Large
deviations in social learning

**Abstract:** Models of information
exchange that originate from economics provide interesting questions in probability.
We will introduce some of these models, discuss open questions, and explain
some recent results.

Joint with Wade
Hann-Caruthers, Matan Harel,
Vadim Martynov, Elchanan Mossel and Philipp Strack

á Thursday, Nov 2, 2017, 10am

**Qiang**** Zeng, Northwestern University**

**Title: **The
Sherrington-Kirkpatrick model is Full-step Replica Symmetry Breaking at zero
temperature

**Abstract****: **Starting in 1979, the
physicist Giorgio Parisi wrote a series of ground
breaking papers introducing the idea of replica symmetry breaking, which
allowed him to predict a solution for the Sherrington-Kirkpatrick (SK) model by
breaking the symmetry of replicas infinitely many times. This is known as
full-step replica symmetry breaking (FRSB). In this talk, we will provide a
mathematical proof of Parisi's FRSB prediction at
zero temperature for the more general mixed p-spin model. More precisely, we
will show that the functional order parameter of this model is not a step
function. This talk is based on joint work with Antonio Auffinger
and Wei-Kuo Chen.

á Thursday, Nov
30, 2017, 10am

**Tom Alberts, University of Utah**

**Title: **Geometric
Methods for Last Passage Percolation

**Abstract: ** In an attempt to generalize beyond solvable methods of analysis
for last passage percolation, recently Eric Cator (Radboud University, Nijmegen) and I have started analyzing
the piecewise linearity of the last passage model. The tools we use to this
point are purely geometric, but have the potential advantage that they can be
used for very general choices of random inputs. I will describe the very pretty
geometry of the last passage model, our work in progress to use it to produce
probabilistic information, and some connections to algebraic geometry.

á Thursday, Dec
7, 2017, 10am

**Anas**** Rahman, University of Melbourne**

**Title: **Random Matrices and Loop
Equations

**Abstract:** I will begin by introducing the Gaussian, Laguerre and Jacobi ensembles
and their corresponding eigenvalue densities. The moments of these eigenvalue densities
are generated by the corresponding resolvent, R(x).
When investigating large matrices of size N, it is natural to expand R(x) as a
series in 1/N, as N tends to infinity. The loop equation formalism enables one
to compute R(x) to any desired order in 1/N via a triangular recursive system.
This formalism is equivalent to the topological recursion, the Schwinger-Dyson
equations and the Virasoro constraints, among other
things. The loop equations provide a relatively accessible entry-point to these
topics and my derivation will rely on nothing more than integration by parts,
as Aomoto applied to the Selberg
integral. Time permitting, I may also explore links to the topological
recursion and/or some combinatorics.

All original results will be from joint work with
Peter Forrester and Nicholas Witte.