**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.)*** **

**Spring 2018**

á Thursday April 19, 2018, 10am

**Nick
Cook, UCLA**

** Title: **The
maximum of the characteristic polynomial for a random permutation matrix

**Abstract: **Let $P$ be a uniform
random permutation matrix of size $N$ and let $\chi_N(z)=
\det(zI - P)$ denote its characteristic polynomial. We prove a
law of large numbers for the maximum modulus of $\chi_N$
on the unit circle, specifically,

\[

\sup_{|z|=1}|\chi_N(z)|= N^{x_c + o(1)}

\]

with probability
tending to one as $N\to \infty$, for a numerical
constant $x_c\approx
0.677$. The main idea of the proof is to uncover an approximate branching
structure in the distribution of (the logarithm of) $\chi_N$,
viewed as a random field on the circle, and to adapt a well-known second moment
argument for the maximum of the branching random walk. Unlike the
well-studied \emph{CUE field} in which $P_N$ is replaced with a Haar unitary, the distribution of $\chi_N(z)$
is sensitive to Diophantine properties of the argument of $z$. To deal with
this we borrow tools from the Hardy--Littlewood circle method in analytic
number theory. Based on joint work with Ofer Zeitouni.

á Thursday May 3, 2018, 10am

**Lucian
Beznea, IMAR**

**Title: **Invariant,
super and quasi-martingale functions of a Markov process

**Abstract: **We
identify the linear space spanned by the real-valued excessive functions
of a Markov process with the set of those functions which are quasimartingales when we compose them with the process.
Applications to semi-Dirichlet forms are given.
We provide a unifying result which clarifies the relations between
harmonic, co-harmonic, invariant, co-invariant, martingale and
co-martingale functions, showing that in the conservative case they are
all the same. The talk is based on joint works with Iulian Cimpean (Bucharest)
and Michael Roeckner (Bielefeld).

á Thursday May 10, 2018, 10am

**Pascal
Maillard, UniversitŽ 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.

á Thursday May 31, 2018, 10am

**Ian
Charlesworth, UCSD**

**Title: **Bi-free
probability and an approach to conjugate variables.

**Abstract: **Free
entropy theory is an analogue of information theory in a non-commutative
setting, which has had great applications to the examination of structural
properties of von Neumann algebras. I will discuss some ongoing joint work with
Paul Skoufranis to extend this approach to the
setting of bi-free probability which attempts to study simultaneously ``left''
and ``right'' non-commutative variables. I will speak in particular of an
approach to a bi-free Fisher information and bi-free conjugate variables --
analogues of Fisher's information measure and the score function of information
theory. The focus will be on constructing these tools in the non-commutative
setting, and time permitting, I will also mention some results such as bi-free
Cramer-Rao and Stam inequalities, and some quirks of
the bi-free setting which are not present in the free setting.

á Thursday June 7, 2018, 10am

**Dan
Romik, UC Davis**

**Title: **Rational probabilities of connectivity events in loop
percolation and fully packed loops

**Abstract: **In this talk I will describe a family of events arising in
two related probability models, one having to do with uniformly random
"fully packed loops" (a family of combinatorial objects which are in
bijection with alternating sign matrices), and another appearing in connection
with a natural random walk on noncrossing matchings.
The connection between the two models is highly nonobvious and was conjectured
by physicists Razumov and Stroganov
in 2001, and given a beautiful proof in 2010 by Cantini
and Sportiello. Another intriguing phenomenon is that
the probabilities of the events in question, known as "connectivity
events", appear to be rational functions of a size parameter N (with the
simplest such formula being 3(N^2-1)/2(4N^2+1)), but this is only conjectured
in all but a few cases. The attempts to prove such formulas by myself and
others have led to interesting algebraic results on a family of multivariate
polynomials known as "wheel polynomials", and to a family of
conjectural constant term identities that is of independent interest and poses
an interesting challenge to algebraic combinatorialists.

á Thursday June 14, 2018, 10am

**Brian
Hall, University of Notre Dame**

**Title: **Eigenvalues
for Brownian motion in the general linear group

**Abstract **is here.** **

**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

**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.