UCSD Probability Seminar   2011-2012

Seminar Information

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

Organizer:    Todd Kemp
E-mail:          tkemp@math.ucsd.edu

Previous years' webpages:


Fall Quarter Schedule

Sep 8    Lucian Beznea (Simion Stoilow Institute of Mathematics of the Romanian Academy)
Special Time: 2pm  
Title:   The semigroup approach for measure-valued branching processes and a nonlinear Dirichlet problem
Abstract:   We use a branching Markov process on the space of finite configurations to solve a Dirichlet problem associated with the operator Laplacian(u) + u^2. We follow the pioneering works of M. Nagasawa, N. Ikeda, S. Watanabe, M.L. Silverstein, and the approach of E.B. Dynkin.
Oct 20    Brian Hall (Notre Dame)
Title:   Analysis on Lie groups from a probabilistic perspective
Abstract:   I will discuss results in analysis on a compact Lie group that can be obtained using Brownian motion. These include a "Hermite expansion" and an analog of the Segal-Bargmann transform. Both results can be understood by lifting Brownian motion in the group to Brownian motion in the Lie algebra. I will also briefly discuss an open problem concerning the infinite-dimensional limit of these results.
Oct 27    Thomas Laetsch (UCSD)
Title:   An L^2 metric limit theorem for Wiener measure on manifolds with non-positive sectional curvature.
Abstract:   An interpretation for an intuitive expression for the path integral prescription of the Hamiltonian on a manifold is given. Following the direction of the informal derivation of the path integral, a limit theorem is proved using finite dimensional approximations to Wiener measure.
Nov 3    William Massey (Princeton ORFE)
Title:   Skewness-Variance Approximations for Dynamic Rate, Multi-Server Queues with Abandonment.
Abstract:   The multi-server queue with non-homogeneous Poisson arrivals and customer abandonment is a fundamental, dynamic rate, queueing model for large scale service systems such as call centers and hospitals. Scaling the arrival rates and number of servers arises naturally as designing a staffing schedule in response to predictable increasing demand. Mathematically, this gives us the fluid and diffusion limits found in Mandelbaum, Massey, and Reiman. The general rate asymptotics used here for Markovian service networks reduce to the Halfin-Whitt scaling for multi-server queues.

The diffusion limit suggests a Gaussian approximation to the stochastic behavior of this queueing process. The mean and variance are easily computed from a two-dimensional dynamical system for the fluid and diffusion limiting processes. Recent work by Gautam and Ko found that a modified version of these differential equations can be used to obtain better Gaussian estimates of the original queueing system distribution. In this paper, we introduce a new three-dimensional dynamical system that improves on both of these approaches by constructing a non-Gaussian estimation of the mean, variance, and third cumulative moment. This is joint work with Jamol Pender of Princeton University.
Nov 10    Lionel Levine (Cornell University & University of Michigan)
Title:   Logarithmic fluctuations from circularity.
Abstract:   Starting with n particles at the origin in Z^d, let each particle in turn perform simple random walk until reaching an unoccupied site. Lawler, Bramson and Griffeath proved that with high probability the resulting random set of n occupied sites is close to a ball. We show that its fluctuations from circularity are, with high probability, at most logarithmic in the radius of the ball, answering a question posed by Lawler in 1995 and confirming a prediction made by chemical physicists in the 1980's. Joint work with David Jerison and Scott Sheffield.
Nov 17    Michael Kelly (UCSD)
Title:   Bounding The Rate of Adaptation In A Large Asexually Reproducing Population With Fast Mutation Rates.
Abstract:   We consider a model of asexually reproducing individuals. The birth and death rates of the individuals are affected by a fitness parameter. The rate of mutations that cause the fitnesses to change is proportional to the population size, N. The mutations may be either beneficial or deleterious. In a paper by Yu, Etheridge and Cuthbertson (2009) it was shown that the average rate at which the mean fitness increases in this model is bounded below by log^{1-d} N for any d > 0. We achieve an upper bound on the average rate at which the mean fitness increases of O(log N / log log N).
Nov 24    Thanksgiving Holiday: no seminar.
Dec 1    Benoit Collins (RIMS & University of Ottawa)
Title:   Applications of Random Matrix Theory to Quantum Information Theory via Free Probability
Abstract:   I will first describe a generalization of a result by Haagerup and Thorbjornsen on the asymptotic norm of non-commutative polynomials of random matrices, in the case of unitary matrices. Then I will show how such results help us refine our understanding of the outputs of random quantum channels. In particular one obtains optimal estimates for the minimum output entropy of a large class of typical quantum channels. The first part of this talk is based on joint work with Camille Male, and the second part is based on joint work with Serban Belinschi and Ion Nechita.

Winter Quarter Schedule

Jan 12    Pat Fitzsimmons (UCSD)
Title:   Two results on Gaussian measures.
Abstract:   1. I'll show that Hunt's hypothesis (H) fails for Leonard Gross' infinite dimensional Brownian motion, by exhibiting a subset of the state space of the motion that is hit exactly once for certain starting points.

