Probability Seminar

Seminars

Previously Posted Seminars

Other UMN Seminars

Probability 2008-2009

Probability 2009-2010

Fall 2011 - Spring 2012

• September 9, 2011 Phillip Wood University of Wisconsin (Madison)

  Random doubly stochastic tridiagonal matrices

  Let T_n be the compact convex set of tridiagonal doubly stochastic matrices. These arise naturally as birth and death chains with a uniform stationary distribution. One can think of a "typical" matrix T_n as one chosen uniformly at random, and this talk will present a simple algorithm to sample uniformly in T_n. Once we have our hands on a 'typical' element of T_n, there are many natural questions to ask: What are the eigenvalues? What is the mixing time? What is the distribution of the entries? This talk will explore these and other questions, with a focus on whether a random element of T_n exhibits a cutoff in its approach to stationarity. Joint work with Persi Diaconis.
• September 16, 2011 Carsten Schütt Christian-Albrechts-Universität of Kiel (Germany)

  Mahler's conjecture and curvature

  Let K be a convex body in R^n with Santalo point at 0. We show that if K has a point on the boundary with positive generalized Gauss curvature, then the volume product |K| |K^o| is not minimal. This means that a body with minimal volume product has Gauss curvature equal to 0 almost everywhere and thus suggests strongly that a minimal body is a polytope.
• September 23, 2011 Prasad Tetali Georgia Institute of Technology (Atlanta)

  Scaling Limits in Sparse Random Graphs via a Combinatorial Interpolation Technique

  We establish the existence of free energy limits for several combinatorial models on Erdos-Renyi graph G(N; [cNc]) and random r-regular graph G(N;r). For a variety of models, including independent sets, MAX-CUT, Coloring and K-SAT, we prove that the free energy both at a positive and zero temperature, appropriately rescaled, converges to a limit as the size of the underlying graph diverges to innity. For example, as a special case we prove that the size of a largest independent set in these graphs, normalized by the number of nodes converges to a limit w.h.p. This resolves an open problem which was proposed by Aldous. It was also mentioned as an open problem in other works by Wormald, Bollobas-Riordan, and Janson-Thomason. Our approach is based on extending and simplifying the interpolation method of Guerra and Toninelli, followed by Franz and Leone. We provide a simpler combinatorial approach and work with the zero temperature case (optimization) directly both in the case of Erdos-Renyi graph G(N;[cN]) and random regular graph G(N;r). In addition we establish the large deviations principle for the satisability property of the constraint satisfaction problems such as Coloring, K-SAT and NAE-K-SAT for the G(N;[Nc]) random graph model. This is joint work with D. Gamarnik (MIT) and M. Bayati (Stanford).
• September 30, 2011 Mokshay Madiman Yale University (New Haven)

  Some combinatorial entropy inequalities, with applications

  We will provide an overview of a variety of inequalities for discrete entropy that are indexed by hypergraphs. The first part of the talk (based on joint work with P. Tetali) will discuss inequalities for entropy of joint distributions, which generalize various inequalities due to Shannon, Han, Fujishige and Shearer. The second part of the talk (based on joint work with A. Marcus and P. Tetali) will discuss inequalities for entropy of sums of independent random variables, and more generally of so-called partition-determined functions. The latter will be used as tools to develop general cardinality inequalities for sumsets in possibly nonabelian groups, with applications to additive combinatorics in mind. In particular, inequalities that generalize those of Gyarmati-Matolcsi-Ruzsa and Balister-Bollobas will be demonstrated, and partial progress reported towards a conjecture of Ruzsa for sumsets in nonabelian groups.
• October 7, 2011 Frank Aurzada Technische Universität Berlin (Germany)

  The one-sided exit problem for integrated processes and fractional Brownian motion

  We study the one-sided exit problem, also known as one-sided barrier problem, that is, given a stochastic process X we would like to find, as T\to\infty, the asymptotic rate of P(sup_{0\leq t\leq T} X_t \leq 1). This question is considered for alpha-fractionally integrated centered Levy processes and 'integrated' centered random walks. The rate of decrease of the above probability is polynomial with exponent \theta=\theta(\alpha)>0 which only depends on \alpha but not on the choice of the Levy process or random walk. This generalizes results of Y.G. Sinai (1991) who considered the simple random walk and \alpha=1. Similar recent results are due to V. Vysotsky (2010 and 2011) and A. Dembo and F. Gao (2011). The results are compared to the corresponding ones for fractional Brownian motion.
• October 14, 2011 No seminar this week (Midwest Probability Colloquium)
• October 21, 2011 Wenbo Li University of Delaware

