University of Chicago Probability Seminar

October 17
Vladas Sidoravicius, IMPA, A one-dimensional DLA problem

October 26
Steve Lalley, U of C :   Random walks on surface groups.

November 2
Robert Masson, U of C :   Growth exponent of loop-erased random walk in two dimensions.

November 16
Greg Lawler, U of C :   Star configurations for radial SLE.
A k-star configuration for radial Schramm-Loewner evolution is a certain measure on k-tuples of nonintersecting curves from an interior point to a boundary. The case k=1 is the usual radial SLE, the case k=2 is two-sided radial SLE, and the case k=3 arises in the study of spanning trees. I will give a description of k-stars in terms of configurational measures and try to describe what kind of lattice measures approach these measures.

January 25
Omer Angel, University of Toronto :   Local limits of graphs.
There has been much recent progress in describing the structure of general dense graphs and their limits, in the spirit of the Szemeredy regularity lemma. I will present a parallel theory for sparse graphs (with bounded degrees). The limits are probability measures on infinite graphs, and are the answer to the following question: What can the neighbourhood of a random vertex in a finite graph look like? I will present a characterization of the ergodic measures and an application of the theory: An Extension of a theorem of Benjamini and Schramm on recurrence of limits planar graph limits to limits of graphs with an excluded minor. TBA

February 1
Julien Dubedat, U of C :   SLE and free fields.
Schramm-Loewner Evolutions (SLE) and (massless, Euclidean) free fields are probability measures on respectively curves and distributions in a planar domain; both satisfy conformal invariance properties. We describe some relations between the two objects, in terms of partition functions and couplings.

February 8
Balint Virag, U of Toronto:   Properties of the Brownian Carousel
The Brownian Carousel is a probabilistic description of the sine point processes, the bulk eigenvalue limits of hermitian random matrix models. I will explain how it arises and how to use it to understand properties of the sine point process. Open problems will be presented. This is joint work with B. Valko.

February 15
Jonathan Mattingly, Duke Univ. :   Troubles with a chain of stochastic oscillators
I will describe a simple chain of coupled oscillators used to model heat conduction. Eventually one would like to understand the structure of the energy flux at equilibrium. I will focus on the less lofty goal of showing that there is an equilibrium. This turns out to be surprisingly difficult. The results I will present make use of a detailed analysis of the multiscale in time structure of the problem.

February 22, no seminar

February 29, Robert Bauer, UIUC, :   Chordal restriction measures on Riemann surfaces and their description using Theta functions
We define chordal restriction measures on Riemann surfaces by introducing the notion of chordal measure-valued differentials of weight $\alpha$. We then show the existence and uniqueness (up to a multiplicative constant) of these measures and study them using period matrices and associated theta functions.

March 7, no seminar

March 14, Mathieu Merle, U of British Columbia :   Scaling limit of invasion percolation cluster on a regular tree
We consider invasion percolation on a regular tree. Recent work of Angel, Goodman, den Hollander and Slade showed a structural representation of the invasion percolation cluster (IPC) as an infinite backbone from which emerge independent subcritical Galton-Watson trees (a similar representation also holds for the incipient infinite cluster, however in that case the off-backbone trees are critical). We use this structural representation of (IPC) to show that the (IPC), when suitably rescaled, converges to a continuous tree. Coding functions of the limiting continuous tree can be expressed in terms of the solution of a certain (SDE). Moreover, the limit can be used to deduce asymptotic properties of the (IPC). (Joint work with Omer Angel (U of Toronto) and Jesse Goodman (UBC)).

March 21, TBA

April 4, Manjunath Krishnapur, U of Toronto :   From Random Matrices to Random Analytic Functions
Peres and Virag proved that the zeros of the power series a_0+za_1+z^2a_2+... with i.i.d. standard complex Gaussian coefficients, is a determinantal point process on the unit disk, invariant in distribution under isometries of the hyperbolic plane. Extending this result, I show that the singular points of the matrix-valued power series A_0+zA_1+z^2A_2+..., where A_i are k x k matrices with i.i.d. standard complex Gaussian entries, is also a determinantal process in the disk. This gives a unified framework in which to view the result of Peres and Virag and a well-known theorem of Ginibre on Gaussian random matrices.

