Berkeley Probability Seminar
Wednesdays, 3:10 - 4:00pm
330 Evans Hall

 Organizers: Sourav Chatterjee Shankar Bhamidi Asaf Nachmias
 Mathematics Statistics Berkeley Probability Group Maps & Directions Previous Seminars

Spring 2007

24 Jan Anna Amirdjanova (U. Michigan at Ann Arbor) Physically inspired stochastic partial differential equations Brian Rider (U. Colorado at Boulder) Beta Tracy-Widom Laws, Random Schroedinger, and Diffusion Parthanil Roy (Cornell) Extreme Values of Stable Random Fields and Long Range Dependence Tom Schmitz (UCLA) Diffusions in random environment and ballistic behavior Asaf Nachmias (Berkeley) The diameter and mixing time of critical random graphs Alice Guionnet (ENS de Lyon & Berkeley) Spectrum of Levy matrices Charles Bordenave (Berkeley) Near-Minimal Spanning Trees: a Scaling Exponent in Probability Models Leonid Kontorovich (CMU) Measure Concentration of Strongly Mixing Processes with Applications Jonathan Mattingly (Duke) Ergodicity in infinite dimensions: SPDEs and the Wasserstein metric No seminar: Spring recess Alexander Soshnikov (UC Davis) Spectral Norm of Random Wigner Matrices with Non-Symmetrically Distributed Entries David Aldous (Berkeley) Probability and Spatial Networks Siamak Sorooshyari (Lucent Technologies - Bell Labs) A Multivariate Statistical Approach to Performance Analysis of Wireless Communication Systems Romain Thibaux (Berkeley) Hierarchical beta processes and the Indian buffet process Sourav Chatterjee (Berkeley) Spin glasses and Stein's method Giambattista Giacomin (Université Paris 7) On the localization transition for pinning models

