### Probability Seminar at University of CincinnatiSpring 2015

 This semester the regular meeting time of our seminar is Wednesday 3:35-4:30pm at 4206 French Hall (Seminar Room). Please email yizao.wang@uc.edu if you want to be added to our email list or have any questions. Jan 28, no meeting. Feb 4, Wed, Quenched invariance principle for random walks in time-dependent random environment. Xiaoqin Guo, Purdue University. In this talk we discuss random walks in a time-dependent zero-drift random environment in Z^d. We prove a quenched invariance principle under an appropriate moment condition. The proof is based on the use of a maximum principle for parabolic difference operators. This is a joint work with Jean-Dominique Deuschel and Alejandro Ramirez. Feb 11, Wed, Random walk on weighted graphs, Yizao Wang, University of Cincinnati. In this talk we review some basic setup for random walks on weighted graphs, focusing on harmonic functions and effective resistances. The talk only assumes basic knowledge of Markov chains. Feb 18 (canceled due to weather) rescheduled on Feb 20, Friday 3:35-3:30pm at Seminar Room 4206 French Hall. On the empirical spectral distribution for sample covariance matrices with long memory, Florence Merlevède, Unversite de Marne-la-Vallée, France. The talk will focus on the empirical eigenvalue distribution of sample covariance matrices. We will show in particular that if the sample covariance matrix is generated by independent copies of a stationary regular sequence then its empirical eigenvalue distribution always has a limiting distribution depending only on the spectral density of the sequence. We characterize this limit in terms of Stieltjes transform via a certain simple equation. No rate of convergence to zero of the covariances is imposed. If the entries of the stationary sequence are functions of independent random variables the result holds without any other additional assumptions. The talk is based on a joint work with M. Peligrad. Feb 25, Wed, On the chemical distance in critical percolation. Michael Damron, Indiana University. In two-dimensional critical percolation, the works of Aizenman-Burchard and Kesten-Zhang imply that macroscopic distances inside percolation clusters are bounded below by a power of the Euclidean distance greater than 1+\epsilon for some positive \epsilon. No more precise lower bound has been given since 1999. Conditional on the existence of an open crossing of a box of side length n, there is a distinguished open path which can be characterized in terms of arm exponents: the lowest open path crossing the box. This clearly gives an upper bound for the shortest path. The lowest crossing was shown by Zhang and Morrow to have volume n^{4/3+o(1)} on the triangular lattice. Addressing a question of Kesten and Zhang from 1992, we compare the length of shortest circuit in an annulus to that of the innermost circuit (defined analogously to the lowest crossing). The main theorem shows that the ratio of the expected length of the shortest circuit to the expected length of the innermost circuit goes to 0 with n. This is nearly completed work with Jack Hanson and Phil Sosoe. Feb 26, Thurs, 4-5pm, Department Colloquium, WCharlton 120. Recent advances in first-passage percolation. Michael Damron, Indiana University. In first-passage percolation (FPP), one places random non-negative weights on the edges of a graph and considers the induced weighted graph metric. Of particular interest is the case where the graph is Z^d, the standard d-dimensional cubic lattice, and many of the questions involve a comparison between the asymptotics of the random metric and the standard Euclidean one. In this talk, I will give an introduction to some of the main lines of research in FPP, concentrating on my recent results with A. Auffinger and J. Hanson. The topics will include the geometry and directional properties of geodesics and the set of possible limiting shapes for balls. Mar 1, 10-11am, CRC3230. Bose-Einstein condensation: from many quantum particles to a quantum superparticle, Kay Kirkpatrick, UIUC Plenary talk at Fifth Ohio River Analysis Meeting Near absolute zero, a gas of quantum particles can condense into an unusual state of matter, called Bose-Einstein condensation (BEC), that behaves like a giant quantum particle. The rigorous connection has recently been made between the physics of the microscopic many-body dynamics and the mathematics of the macroscopic model, the cubic nonlinear Schrodinger equation (NLS). I'll discuss progress with Gerard Ben Arous and Benjamin Schlein on a central limit theorem for the quantum many-body dynamics, a step towards large deviations for Bose-Einstein condensation. March 4, campus closed due to weather. March 11, Wed, Regenerative process Monte Carlo methods, Ju-Yi Yen, University of Cincinnati. Let $f : S \rightarrow\mathbb{R}$ be $\mathcal{S}$ measurable and integrable with respect to $\pi$. Let $\lambda \equiv \int_S fd\pi$. Our goal is to estimate $\lambda$. If $\pi$ is a probability measure (i.e. $\pi(S)=1$), there are currently two well-known statistical procedures for this. One is based on iid sampling from $\pi$, also called IID Monte Carlo (IIDMC); the other method is Markov chain Monte Carlo (MCMC). Both IIDMC and MCMC require that the target distribution $\pi$ is a probability distribution or at least a totally finite measure. In this talk, we discuss Monte Carlo methods to statistically estimate the