2. It is well known that a Lebesgue measurable additive function from R to R is necessarily continuous (and linear). I'll show how D. Stroock's recent proof of L. Schwartz's "Borel graph theorem" can be adapted to show that a "universally Gaussian measurable" and additive map from one Banach space to another is automatically continuous (and linear).
Jan 19    Wei-Kuo Chen (University of California, Irvine)
Title:   Chaos problem in the Sherrington-Kirkpatrick model.
Abstract:   In physics, the main objective in spin glasses is to understand the strange magnetic properties of alloys. Yet the models invented to explain the observed phenomena are of a rather fundamental nature in mathematics. In this talk, we will focus on one of the most important mean field models, called the Sherrington-Kirkpatrick model, and discuss its disorder and temperature chaos problems. Using the Guerra replica symmetry breaking bound and the Ghirlanda-Guerra identities, we will present mathematically rigorous results and proofs for these problems. Most part of the talk is the joint work with Dmitry Panchenko (Texas A&M).
Feb 2    Brian Rider (University of Colorado, Boulder)
Title:   Spiking the random matrix hard edge.
Abstract:   The largest eigenvalue of a rank one perturbation of random hermitian matrix is known to exhibit a phase transition. If the perturbation is small, one sees the famous Tracy-Widom law; if the perturbation is large, the result is simple Gaussian fluctuations. Further, there is a scaling window about a critical value of the perturbation which leads to a new one parameter family of limit laws. The same phenomena exists for random sample covariance matrices in which one of the population eigenvalues is "spiked", or takes a value other than one. Bloemendal-Virag have shown how this picture persists in the context of the general beta ensembles, giving new formulations of the discovered critical limit laws (among other things). Yet another route, explained here, is to go through the random matrix hard edge, perturbing the smallest eigenvalues in the sample covariance set-up. A limiting procedure then recovers all the alluded to limit distributions. (Joint work with Jose Ramirez.)
Feb 9    Tonci Antunovic (UC Berkeley)
Title:   Some path properties of Brownian motion with variable drift.
Abstract:   If B is a Brownian motion and f is a function in the Dirichlet space, then by Cameron-Martin theorem, the process B-f has the same almost sure path properties as B. In this talk we will present some properties of the image and the zero set of Brownian motion perturbed by certain less regular drifts f (examples include Hilbert curves and Cantor functions). Based on joint works with Krzysztof Burdzy, Yuval Peres, Julia Ruscher and Brigitta Vermesi.
Mar 8    Maria Eulalia Vares (Universidade Federal do Rio de Janeiro)
Title:   First passage percolation and escape stragegies
Abstract:   Consider first passage percolation on Z^d with passage times given by i.i.d. random variables with common distribution F. Let t_\pi(u,v) be the time from u to v for a path \pi and t(u,v) the minimal time among all such paths from u to v. We ask whether or not there exist points x,y in Z^d and a semi-infinite path pi=(y_0=y,y_1,...) such that t_\pi(y,y_{n+1}) < t(x,y_n) for all n. Necessary and sufficient conditions on F are given for this to occur. Based on joint work with Enrique D. Andjel.
Mar 15    Kay Kirkpatrick (UIUC)
Title:   Bose-Einstein condensation and quantum many-body systems
Abstract:   Near absolute zero, a gas of quantum particles can condense into an unusual state of matter, called Bose-Einstein condensation (BEC), that behaves like a giant quantum particle. The rigorous connection has recently been made between the physics of the microscopic many-body dynamics and the mathematics of the macroscopic model, the cubic nonlinear Schrodinger equation (NLS). I'll discuss recent progress with Gerard Ben Arous and Benjamin Schlein on a central limit theorem for the quantum many-body systems, a step towards large deviations for Bose-Einstein condensation.

Spring Quarter Schedule

April 19    Michael Kozdron (University of Regina & MSRI)
Title:   On the convergence of loop-erased random walk to SLE(2) in the natural parametrization.
Abstract:   The Schramm-Loewner evolution is a one-parameter family of random growth processes in the complex plane introduced by Oded Schramm in 1999. In the past decade, SLE has been successfully used to describe the scaling limits of various two-dimensional lattice models. One of the first proofs of convergence was due to Lawler, Schramm, and Werner who gave a precise statement that the scaling limit of loop-erased random walk is SLE with parameter 2. However, their result was only for curves up to reparameterization. There is reason to believe that the scaling limit of loop-erased random walk is SLE(2) with the very specific natural time parameterization that was recently introduced by Lawler and Sheffield, and further studied by Lawler, Zhou, and Rezaei. I will describe several possible choices for the parameterization of the discrete curve that should all give the natural time parameterization in the limit, but with the key difference being that some of these discrete time parameterizations are easier to analyze than the others. This talk is based on joint work in progress with Tom Alberts and Robert Masson.
Apr 26    Nathan Ross (UC Berkeley)
Title:   A probabilistic approach to local limit theorems
Abstract:   We discuss a new method for obtaining a local limit theorem (LLT) from a known distributional limit theorem. The method rests on a simple analytic inequality (essentially due to Hardy, Landau, and Littlewood) which can be applied directly after quantifying the smoothness of the distribution of interest. These smoothness terms are non-trivial to handle and so we also provide new (probabilistic) tools for this purpose. We illustrate our approach by showing LLTs with rates for the magnetization in the Curie-Weiss model at high temperature and for some counts in an Erdos-Renyi random graph. This is joint work with Adrian Roellin.
May 3    Ioana Dumitriu (University of Washington)
Title:   Random Regular Graphs: how local structure determines Wigner-matrix eigenstatistics
Abstract:   Fueled partly by the recent explosive progress in proving universality for random matrix ensembles, spectral graph theory has also made significant advances in the past couple of years.