Abstracts

 Physically inspired stochastic partial differential equations Anna Amirdjanova (U. Michigan at Ann Arbor) The talk is devoted to the study of three fundamental equations of mathematical physics (heat, Burgers' and Navier-Stokes) when the classical models are perturbed by random forces. Questions of existence, uniqueness and regularity of solutions will be discussed and some open problems presented. In the second half of the talk we will focus on the stochastic vorticity model, arising in hydrodynamics via stochastic point vortex methods, and discuss its properties. top of page Beta Tracy-Widom Laws, Random Schroedinger, and Diffusion Brian Rider (U. Colorado at Boulder) The beta ensembles of Random Matrix Theory are natural generalizations of the Gaussian Orthogonal, Unitary, and Symplectic Ensembles (as well as their Wishart counterparts), these classical cases corresponding to beta = 1, 2, or 4. We prove that the largest eigenvalues of these general ensembles have limit laws described by the low lying spectrum of certain random Schroedinger operators. This provides a new characterization of the celebrated Tracy-Widom laws (now for all beta). As a corollary, there is a second, and also new, characterization in terms of the explosion probability of an associated one-dimensional diffusion. These descriptions even have applications, allowing for information on the shape of the "beta Tracy-Widom" distributions. (Based on work with J. Ramirez and B. Virag.) top of page Extreme Values of Stable Random Fields and Long Range Dependence Parthanil Roy (Cornell) This talk will focus on the extreme values of stationary symmetric \alpha-stable random fields over hypercubes of increasing size. We will discuss how the asymptotic behavior of these extremes can be connected to certain ergodic theoretical and group theoretical properties of the integral representation of the random field. This connection helps us to understand the long range dependence of such processes, which is otherwise difficult to analyze due to absence of correlations. (This is a joint work with Gennady Samorodnitsky.) top of page Diffusions in random environment and ballistic behavior Tom Schmitz (UCLA) We study continuous diffusions in random environment in d-dimensional space. Once the environment is chosen, it remains fixed in time. To restore some stationarity, it is common to average both with respect to the path and environment mesure. One then obtains the so-called annealed measures, that are typically non-Markovian measures. Our goal is to study the asymptotic behavior of the diffusion in random environment under the annealed measure, with particular emphasis on the ballistic regime ('ballistic' means that a law of large numbers with non-vanishing limiting velocity holds). In the spirit of Sznitman, who treated the discrete setting, we introduce conditions (T) and (T'), and show that they imply, when d>1, a ballistic law of large numbers and a central limit theorem with non-degenerate covariance matrix. As an application of our results, we highlight condition (T) as a source of new examples of ballistic diffusions in random environment. top of page The diameter and mixing time of critical random graphs Asaf Nachmias (Berkeley) Let C1 denote the largest connected component of the critical Erdos-Renyi random graph G(n,1/n). We show that, typically, the diameter of C1 is of order n1/3 and the mixing time of the lazy simple random walk on C1 is of order n. The latter answers a question of Benjamini, Kozma and Wormald. These results extend to clusters of size n2/3 of p-bond percolation on any d-regular n-vertex graph where such clusters exist, provided that p(d - 1) < 1 + O(n-1/3). Joint work with Yuval Peres. top of page Spectrum of Levy matrices Alice Guionnet (ENS de Lyon & Berkeley) Take a square N × N symmetric matrix XN with real i.i.d random entries above the diagonal with law &mu such that &mu(x2) = 1. Then, if &lambda1,&hellip,&lambdaN denote the eigenvalues of XN, Wigner's theorem asserts that the spectral measure N-1&sum &deltaN-1/2&lambdai converges weakly in expectation and almost surely towards the semi-circle law. We study what happens when &mu has no finite second moment, but belong to the domain of attraction of an &alpha-stable law. In particular, we show the weak convergence of E[N-1&sum &deltaN-1/&alpha&lambdai] towards a symmetric law with heavy tail. top of page Near-Minimal Spanning Trees: a Scaling Exponent in Probability Models Charles Bordenave (Berkeley) We will study the relation between the minimal spanning tree (MST) on many random points and the "near-minimal" tree which is optimal subject to the constraint that a proportion \delta of its edges must be different from those of the MST. Heuristics suggest that, regardless of details of the probability model, the difference of the lengths should scale as the number pf points times \delta squared. We will see why this scaling result is true in the model of the lattice with random edge-lengths. In the 2-dimensional Euclidean model, by exploiting the well-known connection between MSTs and continuum percolation of the Gilbert disc model we will reduce the scaling result to an Ansatz that a known technical result for lattice percolation extends to continuum percolation. This Ansatz will be related to Kesten's work on the critical exponents in two dimensional percolation. This is a joint work with David Aldous (UC Berkeley) and Marc Lelarge (Ecole Normale Superiere and INRIA) top of page Measure Concentration of Strongly Mixing Processes with Applications Leonid Kontorovich (CMU) The concentration of measure phenomenon was first discovered in 1930's by Paul Levy and has been investigated since then, with increasing intensity in recent decades. The probability-theoretic results have been gradually percolating throughout the mathematical community, finding applications in Banach space geometry, analysis of algorithms, statistics and machine learning. There are several approaches to proving concentration of measure results; we shall offer a brief survey of these. The principal contribution of this work is a new concentration inequality for nonproduct measures. The inequality is proved by elementary means, yet enables one, with minimal effort, to recover and generalize the best current results for Markov chains, as well as to obtain (apparently) new results for hidden Markov chains and Markov trees. Applications of these inequalities include a strong law of large numbers for a broad class of non-independent processes. In particular, this allows one to analyze the convergence of inhomogeneous Markov Chain Monte Carlo algorithms. We have also obtained some partial results on extending the Rademacher-type generalization bounds to processes with arbitrary dependence. top of page Ergodicity in infinite dimensions: SPDEs and the Wasserstein metric Jonathan Mattingly (Duke) I will discuss a particular perspective on proving ergodic theorems for infinite dimensional Markov processes. I will begin with finite