# UC San Diego Probability Seminar   2016-2017

#### Seminar Information

Thursdays, 10am   AP&M 6402
University of California, San Diego
La Jolla, CA 92093-0112
Map

Organizer:    Bruce Driver
E-mail:          bdriver@ucsd.edu

Previous years' webpages:

## Fall 2016 Schedule

Sep 29    Bruce Driver,  UCSD
 Title: The Makeenko-Migdal equations for the 2d -Yang-Mills measure Abstract: We will discuss the Makeenko--Migdal equation (MM equation) which relates variations of a "Wilson loop functional" (relative to the Euclidean Yang--Mills measure) in the neighborhood of a simple crossing to the associated Wilson loops on either side of the crossing. We will begin by introducing the 2d -- Yang-Mills measure and explaining the necessary background in order to understand the theorem. The goal is to describe the original heuristic argument of Makeenko and Migdal and then explain how these arguments can be made rigorous using stochastic calculus.

Oct 6    Name, University
 Title: Title here. Abstract: Abstract here.

Oct 13    Name, University
 Title: Title here. Abstract: Abstract here.

Oct 20    Wei Wu, NYU
 Title: Loop erased random walk, uniform spanning tree and bi-Laplacian Gaussian field in the critical dimension. Abstract: Critical lattice models are believed to converge to a free field in the scaling limit, at or above their critical dimension. This has been (partially) established for Ising and Phi^4 models for d \geq 4. We describe a simple spin model from uniform spanning forests in $\Z^d$ whose critical dimension is 4 and prove that the scaling limit is the bi-Laplacian Gaussian field for $d\ge 4$. At dimension 4, there is a logarithmic correction for the spin-spin correlation and the bi-Laplacian Gaussian field is a log correlated field. The proof also improves the known mean field picture of LERW in d=4, by showing that the renormalized escape probability (and arm events) of 4D LERW converge to some "continuum escaping probability". Based on joint works with Greg Lawler and Xin Sun.

Oct 27    Name, University
 Title: Title here. Abstract: Abstract here.

Nov 3     Reza Aghajani, UCSD
 Title: Mean-Field Dynamics of Load-Balancing Networks with General Service Distributions Abstract: We introduce a general framework for studying a class of randomized load balancing models in a system with a large number of servers that have generally distributed service times and use a first-come-first serve policy within each queue. Under fairly general conditions, we use an interacting measure-valued process representation to obtain hydrodynamics limits for these models, and establish a propagation of chaos result. Furthermore, we present a set of partial differential equations (PDEs) whose solution can be used to approximate the transient behavior of such systems. We prove that these PDEs have a unique solution, use a numerical scheme to solve  them, and demonstrate the efficacy of these approximations using Monte Carlo simulations. We also illustrate how the PDE can be used to gain insight into network performance.

Nov 10    Stephen DeSalvo, UCLA
 Title: Poisson approximation of combinatorial assemblies with low rank Abstract: We present a general framework for approximating the component structure of random combinatorial assemblies when both the size $n$ and the number of components $k$ is specified. The approach is an extension of the usual saddle point approximation, and we demonstrate near-universal behavior when the rank $r := n-k$ is small relative to $n$ (hence the name `low rank’). In particular, for $\ell = 1, 2, \ldots$, when $r \asymp n^\alpha$, for $\alpha \in \left(\frac{\ell}{\ell+1}, \frac{\ell+1}{\ell+2}\right)$, the size~$L_1$ of the largest component converges in probability to $\ell+2$. When $r \sim t\, n^{\ell/(\ell+1)}$ for any $t>0$ and any positive integer $\ell$, we have $\P(L_1 \in \{\ell+1, \ell+2\}) \to 1$. We also obtain as a corollary bounds on the number of such combinatorial assemblies, which in the special case of set partitions fills in a countable number of gaps in the asymptotic analysis of Louchard for Stirling numbers of the second kind. This is joint work with Richard Arratia.

Nov 17    Name, University
 Title: Title here. Abstract: Abstract here.

Dec 1    Name, University
 Title: Title here. Abstract: Abstract here.

## "Winter" 2017 Schedule

Jan 12    Konstantin Tikhomirov, Princeton
 Title: The spectral gap of dense random regular graphs Abstract: Let G be uniformly distributed on the set of all simple d-regular graphs on n vertices, and assume d is bigger than some (small) power of n. We show that the second largest eigenvalue of G is of order √d with probability close to one. Combined with earlier results covering the case of sparse random graphs, this settles the problem of estimating the magnitude of the second eigenvalue, up to a multiplicative constant, for all values of n and d, confirming a conjecture of Van Vu. Joint work with Pierre Youssef.

Jan 19    Name, University
 Title: Title here. Abstract: Abstract here.