  Probabilities of all real zeros for random polynomials

  There is a long history on the study of zeros of random polynomials whose coefficients are independent, identically distributed, nondegenerate random variables. We will first provide an overview on zeros of random functions and then show exact and/or asymptotic bounds on probabilities that all zeros of a random polynomial are real under various distributions.
• October 28, 2011 Elisabeth Werner Case Western Reserve University (Clevelend)

  Functional affine-isoperimetry and an inverse logarithmic Sobolev inequality

  We give a functional version of the affine isoperimetric inequality for log-concave functions which may be interpreted as an inverse form of a logarithmic Sobolev inequality inequality for entropy. A linearization of this inequality gives an inverse inequality to the Poincare inequality for the Gaussian measure.
• November 4, 2011 Xin Liu IMA, University of Minnesota (Minneapolis)

  Diffusion approximations for multiscale stochastic networks in heavy traffic

  We study a sequence of nearly critically loaded queueing networks, with time varying arrival and service rates and routing structure. The nth network is described in terms of two independent finite state Markov processes {Xn(t): t = 0} and {Yn(t): t = 0} which can be interpreted as the random environment in which the system is operating. The process Xn changes states at a much higher rate than the typical arrival and service times in the system, while the reverse is true for Yn. The variations in the routing mechanism of the network are governed by Xn, whereas the arrival and service rates at various stations depend on the state process (i.e. queue length process) and both Xn and Yn. Denoting by Zn the suitably normalized queue length process, it is shown that, under appropriate heavy traffic conditions, the pair Markov process (Zn, Yn) converges weakly to the Markov process (Z,Y), where Y is a finite state continuous time Markov process and the process Z is a reflected diffusion with drift and diffusion coefficients depending on (Z,Y) and the stationary distribution of Xn. We also study stability properties of the limit process (Z,Y).
• November 11, 2011 Daniel John Fresen University of Missouri-Columbia

  Concentration inequalities in the space of convex bodies

  We discuss the following phenomenon: The convex hull of a large i.i.d. sample from a log-concave probability measure on Euclidean space approximates a pre-determined body in the logarithmic-Hausdorff distance (and in the Banach-Mazur distance). For p-log-concave distributions with p>1 (such as the normal distribution where p=2) we also have approximation in the Hausdorff distance. Three deterministic bodies are given as approximants to the random body; one is the floating body, another is given as a contour of the density function and the third is given in terms of the Radon transform. Quantitative bounds are given.
• November 18, 2011 Matthew Roberts (Montreal)

  A simple path to asymptotics for the frontier of a branching Brownian motion

  Bramson's 1978 result on the position of the maximal particle in a branching Brownian motion has inspired many related results and generalizations in recent years. We show how modern methods can be used to give a simpler proof of the original result.
• November 25, 2011 No seminar this week (Thanksgiving)
• December 2, 2011 Antti Knowles Harvard University

  Finite-rank deformations of Wigner matrices

  The spectral statistics of large Wigner matrices are by now well-understood. They exhibit the striking phenomenon of universality: under very general assumptions on the matrix entries, the limiting spectral statistics coincide with those of a Gaussian matrix ensemble. I shall talk about Wigner matrices that have been deformed by adding a finite-rank matrix. By Weyl's interlacing inequalities, this deformation does not affect the large-scale statistics of the spectrum. However, it may affect eigenvalues near the spectral edge, causing them to break free from the bulk spectrum. In a series of seminal papers, Baik, Ben Arous, and Peche (2005) and Peche (2006) established a sharp phase transition in the statistics of the extremal eigenvalues of perturbed Gaussian matrices. At the transition, an eigenvalue detaches itself from the bulk and becomes an outlier. I shall report on recent joint work with Jun Yin. We consider an NxN Wigner matrix H perturbed by an arbitrary deterministic finite-rank matrix A. We allow the eigenvalues of A to depend on N. Under optimal (up to factors of log N) conditions on the eigenvalues of A, we identify the limiting distribution of the outliers of H+A. We also prove that the remaining eigenvalues of H+A "stick" to eigenvalues of H, thus establishing the edge universality of H+A. On the other hand, our results show that the distribution of the outliers is not universal, but depends on the distribution of H and on the geometry of the eigenvectors of A. As the outliers approach the bulk spectrum, this dependence is washed out and the distribution of the outliers becomes universal.
• December 9, 2011 Michael Damron Princeton University