dimensional systems and work my way up to infinite dimensional systems. Specifically, I will give a version of Harris's ergodic result in the Wasserstein metric (*not* total variation). I will talk about some recent results in the ergodic theory of stochastic partial differential equations. I will then discuss some tools from Malliavin calculus used in proving the results. top of page Spectral Norm of Random Wigner Matrices with Non-Symmetrically Distributed Entries Alexander Soshnikov (UC Davis) In the first part of the talk, I will show that the spectral radius of an N × N random symmetric matrix with i.i.d. bounded centered but non-symmetrically distributed entries is bounded from above by 2&sigma + o(N-6/11+&epsilon), where &sigma2 is the variance of the matrix entries and &epsilon is an arbitrary small positive number. Our bound improves the earlier results by Z. Füredi and J. Komlós (1981), and Van Vu (2005). Our approach heavily relies on combinatorial considerations. This is a joint paper with Sandrine Peche. In the second part of the talk, I will discuss a resolvent approach to study the universality at the edge of the spectrum. top of page Probability and Spatial Networks David Aldous (Berkeley) Network design and analysis have been studied in many different applied contexts, yet many simple-to-state abstracted mathematical problems have not been studied very systematically. For a road network on n cities, what is the trade-off between total network length and the efficiency of the network in providing short routes? For an airline network on n cities, requiring routes to have an average of no more that 3 hops, how short can network length be? Such questions can involve probability in several ways. First, the average case" model of randomly-distributed cities is a natural counterpart to worst-case analysis. Second, while upper bounds on performance are obtained by explicit construction, lower bounds need more mathematical arguments provided by classical integral geometry. Third, the Poisson line process turns out to be very useful! [Joint work with Wilf Kendall. The two papers discussed are available at http://arxiv.org/abs/cond-mat/0702502 and http://front.math.ucdavis.edu/math.PR/0701140 ] top of page A Multivariate Statistical Approach to Performance Analysis of Wireless Communication Systems Siamak Sorooshyari (Lucent Technologies - Bell Labs) The explosive growth of wireless communication technologies has placed paramount importance on accurate performance analysis of the fidelity of a service offered by a system to a user. Unlike the channels of wireline systems, a wireless medium subjects a user to time-varying detriments such as multipath fading, cochannel interference, and thermal receiver noise. As a countermeasure, structured redundancy in the form of diversity has been instrumental in ensuring reliable wireless communication characterized by a low bit error probability (BEP). In the performance analysis of diversity systems the common assumption of uncorrelated fading among distinct branches of system diversity tends to exaggerate diversity gain resulting in an overly optimistic view of performance. A limited number of works take into account the problem of statistical dependence. This is primarily due to the mathematical complication brought on by relaxing the unrealistic assumption of independent fading among degrees of system diversity. We present a multivariate statistical approach to the performance analysis of wireless communication systems employing diversity. We show how such a framework allows for the statistical modeling of the correlated fading among the diversity branches of the system users. Analytical results are derived for the performance of maximal-ratio combining (MRC) over correlated Gaussian vector channels. Generality is maintained by assuming arbitrary power users and no specific form for the covariance matrices of the received faded signals. The analysis and results are applicable to binary signaling over a multiuser single-input multiple-output (SIMO) channel. In the second half of the presentation, attention is given to the performance analysis of a frequency diversity system known as multicarrier code-division multiple-access (MC-CDMA). With the promising prospects of MC-CDMA as a predominant wireless technology, analytical results are presented for the performance of MC-CDMA in the presence of correlated Rayleigh fading. In general, the empirical results presented in our work show the effects of correlated fading to be non-negligible, and most pronounced for lightly-loaded communication systems. top of page Hierarchical beta processes and the Indian buffet process Romain Thibaux (Berkeley) The Indian buffet process is an exchangeable process that generates binary matrices. It has been used in machine learning applications as a prior over models where objects can belong to overlapping categories (factorial models). I will show its close relationship with the beta process, a nonparametric prior originally developed for survival analysis. For people familiar with the Dirichlet process, the beta process and the Indian buffet process are close analogs of the Dirichlet process and the Chinese restaurant process. I will explain and show examples of all these objects. I will then use this link to derive algorithms and define hierarchies of beta processes. top of page Spin glasses and Stein's method Sourav Chatterjee (Berkeley) The high temperature phase of the Sherrington-Kirkpatrick model of spin glasses is solved by the famous Thouless-Anderson-Palmer (TAP) system of equations. The only rigorous proof of the TAP equations, based on the cavity method, is due to Michel Talagrand. The basic premise of the cavity argument is that in the high temperature regime, certain objects known as local fields' are approximately gaussian in the presence of a cavity'. In this talk, I will show how to use the classical Stein's method from probability theory to discover that under the usual Gibbs measure with no cavity, the local fields are asymptotically distributed as asymmetric mixtures of pairs of gaussian random variables. An alternative (and seemingly more transparent) proof of the TAP equations automatically drops out of this new result, bypassing the cavity method. top of page On the localization transition for pinning models Giambattista Giacomin (Université Paris 7) Pinning models and are Gibbs measures built on renewal processes. Alternatively they can be seen as one dimensional statistical mechanics systems with long tail two-body interactions. This class of models includes and gives a unitary vision of a large number of systems modeling phenomena like polymer pinning on a defect line, interface wetting, DNA denaturation, and more. We will review basic features of pinning models and we will present some recent progress, notably in the analysis of the behavior of the correlation length of the system near criticality. top of page
 Mathematics Statistics Berkeley Probability Group Maps & Directions Previous Seminars