integrals of a class of functions with respect to some distributions that may not be finite (i.e. $\pi(S) = \infty$) based on regenerative stochastic processes in continuous time such as Brownian motion. March 18, Spring break, no meeting. March 25, Wed, Yizao Wang. This will be an expository talk on hydrodynamic limit of interacting particle systems. April 1, Wed, Analysis on fractals and differential forms on Dirichlet spaces, Dan Kelleher, Purdue University. Analysis on fractals is a subject which lies at the intersection of probability, analysis and geometry. I will begin by talking about some of the advances and problems in the area, such as convergence of Laplacians/central limit theorems for random walks on approximating graphs, or calculating the spectrum and heat kernel estimates on the limiting fractal. I will also talk about the development of geometric objects on spaces with Dirichlet forms, such as intrinsic metrics, differential forms and Dirac operators. April 3, Friday, 3:30-4:30pm, 130 WCharlton, Colloquium. Hypocoercive diffusions, Fabrice Baudoin, Purdue University. In this talk, we will present a new method to study the convergence to equilibrium of hypocoercive diffusions. The method is based on local computations and parallels the Bakry-Emery approach to hypercontractivity. Refreshments will be served 2:45-3:15pm in the Faculty & Graduate Student Lounge, 4118 French Hall. April 8, Wed, Quenched Invariance Principles for the Fourier Transforms of a Stationary Process, David Barrera, University of Cincinnati. In these talks I will present the latest chapter in the series of investigations related to the Fourier Transforms of a Stationary Process started about 18 months ago with professor M.Peligrad. The first part of this cycle was devoted to present the quenched version of Peligrad-Wu's CLT for the Discrete Fourier Transforms of a Stationary process, and later, on the second part, we addressed the question of whether certain hypothesis of "random centering" was a necessary one (which required to construct an appropriate example), discussing also the meaning of our quenched CLT for linear processes in L2. In this third part I will address the question of extending the quenched CLT for Fourier Transforms to a quenched functional CLT or "invariance principle". I will present the results already obtained and the generalities of their proofs, emphasizing the novelties that such proofs require when compared to those of their annealed (classical) counterparts and the measure/ergodic-theoretic subtleties that make them appealing from a more abstract point of view. April 15, Wed, David Barrera, University of Cincinnati. April 22, Wed, Matrix polynomial generalizations of the sample variance covariance matrix and their large dimensional behavior. Arup Bose, Indian Statistical Institute, Kolkata. Suppose $Z_{u} = ((\varepsilon_{u,i,j}))_{p \times n},\ u \geq 1$, are independent (ID) matrices where $\{\varepsilon_{u,i,j}: u,i,j \geq 1\}$ are independent with mean zero and variance one. Also suppose $\{B_{2u-1}\}$ and $\{B_{2u}\}$ are constant matrices of order $p \times p$ and $n \times n$ respectively. Consider all $p \times p$ matrices of the form $$\label{eqn: pmatrix} P_{l,(u_{l,1},u_{l,2},\ldots, u_{l,k_{l}})} = \prod_{i=1}^{k_l} \left(n^{-1}A_{l,2i-1}Z_{u_{l,i}}A_{l,2i}Z_{u_{l,i}}^{*}\right) A_{l,2k_{l}+1}$$ where $\{A_{l,2i-1}\}$ and $\{A_{l,2i}\}$ and $\{Z_{u_{l,i}}\}$ are matrices from the collection $\{B_{2i-1}\}$, $\{B_{2i}\}$ and $\{Z_i\}$ respectively. These types of matrices often appear in high-dimensional statistical analysis. Two different regimes are: I. $p/n \to y \in(0,\infty)$ i.e. $p$ and $n$ increase at the same rate, and II. $p/n \to 0$, i.e. $n$ grows faster than $p$. We shows that in either regimes, the non-commutative $*$ probability space (NCP)\ $\text{Span}\{P_{l,(u_{l,1},u_{l,2},\ldots, u_{l,k_{l}})}: l \geq 1 \}$ (with an additional centering in Regime II) converges with state $p^{-1}E\text{Tr}$. In Regime I, the limit NCP can be expressed in terms of some free semi-circle variables and the limits of the deterministic matrices $\{B_{2i-1}\}$ and $\{B_{2i}\}$. In this regime, for all $i,j \geq 1$, the limits of $B_{2i-1}$ and $B_{2j}$ behave as free independent variables. In Regime II, the limit NCP can be described in terms of some free semi-circle families which are free of the limits of $\{B_{2i-1}\}$ and the matrices $B_ {2i}$ appear in the limit only via $\lim n^{-1}\text{Tr}(B_{2i}^{2})$. This has consequences for the asymptotic eigenvalue distribution in the bulk for autocovariance matrices in large dimensional time series models. May 26, Tuesday, 4pm. Some invariance principles for self-similar random fields. Olivier Durieu, Université de Tours, France. For a stationary random field $(X_j)_{j\in\mathbb Z^d}$ and some measure $\mu$ on $\mathbb R^d$, we consider the set-indexed weighted sum process $S_n(A)=\sum_{j\in\mathbb Z^d} \mu(nA\cap R_j)^\frac12 X_j$, where $R_j$ is the unit cube with lower corner $j$. We establish a general invariance principle under a weak dependance condition on the $X_j$'s and an entropy condition on the class of sets $A$. The limit processes are self-similar set-indexed Gaussian processes with continuous sample paths. Using Chentsov's type representations to choose appropriate measures $\mu$ and particular sets $A$, we obtain Lévy (fractional) Brownian fields or (fractional) Brownian sheets. Joint work with Hermine Biermé.