Probability Seminar

Seminars

Previously Posted Seminars

Other UMN Seminars

Spring 2009

• January 23 Dapeng Zhan Yale University

  Reversibility and duality of SLE

  Stochastic Loewner evolution (SLE) introduced by Oded Schramm is a breakthrough in studying the scaling limits of many two-dimensional lattice models from statistical physics. In this talk, I will discuss the proofs of the reversibility conjecture and duality conjecture about SLE. The proofs of these two conjectures use the same idea, which is to use a coupling technique to lift local couplings of two SLE processes that locally commute with each other to a global coupling. And from the global coupling, we can clearly see that the two conjectures hold.

• January 30 John Baxter University of Minnesota

  Work of Michael Hochman on Upcrossings

  It is a striking fact, proved first by Errett Bishop in 1966, that the ergodic averages of a stationary process satisfiy upcrossing inequalities similar to, but more difficult than, the upcrossing inequalities for martingales. Later versions of this result have been given by Ivanov, by Kalikow and Weiss, and by Jones, Kaufman, Rosenblatt and Wierdl, among others, which include exponential estimates for upcrossings under some conditions. Hochman considers upcrossings for a sequence of random variables obtained by applying a function to the initial segments of a stationary sequence. Hochman's inequalities imply in particular that for a bounded subadditive process the probability of having at least L upcrossings decays exponentially in L.

• February 6 William Sudderth University of Minnesota

  How to control a process to a goal

  Abstract in PDF

• February 13 Bert Fristedt University of Minnesota

  The Maximum Average Gain in a Sequence of Bernoulli Games, a paper by Wolfgang Stadje

  Abstract in PDF

• February 20 No seminar this week.

• February 27 Friedrich Götze University of Bielefeld

  Correlations of Characteristic Polynomials of Random Matrices

  The local spectral distribution of eigenvalues of Wigner random matrices in the bulk of the spectrum is conjectured to universal after suitable scaling with a distribution determined by Gaussian Wigner matrices. Here we discuss results concerning the correlation coefficient of chacteristic polynomials of Wigner matrices which yields in the limit the correlation kernel of the spectrum of Gaussian matrices. Analogies with results for other ensembles of random matrices and correlations of zeta-functions are discussed. This is joint work with H. Koesters.

• March 6 No seminar this week.

• March 13 Nicolai Krylov University of Minnesota

  Applying quasiderivatives to estimating the smoothness of harmonic functions corresponding to degenerate diffusion processes

• March 20 HOLIDAY

• March 27 Mokshay Madiman Yale University

  A New Look at the Compound Poisson Distribution and Compound Poisson Approximation using Entropy

  We develop an information-theoretic foundation for compound Poisson approximation and limit theorems (analogous to the corresponding developments for the central limit theorem and for simple Poisson approximation). First, sufficient conditions are given under which the compound Poisson distribution has maximal entropy within a natural class of probability measures on the nonnegative integers. In particular, it is shown that a maximum entropy property is valid if the measures under consideration are log-concave, but that it fails in general. Second, approximation bounds in the (strong) relative entropy sense are given for distributional approximation of sums of independent nonnegative integer valued random variables by compound Poisson distributions. The proof techniques involve the use of a notion of local information quantities that generalize the classical Fisher information used for normal approximation, as well as the use of ingredients from Stein's method for compound Poisson approximation. This work is joint with Andrew Barbour (Zurich), Oliver Johnson (Bristol) and Ioannis Kontoyiannis (AUEB).

• April 3 Cheng Ouyang Northwestern University

  The Cameron-Martin Theorem for Riemannian Manifolds

  Abstract in PDF

• April 10 Timo Seppäläinen University of Wisconsin-Madison

  Fluctuation bounds for a class of zero range processes

  Abstract in PDF

• April 17 John Kieffer University of Minnesota

  Recent Progress Concerning Information Theory Aspects of Some Probability Models

  Information theory aspects of two types of probability models are discussed. The first type of probability model is a random data structure generated using a stochastic graph grammar. Information theory addresses the asymptotics of the entropy of the structure as the structure becomes arbitrarily large. A recent such result concerning the entropy of random binary trees is cited. The second type of probability model is a pair process (X,Y), in which X is a given IID process and Y is an IID process obtained by passing X through a (typically unique) channel so that Y is the closest process to X in a distortion sense, subject to a mutual information constraint. Since the 1950's, information theory has addressed the difficult problem of replacing the channel linking X to Y with a deterministic code linking X to a process close to Y. We outline a new approach to this old problem.

• April 24 Soumik Pal University of Washington

  Interacting diffusion models of capital markets

  Consider a multidimensional interacting diffusion model where the drift and the diffusion coefficients for individual coordinates are dependent on the relative sizes of their current values compared to the others. The coordinate processes are otherwise exchangeable. Two such models were introduced by Fernholz and Karatzas as models for equity markets where it has long been empirically observed that the rate of growth and the fluctuation in the value of a stock capital depends on its relative value with respect to the entire market. In one model, called the rank-based model, the time-dynamics is determined by the ordering in which the coordinate processes can be arranged at any time. In the other, named the volatility-stabilized market model, the drift and the diffusion parameters are functions of the ratio of the current value to the total sum over all the coordinates. We show some remarkable properties of these models. The rank-based models, which also appear in queueing theory and dynamic models of spin glasses, show phase transition in its equilibrium behavior as the number of coordinates go to infinity. The volatility-stabilized models, on the other hand, are deeply connected to the multi-allele Wright-Fisher diffusions in genetics. We will try to get a view of the beautiful world of these interacting diffusions and how one can use tools like reflected Brownian motions and the Bessel processes to analyze their behavior.

• May 1 No seminar this week.

