UC San Diego Probability Seminar   2015-2016

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 2015 Schedule

Oct 8    Joscha Diehl, Technical University of Berlin
Title:   Weakly asymmetric Ginzburg-Landau lattice model and stochastic Burgers.
Abstract:   The Ginzburg-Landau lattice model is a system of stochastic differential equations that is known to converge, if properly rescaled, to a linear SPDE. When introducing an asymmetry in the model that vanishes in the limit, we prove convergence to the (semilinear) stochastic Burgers equation. The theory for this stochastic PDE is non-trivial, and we apply the theory of an energy solution. The latter was introduced by Goncalves-Jara and under a slight reformulation Gubinelli-Perkowski were recently able to show well-posedness for this equation.
Oct 15    Eviatar Procaccia, Texas A&M University
Title:   The boy who cried Wulff.
Abstract:   We consider a Gibbs distribution over random walk paths on the square lattice, proportional to the cardinality of the path's boundary. We show that in the zero temperature limit, the paths condensate around an asymptotic shape. This limit shape is characterized as the minimizer of the functional, mapping open connected subsets of the plane to the sum of their principle eigenvalue and perimeter (with respect to some norm). A prime novel feature of this limit shape is that it is not in the class of Wulff shapes.
Oct 22    David Renfrew, UCLA
Title:   Spectral properties of large Non-Hermitian Random Matrices.
Abstract:   The study of the spectrum of non-Hermitian random matrices with independent, identically distributed entries was introduced by Ginibre and Girko. I will present two generalizations of the iid model where the independence and identical distribution assumptions are relaxed.
Oct 29    Xue-Mei Li, University of Warwick
Title:   Limits theorems on Random ODEs on Manifolds and Examples.
Abstract:   We explain limit theorems associated with a family of random ordinary differential equations on manifolds, driven by randomly perturbed vector fields. After rescaling, the differentiable random curves converge to a Markov process whose Markov generator can be written explicitly in Hormander form. We also give rates of convergence in the Wasserstein distance.

Example 1. A unit speed geodesic, which chooses a direction randomly and uniformly at every instant of order 1/epsilon, converges to a Brownian motion as epsilon tends to 0. Furthermore their horizontal lifts converge to the Horizontal Brownian motion.

Examples 2. Inspired by the problem of the convergence of Berger's spheres to a S^(1/2), we introduce a family of Interpolation equations on a Lie group G. These are stochastic differential equations on a Lie group driven by diffusion vector fields in the direction of a subgroup H rescaled by 1/epsilon, and a drift vector field in a transversal direction. If there is a reductive structure, we identify a family of slow variables which, after rescaling, converges to a Markov process on G. Furthermore, the projection of the limiting Markov process to the orbit manifolds G/H is Markov. The limits can be identified in terms of the eigenvalue of a second order differential operator on the subgroup and the Ad(H) invariant decomposition of the Lie algebra.
Nov 5    Steven Heilman, UCLA
Title:   Low Correlation Noise Stability of Euclidean Sets.
Abstract:   The noise stability of a Euclidean set is a well-studied quantity. This quantity uses the Ornstein-Uhlenbeck semigroup to generalize the Gaussian perimeter of a set. The noise stability of a set is large if two correlated Gaussian random vectors have a large probability of both being in the set. We will first survey old and new results for maximizing the noise stability of a set of fixed Gaussian measure. We will then discuss some recent results for maximizing the low-correlation noise stability of three sets of fixed Gaussian measures which partition Euclidean space. Finally, we discuss more recent results for maximizing the low-correlation noise stability of symmetric subsets of Euclidean space of fixed Gaussian measure. All of these problems are motivated by applications to theoretical computer science.
Nov 12    Lionel Levine, Cornell University
Title:   Threshold state of the abelian sandpile.
Abstract:   A sandpile on a graph is an integer-valued function on the vertices. It evolves according to local moves called topplings. Some sandpiles stabilize after a finite number of topplings, while others topple forever. For any sandpile s_0 if we repeatedly add a grain of sand at an independent random vertex, we eventually reach a "threshold state'' s_T that topples forever. Poghosyan, Poghosyan, Priezzhev and Ruelle conjectured a precise value for the expected amount of sand in s_T in the limit as s_0 tends to negative infinity. I will outline how this conjecture was proved here by means of a Markov renewal theorem.
Nov 26    Thanksgiving Holiday: no seminar.
Dec 4, 3:00pm    Tianyi Zheng, Stanford University
Title:   Random walk parameters and the geometry of groups
Abstract:   The first characterization of groups by an asymptotic description of random walks on their Cayley graphs dates back to Kesten's criterion of amenability. I will first review some connections between the random walk parameters and the geometry of the underlying groups. I will then discuss a flexible construction that gives solution to the inverse problem (given a function, find a corresponding group) for large classes of speed, entropy and return probability of simple random walks on groups of exponential volume growth. Based on joint work with Jeremie Brieussel.