For the Erdos-Renyi random graph model, p >> log n / n, short-scale semicircle laws, spacings in the bulk, extremal eigenvalue distributions, and complete delocalization of eigenvectors have been shown by Erdos-Knowles-Yau-Yin ('11-'12) via a treatment similar to the Wigner matrix case (and based on "hard" analysis).

Regular random graphs, being less Wigner-like, are harder to tackle with such methods, but recent combinatorial treatments have yielded much success. Through a careful estimation of probabilities and a reduction to the Erdos-Renyi model, Tran-Vu-Wang ('10) have been able to establish short-scale semicircle laws for the d >> log n regime in the case of d-regular graphs. Also, a certain type of delocalization has been established for an extended growth regime for d by Brooks-Lindenstrauss ('09). Somewhat complementarily, for a polylogarithmic regime of d, D.-Pal proved both short-scale semicircle laws, and a slightly weaker delocalization ('09) using the local tree-like structure of the graph.

I will sketch the method of proof, and explain how the same ideas can be used to obtain similar results for bipartite, biregular graphs (D.-Johnson, '12), global fluctuations for the permutation model (D.-Johnson-Pal-Paquette, '11), and global fluctuations for the uniform model in polylogarithmic d regime (Johnson '12).
May 17    Ruth Williams (UCSD)
Special Time: 4pm  
Title:   Resource Sharing in Stochastic Networks
Abstract:   Stochastic networks are used as models for complex processing systems involving dynamic interactions subject to uncertainty. Applications arise in high-tech manufacturing, the service industry, telecommunications, computer systems and bioengineering. The control and analysis of such networks present challenging mathematical problems. In this talk, a concrete application will be used to illustrate a general approach to the study of stochastic processing networks based on deriving more tractable approximate models. Specically, we will consider a model of Internet congestion control in which processing can involve the simultaneous use of several resources (or links), a phenomenon that is not well understood. Elegant fluid and diffusion approximations will be derived and used to study the performance of this model. A key insight from this analysis is a geometric representation of the consequences of using a "fair" policy for the sharing of resources. The talk will conclude with a summary of the current status and description of open problems associated with approximate models for general stochastic processing networks.
May 24    Konstantinos Spiliopoulos (Brown University & Boston University)
Location: AP&M 7421  
Title:   Large Deviations and Monte Carlo Methods for Problems with Multiple Scales
Abstract:   The need to simulate rare events occurs in many application areas, including telecommunication, finance, insurance, computational physics and chemistry. However, virtually any simulation problem involving rare events will have a number of mathematical and computational challenges. As it is well known, standard Monte Carlo sampling techniques perform very poorly in that the relative errors under a fixed computational effort grow rapidly as the event becomes more rare. In this talk, I will discuss large deviations, rare events and Monte Carlo methods for systems that have multiple scales and that are stochastically perturbed by small noise. Depending on the type of interaction of the fast scales with the strength of the noise we get different behavior, both for the large deviations and for the corresponding Monte Carlo methods. Using stochastic control arguments we identify the large deviations principle for each regime of interaction. Furthermore, we derive a control (equivalently a change of measure) that allows to design asymptotically efficient importance sampling schemes for the estimation of associated rare event probabilities and expectations of functionals of interest. Standard Monte Carlo methods perform poorly in these kind of problems in the small noise limit. In the presence of multiple scales one faces additional difficulties and straightforward adaptation of importance sampling schemes for standard small noise diffusions will not produce efficient schemes. We resolve this issue and demonstrate the theoretical results by examples and simulation studies. Applications of these results in chemistry problems and in mathematical finance will also be discussed.
June 7    Tom Alberts (Caltech)
Title:   Diffusions of Multiplicative Cascades
Abstract:   A multiplicative cascade is a randomization of any measure on the unit interval, constructed from an iid collection of random variables indexed by the dyadic intervals. Given an arbitrary initial measure I will describe a method for constructing a continuous time, measure valued process whose value at each time is a cascade of the initial one. The process also has the Markov property, namely at any given time it is a cascade of the process at any earlier time. It has the further advantage of being a martingale and, under certain extra conditions, it is also continuous. I will discuss applications of this process to models of tree polymers and one-dimensional random geometry.

Joint work with Ben Rifkind (University of Toronto).