Seminars & Colloquia

Thursday, February 9, 2017

TBA by Christopher Hoffman

Thu, 02/09/2017 - 3:05pm, Skiles 006

Christopher Hoffman, University of Washington        Organizer: Michael Damron

Thursday, September 29, 2016

Can one hear the shape of a random walk?

Thu, 09/29/2016 - 3:05pm, Skiles 006

Eviatar Procaccia, Texas A&M University, email        Organizer: Michael Damron

We consider a Gibbs distribution over random walk paths on the square lattice, proportional to a random weight of the path’s boundary. We show that in the zero temperature limit, the paths condensate around an asymptotic shape. This limit shape is characterized as the minimizer of the functional, mapping open connected subsets of the plane to the sum of their principle eigenvalue and perimeter (with respect to the first passage percolation norm). A prime novel feature of this limit shape is that it is not in the class of Wulff shapes. This is joint work with Marek Biskup.

Thursday, September 15, 2016

The size of the boundary in the Eden model

Thu, 09/15/2016 - 3:05pm, Skiles 006

Michael Damron, School of Mathematics, Georgia Tech        Organizer: Christian Houdré

The Eden model, a special case of first-passage percolation, is a stochastic growth model in which an infection that initially occupies the origin of Z^d spreads to neighboring sites at rate 1. Infected sites are colonized permanently; that is, an infected site never heals. It is known that at time t, the infection occupies a set B(t) of vertices with volume of order t^d, and the rescaled set B(t)/t converges to a convex, compact limiting shape. In joint work with J. Hanson and W.-K. Lam, we partially answer a question of K. Burdzy, concerning the order of the size of the boundary of B(t). We show that, in various senses, the boundary is relatively smooth, being typically of order t^{d-1}. This is in contrast to the fractal behavior of interfaces characteristic of percolation models.

Thursday, September 8, 2016

The invariable Ewens distribution

Thu, 09/08/2016 - 3:05pm, Skiles 006

Matthew Junge, Duke University, email        Organizer: Michael Damron

Form a multiset by including Poisson(1/k) copies of each positive integer k, and consider the sumset---the set of all finite sums from the Poisson multiset. It was shown recently that four such (independent) sumsets have a finite intersection, while three have infinitely many common elements. Uncoincidentally, four uniformly random permutations will invariably generate S_n with asymptotically positive probability, while three will not. What is so special about four? Not much. We show that this result is a special case of the "ubiqituous" Ewens sampling formula. By varying the distribution's parameter we can vary the number of random permutations needed to invariably generate S_n, and, relatedly, the number of Poisson sumsets to have finite intersection. *Joint with Gerandy Brita Montes de Oca, Christopher Fowler, and Avi Levy.

Thursday, April 28, 2016

Some new non-asymptotic results about the accuracy of the weighted bootstrap

Thu, 04/28/2016 - 3:05pm, Skiles 006

Mayya Zhilova, School of Mathematics, Georgia Tech        Organizer: Christian Houdré

The bootstrap procedure is well known for its good finite-sample performance, though the majority of the present results about its accuracy are asymptotic. I will study the accuracy of the weighted (or multiplier) bootstrap procedure for estimation of quantiles of a likelihood ratio statistic. The set-up is the following: the sample size is bounded, random observations are independent, but not necessarily identically distributed, and a parametric model can be misspecified. This problem had been considered in the recent work of Spokoiny and Zhilova (2015) with non-optimal results. I will present a new approach improving the existing results.

Thursday, April 21, 2016

Levy-Khintchine random matrices and the Poisson weighted infinite skeleton tree

Thu, 04/21/2016 - 3:05pm, Skiles 006

Paul Jung, University of Alabama Birmingham        Organizer: Christian Houdré

We look at a class of Hermitian random matrices which includes Wigner matrices, heavy-tailed random matrices, and sparse random matrices such as adjacency matrices of Erdos-Renyi graphs with p=1/N. Our matrices have real entries which are i.i.d. up to symmetry. The distribution of entries depends on N, and we require sums of rows to converge in distribution; it is then well-known that the limit must be infinitely divisible. We show that a limiting empirical spectral distribution (LSD) exists, and via local weak convergence of associated graphs, the LSD corresponds to the spectral measure associated to the root of a graph which is formed by connecting infinitely many Poisson weighted infinite trees using a backbone structure of special edges. One example covered are matrices with i.i.d. entries having infinite second moments, but normalized to be in the Gaussian domain of attraction. In this case, the LSD is a semi-circle law.