"Winter" 2016 Schedule

Jan 7    Alex Hening, University of Oxford
Title:   The free path in a high velocity random flight process associated to a Lorentz gas in an external field.
Abstract:   We investigate the asymptotic behavior of the free path of a variable density random flight model in an external field as the initial velocity of the particle goes to infinity. The random flight models we study arise naturally as the Boltzmann-Grad limit of a random Lorentz gas in the presence of an external field. By analyzing the time duration of the free path, we obtain exact forms for the asymptotic mean and variance of the free path in terms of the external field and the density of scatterers. As a consequence, we obtain a diffusion approximation for the joint process of the particle observed at reflection times and the amount of time spent in free flight. This is based on joint work with Doug Rizzolo and Eric Wayman.
Jan 25, 10:00am    Fabrice Baudoin, Purdue University
Title:   Topics in Stochastic Analysis
Abstract:   Starting from basic principles, we will present some recent developments in the theory of rough paths and in the theory of sub-Riemannian diffusions. The first part of the talk will be devoted to the theory of rough paths. This theory was developed in the 1990s by T. Lyons, and allows to give a sense to solutions of differential equations driven by irregular paths. The theory itself has nothing to do with probability theory but has had a tremendous impact on several recent developments in stochastic analysis; it served as an inspiration to Hairer's regularity structure theory, for which he was awarded the Fields medal in 2014. In the second part of the talk, we will address several problems in the geometric analysis of some sub-Riemannian manifolds, which can (surprisingly) be solved using diffusion semigroups techniques.
Jan 28    Prasad Tetali, Georgia Tech
Title:   Displacement convexity of entropy and discrete curvature
Abstract:   Inspired by the recent developments and mature understanding of the notion of lower-boundedness of Ricci curvature in continuous settings (such as Riemannian manifolds), several independent groups of researchers have proposed intriguing analogs of such a curvature in discrete settings (such as graphs). The proposals depend on the perspective being probabilistic, analytical or combinatorial. In this talk, I will briefly mention a few of these approaches, consequences, and state some open problems.
Feb 4    Michael Anshelevich, Texas A&M University
Title:   The exponential homomorphism in non-commutative probability
Abstract:   The wrapping transformation is easily seen to intertwine convolutions of probability measures on the real line and the circle. It is also easily seen to not transform additive free convolution into the multiplicative one. However, we show that on a large class L of probability measures on the line, wrapping does transform not only the free but also Boolean and monotone convolutions into their multiplicative counterparts on the circle. This allows us to prove various identities between multiplicative convolutions by simple applications of the additive ones. The restriction of the wrapping to L has several other unexpected nice properties, for example preserving the number of atoms. This is joint work with Octavio Arizmendi.
Feb 18    Solesne Bourguin, Boston University
Title:   Portmanteau inequalities on the Poisson space
Abstract:   In this talk, we present some results originating from a new general inequality obtained by combining the Chen-Stein method with Malliavin calculus on the Poisson space, such as multidimensional Poisson approximations, mixed limit theorems, as well as a characterization of asymptotic independence for U-statistics. Applications to stochastic geometry through limit theorems involving the joint convergence of vectors of subgraph-counting statistics exhibiting both a Poisson and a Gaussian behavior will also be discussed.
Feb 25    Tai Melcher, University of Virginia
Title:   Small-time asymptotics of subRiemannian Hermite functions
Abstract:   As in the Riemannian setting, a subRiemannian heat kernel is controlled by the geometry of the underlying manifold. In particular, the asymptotic behavior of the kernel can reveal certain geometric and topological data. We study the logarithmic derivatives of subRiemannian heat kernels in some cases and show that, under appropriate scaling, they converge to their analogues on stratified groups. This gives one quantification of the now standard idea that stratified groups play the role of the tangent space to subRiemannian manifolds. This is joint work with Joshua Campbell.
Mar 10    Steve Zelditch, Northwestern University
Title:   Large N limit of heat kernel measure on positive Hermitian matrices and random metrics.
Abstract:   Heat kernel measure K(t, I, A) dA on positive Hermitian NxN matrices is a probability measure whose large N limit is important for several different types of problems in mathematical physics. My talk introduces a new application: to random Kahler metrics on any Kahler manifold. The pair correlation function of random metrics is explicitly calculated for each N. The large N asymptotics are closely related to zero sets of random holomorphic functions.