  A simplified proof of the relation between scaling exponents in first-passage percolation

  In first passage percolation, we place i.i.d. non-negative weights on the nearest-neighbor edges of Z^d and study the induced random metric. A long-standing conjecture gives a relation between two "scaling exponents": one describes the variance of the distance between two points and the other describes the transversal fluctuations of optimizing paths between the same points. This is sometimes referred to as the "KPZ relation." In a recent breakthrough work, Sourav Chatterjee proved this conjecture using a strong definition of the exponents. I will discuss work I just completed with Tuca Auffinger, in which we introduce a new and intuitive idea that replaces Chatterjee's main argument and gives an alternative proof of the relation. One advantage of our argument is that it does not require a certain non-trivial technical assumption of Chatterjee on the weight distribution.
• December 16, 2011 - January 7, 2012 No seminar this period
• January 13, 2012 Wei-Kuo Chen Department of Mathematics, UC Irvine

  The Chaos problem in the Sherrington-Kirkpatrick model

  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 symmetric-breaking bound and Ghirlanda-Guerra identities, we will present mathematically rigorous results and proofs for these problems.
• January 20, 2012 Arnab Sen University of Cambridge (UK)

  Random Toeplitz matrices

  Random Toeplitz matrices belong to the exciting area that lies at the intersection of the usual Wigner random matrices and random Schrodinger operators. In this talk I will describe two recent results on random Toeplitz matrices. First, the maximum eigenvalue, suitably normalized, converges to the 2-4 operator norm of the well-known Sine kernel. Second, the limiting eigenvalue distribution is absolutely continuous, which partially settles a conjecture made by Bryc, Dembo and Jiang (2006). I will also present several open questions and conjectures towards the end of the talk. This is joint work with Balint Virag (Toronto).
• January 27, 2012 Xiaoqin Guo University of Minnesota

  Einstein relation for random walks in random environment

  In this talk, we will prove the Einstein relation for random walks in a uniformly elliptic and iid random environment. We consider the velocity of the random walks in a perturbation of a balanced random environment (i.e., environment with zero drift). We will show that the ratio between the velocity and the size of the perturbation converges to a diffusivity constant of the balanced environment. This talk is based on a joint work with my advisor Ofer Zeitouni.
• February 3, 2012 Dmitriy Bilyk University of Minnesota

  Small ball probabilities for the Brownian sheet, uniform distributions, and related problems

  We shall discuss recent progress on the asymptotic estimates for multiparameter Gaussian processes, such as the Brownian sheet. As shown by Kuelbs and Li, 1993, these inequalities are equivalent to the entropy estimates of Sobolev spaces with mixed smoothness. The methods of obtaining lower bounds are rooted in the work of Talagrand, 1994, and connect this problem to interesting questions in multiparameter wavelet analysis. Higher dimensional analogues of Talagrand's theorem turned out to be extremely proof-resistant, and only partial results are availble in dimensions three and higher (Beck, 1989; Bilyk, Lacey, Vagharshakyan, 2008-10). This problem is also connected to the grand open problem of the theory of irregularities of distribution -- the exact rate of growth of the discrepancy function. We shall examine this relation as well as the use of probabilistic methods in numerous other questions of discrepancy theory.
• February 10, 2012 Krzysztof Burdzy University of Washington (Seattle)

  Deterministic approximations of random reflectors

  Every random reflector on a line which satisfies a natural condition (well established in the theory of billiards) can be approximated by a sequence of deterministic reflectors represented by families of mirrors. Joint work with O. Angel and S. Sheffield.
• February 17, 2012 Nick Krylov University of Minnesota

  Accelerated finite-difference schemes for linear stochastic partial differential equations in the whole space (Joint with Istv\'an Gy\"ongy)

  We give sufficient conditions under which the convergence of finite difference approximations in the space variable of the solution to the Cauchy problem for linear stochastic PDEs of parabolic type can be accelerated to any given order of convergence by Richardson's method.
• February 24, 2012 John Baxter University of Minnesota