Thursday, April 7, 2016

Dynamics of geodesic flows with random forcing on Lie groups with left-invariant metrics

Thu, 04/07/2016 - 3:05pm, Skiles 006

Wenqing Hu, University of Minnesota, Twin Cities        Organizer: Christian Houdré

Motivated by problems in turbulent mixing, we consider stochastic perturbations of geodesic flow for left-invariant metrics on finite-dimensional Lie groups. We study the ergodic properties and provide criteria that ensure the Hormander condition for the corresponding Markov processes on phase space. Two different types of models are considered: the first one is a classical Langevin type perturbation and the second one is a perturbation by a “conservative noise”. We also study an example of a non-compact group. Joint work with Vladimir Sverak.

Thursday, March 31, 2016

Random walks on abelian sandpiles

Thu, 03/31/2016 - 3:05pm, Skiles 006

John Pike, Cornell University        Organizer: Christian Houdré

Given a simple connected graph G=(V,E), the abelian sandpile Markov chain evolves by adding chips to random vertices and then stabilizing according to certain toppling rules. The recurrent states form an abelian group \Gamma, the sandpile group of G. I will discuss joint work with Dan Jerison and Lionel Levine in which we characterize the eigenvalues and eigenfunctions of the chain restricted to \Gamma in terms of "multiplicative harmonic functions'' on V. We show that the moduli of the eigenvalues are determined up to a constant factor by the lengths of vectors in an appropriate dual Laplacian lattice and use this observation to bound the mixing time of the sandpile chain in terms of the number of vertices and maximum vertex degree of G. We also derive a surprising inverse relationship between the spectral gap of the sandpile chain and that of simple random walk on G.

Tuesday, March 8, 2016

Randomized Isoperimetric Inequalities

Tue, 03/08/2016 - 3:05pm, Skiles 005

Peter Pivovarov, University of Missouri        Organizer: Galyna Livshyts

The focus of my talk will be stochastic forms of isoperimetric inequalities for convex sets. I will review some fundamental inequalities including the classical isoperimetric inequality and those of Brunn-Minkowski and Blaschke-Santalo on the product of volumes of a convex body and its polar dual. I will show how one can view these as global inequalities that arise via random approximation procedures in which stochastic dominance holds at each stage. By laws of large numbers, these randomized versions recover the classical inequalities. I will discuss when such stochastic dominance arises and its applications in convex geometry and probability. The talk will be expository and based on several joint works with G. Paouris, D. Cordero-Erausquin, M. Fradelizi, S. Dann and G. Livshyts.

Thursday, March 3, 2016

On the analogue of the monotonicity of entropy in the Brunn-Minkowski theory

Thu, 03/03/2016 - 3:05pm, Skiles 006

Arnaud Marsiglietti, IMA, University of Minnesota        Organizer: Galyna Livshyts

In the late 80's, several relationships have been established between the Information Theory and Convex Geometry, notably through the pioneering work of Costa, Cover, Dembo and Thomas. In this talk, we will focus on one particular relationship. More precisely, we will focus on the following conjecture of Bobkov, Madiman, and Wang (2011), seen as the analogue of the monotonicity of entropy in the Brunn-Minkowski theory: The inequality $$ |A_1 + \cdots + A_k|^{1/n} \geq \frac{1}{k-1} \sum_{i=1}^k |\sum_{j \in \{1, \dots, k\} \setminus \{i\}} A_j |^{1/n}, $$ holds for every compact sets $A_1, \dots, A_k \subset \mathbb{R}^n$. Here, $|\cdot|$ denotes Lebesgue measure in $\mathbb{R}^n$ and $A + B = \{a+b : a \in A, b \in B \}$ denotes the Minkowski sum of $A$ and $B$. (Based on a joint work with M. Fradelizi, M. Madiman, and A. Zvavitch.)

