TechTalks from event: Columbia-Princeton Probability Day 2013

  • About Heavy Tails Random Matrices Authors: Alice Guionnet (MIT)
    Wigner's matrices are Hermitian matrices with independent entries modulo the symmetry constraint. In the last few years, it was shown that the properties of such matrices are similar to those of a matrix with Gaussian entries propiding the entries have enough finite moments. In this talk we will investigate the properties of matrices which do not belong to the universality class of Wigner matrices because their entries have heavy tails.
  • Heat Flow, Harnack Inequalities, and Optimal Transportation Authors: Michel Ledoux (Toulouse)
    The talk will develop connections between Harnack inequalities for the heat flow of diffusion operators with curvature bounded from below and optimal transportation. Through heat kernel inequalities, a new isoperimetric-type Harnack inequality is emphasized. Commutation properties between the heat and Hopf-Lax semigroups are developed consequently, providing direct access to the heat flow contraction property along Wasserstein distances.
  • Robust Optimality of Gaussian Noise Stability Authors: Elchanan Mossel (Berkeley)
    In 1985 C. Borell proved that under the Gaussian measure, half-spaces are the most stable sets. While a number of proofs of this result were discovered over the years, it was not known if half-spaces are the unique optimizers. The talk will survey recent results with Joe Neeman establishing that half-spaces are uniquely the most noise stable sets. Furthermore, we prove a quantitative dimension independent versions of uniqueness, showing that a set which is almost optimally noise stable must be close to a half-space. Our work answers a question of Ledoux from 1994 and has numerous applications in theoretical computer science and social choice.
  • Geometric Applications of Markov Chains Authors: Assaf Naor (NYU)
    The deep and fruitful interactions between probability and geometry are well-established, including the powerful use of probabilistic constructions to prove existence of important objects (e.g., Dvoretzky's theorem and numerous other applications of the probabilistic method to prove existence statements), and the study of the behavior of a variety of important stochastic processes in a geometric setting. In this talk we will describe a useful paradigm in metric geometry (originating from the work of Keith Ball) that allows for a probabilistic interpretation of certain geometric questions whose statement does not have any a priori connection to probability. In particular, we will address the following topics.
    1. Using Markov chains to show that it is possible to extend Lipschitz functions between certain metric spaces.
    2. Using Markov chains to prove impossibility results for Lipschitz extension problems.
    3. Using Markov chains to prove that for certain pairs of metric spaces X, Y, any embedding of X into Y must significantly distort distances.
    4. Using Markov chains to show that good embeddings do exist between certain metric spaces.
    5. Using Markov chains in metric Ramsey theory.
    6. Markov chains as an invariant for Lipschitz quotients and a tool to understand isomorphic uniform convexity.
    7. Markov chains as a tool to prove nonlinear spectral calculus inequalities.
    All of these applications involve geometric insights that introduce an unexpected link with probability theory, and then proving new probabilistic results that complete the solution of the problem at hand.
  • A Stochastic Game of Control and Stopping Authors: Marcel Nutz (Columbia)
    We study the existence of optimal actions in a zero-sum game infτsupPEP[Xτ] between a stopper and a controller choosing the probability measure. We define a nonlinear Snell envelope Y via the theory of sublinear expectations and show that the first hitting time inf{t:Yt=Xt} is an optimal stopping time. The existence of a saddle point is obtained under a compactness condition. (Joint work with Jianfeng Zhang.)
  • Regularity Conditions in the CLT for Linear Eigenvalue Statistics of Wigner Matrices Authors: Percy Wong (Princeton)
    We show that the variance of centred linear statistics of eigenvalues of GUE matrices remains bounded for large n for some classes of test functions less regular than Lipschitz functions. This observation is suggested by the limiting form of the variance (which has previously been computed explicitly), but it does not seem to appear in the literature. We combine this fact with comparison techniques following Tao-Vu and Erdős, Yau, et al. and a Littlewood-Paley type decomposition to extend the central limit theorem for linear eigenvalue statistics to functions in the Hölder class C1/2+ε in the case of matrices of Gaussian convolution type. We also give a variance bound which implies the CLT for test functions in the Sobolev space H1+ε and C1-ε for general Wigner matrices satisfying moment conditions. If the additional assumption of the test function being supported away from the edge of the spectrum is made, we prove the CLT for test functions of regularity 1/2 ∩ L and H1/2+ for GUE and Johansson matrices respectively. Previous results on the CLT impose the existence and continuity of at least one classical derivative.