Spring 2016 Schedule

Apr 14    Tom Alberts, University of Utah
Location:   AP&M 7421
Title:   Random Geometry in the Spectral Measure of the Circular Beta Ensemble
Abstract:   The Circular Beta Ensemble is a family of random unitary matrices whose eigenvalue distribution plays an important role in statistical physics. The spectral measure is a canonical way of describing the unitary matrix that takes into account the full operator, not just its eigenvalues. When the matrix is infinitely large (i.e. an operator on some infinite-dimensional Hilbert space) the spectral measure is supported on a fractal set and has a rough geometry on all scales. This talk will describe the analysis of these fractal properties. Joint work with Raoul Normand and Balint Virag.
May 19    Tom Liggett, UCLA
Title:   k-dependent q-colorings of the integers
Abstract:   In 2008, Oded Schramm asked the following question: For what values of k and q does there exist a stationary, k-dependent sequence of random variables with values in {1,2,...,q} assigning different values to consecutive integers? Schramm proved a number of results related to this question, and speculated about what the answer might be in general. As it turns out, the truth is quite different from his informed guess. This is joint work with A. E. Holroyd.
May 26    Amber Puha, California State University San Marcos
Location:   AP&M 5402
Title:   Analysis of Processor Sharing Queues via Relative Entropy
Abstract:   Processor sharing is a mathematical idealization of round-robin scheduling algorithms commonly used in computer time-sharing. It is a fundamental example of a non-head-of-the-line service discipline. For such disciplines, it is typical that any Markov description of the system state is infinite dimensional. Due to this, measure-valued stochastic processes are becoming a key tool used in the modeling and analysis of stochastic network models operating under various non-head-of-the-line service disciplines.

In this talk, we discuss a new approach to studying the asymptotic behavior of fluid model solutions (formal functional law of large numbers limits) for critically loaded processor sharing queues. For this, we introduce a notion of relative entropy associated with measure-valued fluid model solutions. This approach is developed with idea that similar notions involving relative entropy may be helpful for understanding the asymptotic behavior of critical fluid model solutions for stochastic networks operating under protocols naturally described by measure-valued processes.
June 2    Georg Menz, UCLA
Title:   The log-Sobolev inequality for unbounded spin systems
Abstract:   The log-Sobolev inequality (LSI) is a very useful tool for analyzing high-dimensional situations. For example, the LSI can be used for deriving hydrodynamic limits, for estimating the error in stochastic homogenization, for deducing upper bounds on the mixing times of Markov chains, and even in the proof of the Poincaré conjecture by Perelman. For most applications, it is crucial that the constant in the LSI is uniform in the size of the underlying system. In this talk, we discuss when to expect a uniform LSI in the setting of unbounded spin systems.