Jan 26    Name, University
 Title: Title here. Abstract: Abstract here.

Feb 2    Name, University
 Title: Title here. Abstract: Abstract here.

Feb 9    Masha Gordina, University of Connecticut
 Title: Couplings for hypoelliptic diffusions Abstract: Coupling is a way of constructing Markov processes with prescribed laws on the same probability space. It is known that the rate of coupling (how fast you can make two processes meet) of elliptic/Riemannian diffusions is connected to the geometry of the underlying space. In this talk we consider coupling of hypoelliptic diffusions (diffusions driven by vector fields satisfying Hoermander's condition). S. Banerjee and W. Kendall constructed successful Markovian couplings for a large class of hypoelliptic diffusions. We use a non-Markovian coupling of Brownian motions on the Heisenberg group, and then use this coupling to prove analytic gradient estimates for harmonic functions for the sub-Laplacian. This talk is based on the joint work with Sayan Banerjee and Phanuel Mariano.

Feb 16    Name, University
 Title: Title here. Abstract: Abstract here.

Feb 23    Douglas Rizzolo, University of Deleware
 Title: Diffusions on the space of interval partitions with Poisson-Dirichlet stationary distributions Abstract: We construct a pair of related diffusions on a space of partitions of the unit interval whose stationary distributions are the complements of the zero sets of Brownian motion and Brownian bridge respectively. Our methods can be extended to construct a class of partition-valued diffusions obtained by decorating the jumps of a spectrally positive Levy process with independent squared Bessel excursions. The processes of ranked interval lengths of our partition-valued diffusions are members of a two parameter family of infinitely many neutral allele diffusion models introduced by Ethier and Kurtz (1981) and Petrov (2009). Our construction is a step towards describing a diffusion on the space of real trees, stationary with respect to the law of the Brownian CRT, whose existence has been conjectured by Aldous. Based on joint work with N. Forman, S. Pal, and M. Winkel.

Mar 2    Professor Jiangang Ying, Fudan University
 Title: On symmetric linear diffusions and related problems. Abstract: In this talk, a representation of local and regular Dirichlet forms on real line, which are associated with symmetric linear diffusions, will be given and based on this, several related problems will be discussed.

Mar 9    Name, University
 Title: Title here. Abstract: Abstract here.

Mar 16    Laurent Sallof-Coste, Cornell University
 Title: Convolution powers of complex valued functions Abstract: The study of partial sums of iid sequences is tightly connected to that of iterated convolutions. In this talk, I will discuss results that resemble local limit theorems for iterated convolution of complex valued functions in the case of $\mathbb Z$ and $\mathbb Z^d$. Similarities and differences with the probability densities will be in the spotlight.

## Spring 2017 Schedule

Apr 6    Name, University
 Title: Title here. Abstract: Abstract here.

Apr 13    Name, University
 Title: Title here. Abstract: Abstract here.

Apr 20    Name, University
 Title: Title here. Abstract: Abstract here.

Apr 27    Name, University
 Title: Title here. Abstract: Abstract here.

May 4    Name, University
 Title: Title here. Abstract: Abstract here.

May 11    Name, University
 Title: Title here. Abstract: Abstract here.

May 18    Martin Tassy, UCLA
 Title: Variational principles for discrete maps Abstract: Previous works have shown that arctic circle phenomenons and limiting behaviors of some integrable discrete systems can be explained by a variational principle. In this talk we will present the first results of the same type for a non-integrable discrete system: graph homomorphisms form Z^d to a regular tree. We will also explain how the technique used could be applied to other non-integrable models.

May 25    Name, University
 Title: Title here. Abstract: Abstract here.

Jun 1    Amber L. Puha, California State University San Marcos
 Title: Asymptotically Optimal Policies for Many Server Queues with Reneging Abstract: The aim of this work (joint with Amy Ward (USC, Marshall School of Business)) is to determine fluid asymptotically optimal policies for many server queues with general reneging distributions. For exponential reneging distributions, it has been shown that static priority policies are optimal in a variety of settings, that include generally distributed interarrival and service times. Moreover, in these cases, the priority ranking is determined by a simple rule known as the c-mu-theta rule. For non-exponential reneging distributions, the story is more complex. We study reneging distributions with monotone hazard rates. For reneging distributions with bounded, nonincreasing hazard rates, we prove that static priority is not necessarily asymptotically optimal. We identify a new class of policies, which we are calling Random Buffer Selection and prove that these are asymptotically optimal in the fluid limit. We further identify a fluid approximation for the limiting cost as the optimal value of a certain optimization problem. For reneging distributions with nondecreasing hazard rates, our work suggests that static priority policies are in fact optimal, but the rule for determining the priority ranking seems more complex in general. It is work in progress to prove this.

Jun 8    Name, University
 Title: Title here. Abstract: Abstract here.