  Random Equilibrium Measures

• March 2, 2012 Rafail Khasminskii Wayne State University (emeritus)

  Stability of Stochastic Differential Equations with Markovian Switching

  The main aim of this talk is to present a result concerning reducing asymptotic stability problem for stochastic differential equation (SDE) with sufficiently rapid Markovian switching to the well-studied analogous problem for the "averaged" SDE without switching. Application to the problem of stabilization by switching and to ordinary differential equation (ODE) with switching will be also presented.
• March 9, 2012 Sergey Bobkov University of Minnesota

  Rate of convergence in the entropic central limit theorem

  For the scheme of i.i.d. random variables, we discuss rates of convergence of entropy of the normalized sums to the entropy of gaussian limit. Joint work with G. P. Chistyakov and F. Götze.
• Mach 16, 2012 No seminar this week (Spring Break)
• March 23, 2012 Greg Anderson University of Minnesota

  Joint cumulants of functions of a Gaussian random vector and free-probabilistic counting of colored planar maps

  Our main result is an explicit operator-theoretic formula for the number of colored planar maps with a fixed set of stars each of which has a fixed set of half-edges with fixed coloration. The formula transparently bounds the number of such colored planar maps and does so well enough to prove convergence near the origin of generating functions arising naturally in the matrix model context. Such convergence is known but the proof of convergence proceeding by way of our main result is relatively simple. Our main technical tool is an integration identity representing the joint cumulant of several functions of a Gaussian random vector. In the case of cumulants of order 2 the identity reduces to one well-known as a means to prove the Poincare inequality.
• March 30, 2012 Wenbo Li University of Delaware and IMA

  Small value probabilities and branching related processes

  We first provide an overview on fundamental roles of small value probability (estimates of rare events that positive random variables take smaller values than typical ones) in the theory of stochastic processes. Then we focus on estimates associated with variants of Branching processes and their martingale limits. Relevant techniques and tools will be discussed in the simplest setting.
• April 6, 2012 Tiefeng Jiang School of Statistics, University of Minnesota

  Distributions of Angles in Random Packing on Spheres

  We study the asymptotic behaviors of the pairwise angles among n randomly and uniformly distributed unit vectors in p-dimensional spaces as the number of points n goes to infinity, while the dimension p is either fixed or growing with n. For both settings, we derive the limiting empirical distribution of the random angles and the limiting distributions of the extreme angles. The results reveal interesting differences in the two settings and provide a precise characterization of the folklore that "all high-dimensional random vectors are almost always nearly orthogonal to each other". Applications to statistics and connections with some open problems in physics and mathematics are also discussed. This is a joint work with Tony Cai and Jianqing Fan.
• April 13, 2012 Bert Fristedt University of Minnesota

  Random sets and random discrete distributions

  Let Z be a random closed subset of the nonnegative reals. Suppose that it has measure 0 and that it is self-similar in the sense that cZ has the same distribution as Z for all positive constants c. The set [0,1]\Z is an open set with measure 1. The distribution of Z is characterized by the distribution of the length of the rightmost open interval in [0,1]\Z. This distribution has a density f(u) satisfying: (1-u)f(u) is a decreasing function of u. Moreover, any such distribution on (0,1) can arise in this manner. Two examples of Z are: (1) the set of zeroes of Brownian motion; (2) The set of values of a Levy process with Levy measure x^{-1} dx. The main reference is a paper by Jim Pitman and Marc Yor entitled "Random Discrete Distributions Derived from Self-Similar Random Sets" which appeared in The Electronic Journal of Probability, 1996, Vol. 1, pages 1-28.
• April 20, 2012 No seminar this week (Rivière-Fabes Symposium)
• April 27, 2012 Maury Bramson University of Minnesota

  Pursuit Games and Shy Couplings. Part I

  We are interested in the question of when a shy coupling, for a pair of Brownian motions, exists in a given bounded domain. That is, is it possible to construct a (non-anticipating) coupling between a pair of Brownian motions so that, with positive probability, they are always at least some assigned distance apart? This question is related to the deterministic Lion and Man problem, with the Lion attempting to capture the Man when both are allowed to move at the same rate. This talk is based on joint work with K. Burdzy and W. Kendall.
• May 4, 2012 Maury Bramson University of Minnesota

  Pursuit Games and Shy Couplings. Part II

  This is a continuation of Part I.