• May 8 No seminar this week.

Fall 2008

• September, 12 Sergey Bobkov University of Minnesota

  Isotropic positions of convex measures.

  We will be discussing extensions of Hensley's theorem on sections of isotropic convex bodies to the class of convex measures that are put in a specific position.
• September, 19 Ofer Zeitouni University of Minnesota

  A fragmentation-coagulation chain, random walk on permutations, and Schramm's coupling.

  Abstract in PDF
• September, 26 Vasileios Maroulas IMA, University of Minnesota

  Small noise large deviations for infinite dimensional stochastic dynamical systems.

  Abstract in PDF
• October, 3 Ramon Van Handle Princeton University

  A qualitative long time asymptotic theory for nonlinear filtering

  The asymptotic properties of nonlinear filters, such as stability and unique ergodicity, are of significant importance in proving the convergence of approximate filtering algorithms and the consistency of likelihood-based inference in hidden Markov models. A variety of ingenious methods have been proposed to study these long time properties quantitatively. On a qualitative level, however, none of these results are able to reproduce and extend the powerful observability/detectability criteria which are known in the special case of linear Gaussian models.

  In this talk, I will outline some recent qualitative results which shed light on the fundamental mechanisms that lead to stability of nonlinear filters. The results are very general in nature and subsume and extend to nonlinear models various known results in the linear Gaussian case. This qualitative theory is based on connections between nonlinear filtering on the one hand, and Markov chains in random environments and uniform approximations on the other hand. Along the way, a long-standing gap in a classic paper by H. Kunita (1971) is largely resolved.

• October, 10 Jon Peterson University of Wisconsin

  Properties of the averaged large deviation rate function for random walks in random environments

  An averaged (annealed) large deviation principle for multi-dimensional random walks in random environments (RWRE) was proved by Varadhan in 2003. However, Varadhan's proof is quite complicated and thus it is difficult to derive much qualitative information about the large deviation rate function from the formulation given in his proof. In this talk I will describe a different, much simpler, approach to large deviations of multidimensional RWRE using regeneration times. I will use this approach to show that when the distribution on environments is ``non-nestling,'' the averaged large deviation rate function is analytic in a neighborhood of the limiting velocity of the RWRE.
• October, 17 Brian Rider University of Colorado

  Diffusion at RMT's hard edge

  With J. Ramirez and B. Virag we recently proved that the limiting soft edge eigenvalues of the general beta ensembles have laws shared by the spectral points of a certain random Schroedinger operator. After recalling this fact I'll prove there is a similar picture at the random matrix hard edge. That is, the spectrum of random sample covariance type matrices is described in terms of a (random) differential operator in the large dimensional limit. And thus also, via a Riccati transformation, by the hitting distribution of a simple diffusion. This second description allows us to prove the anticipated transition between the hard and soft edge laws.
• October, 24 Midwest Probability Colloqium Northwestern University

  No seminar

• October, 31 Houman Owhadi Caltech

  Noisy mechanical systems and Stochastic Variational Integrators

  We present a continuous and discrete Lagrangian theory for stochastic Hamiltonian systems on manifolds. Analogously to discrete mechanics and variational integrators, this theory leads to structure-preserving numerical integrators for noisy mechanical systems by extremizing a discrete stochastic action. As an example of application we will examine the (non equilibrium) behavior of a magnetic motor at uniform temperature, submitted to degenerate noise and dissipation. This is a joint work with Nawaf Bou-Rabee.
• November, 7 Hye-Won Kang University of Minnesota

  Multiple scaling methods in chemical reaction networks

  Abstract in PDF
• November, 14
  Manjunath Krishnapur Toronto

  Singular points of matrix-valued analytic functions

  We consider the set of points in the complex plane where a matrix-valued analytic function becomes singular. As the size of the matrix goes to infinity, we find a limiting measure in the complex plane, in the case when the entries of the matrix are iid random analytic functions. This generalizes the circular law of random matrix theory to a much larger family of laws.
• November, 21
  Gabor Pete Toronto & MSRI

  The exact noise and dynamical sensitivity of critical percolation, via the Fourier spectrum

  Let each site of the triangular lattice (or edge of the \Z^2 lattice) have an independent Poisson clock switching between open and closed. So, at any given moment, the configuration is just critical percolation. In particular, the probability of a left-right open crossing in an n*n box is roughly 1/2, and, on the infinite lattice, almost surely there are only finite open clusters.

  In the box, how long do we have to wait before we lose essentially all information about having a left-right open crossing? In the infinite lattice, are there random exceptional times when there are infinite clusters? In joint work with Christophe Garban and Oded Schramm, we gave quite complete answers: exceptional times do exist on both lattices, and the Hausdorff dimension of their set is computed to be 31/36 for the triangular lattice.

  The indicator function of a percolation crossing event is a function on the hypercube {-1,+1}^{sites or edges}, and thus it has a Fourier-Walsh expansion. Our proofs are based on giving sharp concentration results for the ``weight'' of the Fourier coefficients at different frequencies.

• December 5
  Maury Bramson University of Minnesota

  Randomized load balancing for general service times with the FIFO discipline

  We consider the randomized load balancing scheme where each arriving job joins the shortest of d randomly chosen queues from among a pool of n queues, where d is fixed and n goes to infinity. Works by Mitzenmacher (1996) and Vvedenskaya et al. (1996) considered the case with Poisson input and exponentially distributed service times, where an explicit formula for the equilibrium distribution was derived when the system is subcritical. For general service times, the service discipline can affect the nature of the equilibrium distribution. Here, we examine its behavior for the FIFO service discipline and service distributions with fat tails. This is joint work with Y. Lu and B. Prabhakar of Stanford University.