April 9 (Unusual time and place --- 1:30 in 352 Ryerson) , Anna Amirdjanov, University of Michigan :   Nonlinear filtering of random fields in the presence of long-memory noise
An interesting estimation problem, arising in many dynamical systems, is that of filtering; namely, one wishes to estimate a trajectory of a "signal" process (which is not observed) from a given path of an observation process, where the latter is a nonlinear functional of the signal plus noise. In the classical mathematical framework, the stochastic processes are parameterized by a single parameter (interpreted as "time"), the observation noise is a martingale (say, a Brownian motion), and the best mean-square estimate of the signal, called the optimal filter, has a number of useful representations and satisfies the well-known Kushner-FKK and Duncan-Mortensen-Zakai stochastic partial differential equations. However, there are many applications, arising, for example, in connection with denoising and filtering of images and video-streams, where the parameter space has to be multidimensional. Another level of difficulty is added if the observation noise has a long-memory structure, which leads to ``nonstandard'' filtering evolution equations. Each of the two features (multidimensional parameter space and long-memory observation noise) does not permit the use of the classical theory of filtering and the combination of the two has not been previously explored in mathematical literature on stochastic filtering. This talk focuses on nonlinear filtering of a signal in the presence of long-memory fractional Gaussian noise. We will start by introducing first the evolution equations and integral representations of the optimal filter in the one-parameter case, when the noise driving the observation is represented by a fractional Brownian motion. Next, using fraAbstract: The heat kernel for Dirichlet fractional Laplacian $-(-\Delta)^{\alpha/2}|_D$, $\alpha\in (0, 2)$, with zero exterior condition is the transition density of a symmetric $\alpha$-stable process killed upon exiting $D$. In this talk, I will present recent results on sharp two-sided estimates for the heat kernel of Dirichlet fractional Laplacian in $C^{1, 1}$ open sets. I will also present results on the estimates of transition density of other related processes. This talk is based on two joint papers with Zhen-Qing Chen and Panki Kim. ctional calculus and multiparameter martingale theory, the case of spatial nonlinear filtering of a random field observed in the presence of a persistent fractional Brownian sheet will be explored. In addition several new properties of multiple integrals with respect to Gaussian random fields will be discussed and their applications to filtering will be shown. The talk is based in part on a joint work with Matt Linn and Sebastien Chivoret.

April 18, Vlada Limic, University of Provence, Marseille :   TBA
Consider a $\Lambda$-coalescent that  comes down from infinity. Informally, it is an exchangeable coalescent process with possible multiple mergers (but no two merger events are simultaneous) that starts from a configuration containing infinitely many blocks at time $0$ and attains a configuration containing a finite number $N_t$ of blocks at any time $t>0$,  almost surely. We   exhibit a deterministic function $v:(0,\infty)\to (0,\infty)$,such that $N_t/v(t)\to 1$, almost surely, as $t\to 0$. Our approach relies on martingale methods. If time permits we discuss connections to generalized Fleming-Viot processes and continuous-state branching processes. Based on a joint work with Julien and Nathanael Berestycki.

April 25, Noureddine El Karoui, Berkeley :   Spectra of large random matrices and concentration of measure
Widely used techniques of multivariate statistical analysis rely on the spectral properties of matrices of the type $X'X$, where $X$ is a $n\times p$ data matrix. Often, the rows of $X$, $X_i$, are assumed to be i.i.d, with a distribution that may vary with $p$. Nowadays, it is not uncommon for $p$, the number of variables to be of the same order of magnitude as $n$, and it is therefore natural to ask what are the limiting spectral properties of $X'X/n$ when $n$ and $p$ go to infinity and their ratio has a non-zero limit. In this talk, I will describe how concentration of measure results help answer these questions and extend the domain of validity of known results. I will also describe the limiting spectral properties of $n\times n$ "kernel" random matrices with entries $f(\gamma_p X_i'X_j)$, where $f$ is a smooth function and $\gamma_p$ is a parameter that is allowed to vary with $p$. (The motivation for these questions come from questions in statistics and computer science.) I will focus on the high-dimensional case (i.e $p$ goes to infinity with $n$), and show that for "standard" random matrix models, the results are very different from the low-dimensional setting ($p$ fixed), which are often used by practitioners to justify the use of corresponding statistical methods.

May 2, Renming Song, UIUC:  
The heat kernel for Dirichlet fractional Laplacian $-(-\Delta)^{\alpha/2}|_D$, $\alpha\in (0, 2)$, with zero exterior condition is the transition density of a symmetric $\alpha$-stable process killed upon exiting $D$. In this talk, I will present recent results on sharp two-sided estimates for the heat kernel of Dirichlet fractional Laplacian in $C^{1, 1}$ open sets. I will also present results on the estimates of transition density of other related processes. This talk is based on two joint papers with Zhen-Qing Chen and Panki Kim.

May 9, Ashkan Nikeghbali, Zurich    TBA

May 23, Joan Lind, Belmont U    TBA