Thursday, February 25, 2016

Central Limit Theorems and Bootstrap in High Dimensions

Thu, 02/25/2016 - 3:05pm, Skiles 006

Victor Chernozhukov, MIT        Organizer: Karim Lounici

This paper derives central limit and bootstrap theorems for probabilities that sums of centered high-dimensional random vectors hit hyperrectangles and sparsely convex sets. Specifically, we derive Gaussian and bootstrap approximations for probabilities  Pr(n−1/2∑ni=1Xi∈A) where X1,…,Xn are independent random vectors in ℝp and Ais a hyperrectangle, or, more generally, a sparsely convex set, and show that the approximation error converges to zero even if p=pn→∞ as n→∞ and p≫n; in particular, p can be as large as O(eCnc) for some constants c,C>0. The result holds uniformly over all hyperrectangles, or more generally, sparsely convex sets, and does not require any restriction on the correlation structure among coordinates of Xi. Sparsely convex sets are sets that can be represented as intersections of many convex sets whose indicator functions depend only on a small subset of their arguments, with hyperrectangles being a special case.

Paper available on arXiv:1412.3661

Thursday, February 18, 2016

On the infinitesimal versions of Log Brunn Minkowski and Gaussian Brunn Minkowski conjectures

Thu, 02/18/2016 - 3:05pm, Skiles 006

Galyna Livshyts, School of Mathematics, Georgia Tech        Organizer: Galyna Livshyts

Log Brunn-Minkowski conjecture was proposed by Boroczky, Lutwak, Yang and Zhang in 2013. It states that in the case of symmetric convex sets the classical Brunn-MInkowski inequality may be improved. The Gaussian Brunn-MInkowski inequality was proposed by Gardner and Zvavitch in 2007. It states that for the standard Gaussian measure an inequality analogous to the additive form of Brunn_minkowski inequality holds true for symmetric convex sets. In this talk we shall discuss a derivation of an equivalent infinitesimal versions of these inequalities for rotation invariant measures and a few partial results related to both of them as well as to the classical Alexander-Fenchel inequality.

Thursday, February 11, 2016

On the CLT for sample covariance matrices without independent structure in columns

Thu, 02/11/2016 - 3:05pm, Skiles 006

Anna Lytova, University of Alberta        Organizer: Galyna Livshyts

Abstract

Thursday, February 4, 2016

Random zero sets under repeated differentiation of an analytic function

Thu, 02/04/2016 - 3:05pm, Skiles 006

Sneha Subramanian, School of Mathematics, Georgia Tech        Organizer: Christian Houdré

For a random (complex) entire function, what can we say about the behavior of the zero set of its N-th derivative, as N goes to infinity? In this talk, we shall discuss the result of repeatedly differentiating a certain class of random entire functions whose zeros are the points of a Poisson process of intensity 1 on the real line. We shall also discuss the asymptotic behavior of the coefficients of these entire functions. Based on joint work with Robin Pemantle.

Thursday, January 28, 2016

A central limit theorem for temporally non-homogenous Markov chains with applications to dynamic programming

Thu, 01/28/2016 - 3:05pm, Skiles 006

Alessandro Arlotto, Duke University        Organizer: Christian Houdré

We prove a central limit theorem for a class of additive processes that arise naturally in the theory of finite horizon Markov decision problems. The main theorem generalizes a classic result of Dobrushin (1956) for temporally non-homogeneous Markov chains, and the principal innovation is that here the summands are permitted to depend on both the current state and a bounded number of future states of the chain. We show through several examples that this added flexibility gives one a direct path to asymptotic normality of the optimal total reward of finite horizon Markov decision problems. The same examples also explain why such results are not easily obtained by alternative Markovian techniques such as enlargement of the state space. (Joint work with J. M. Steele.)