We meet every Friday, 3pm - 4pm, Room 202 WWH. All students are welcome. Every week one student chooses and presents a topic at an introductory level. Anyone is welcome and encouraged to speak, regardless of their probability background. If you want to give a talk, contact Tuca by email at auffing at cims...
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Ivan Corwin
Abstract: What happens when rush hour traffic comes up against the afternoon joy riders? How long will it take to get home? We answer these and related questions by considering a classical model for traffic known as the Totally Asymmetric Simple Exclusion Process (TASEP). We relate this two-sided density version to a directed polymer growth model known as Last Passage Percolation with two-sided boundary conditions. Considering the distribution of last passage times we demonstrate a connection to the largest eigenvalue distribution for perturbed Wishart (Covariance) Ensembles of Random Matrix Theory. No background necessary!!!
Oren Louidor
Abstract (from Sheffield's paper - GFF for Mathematicians) The d-dimensional Gaussian free field (GFF), also called the (Euclidean bosonic) massless free field, is a d-dimensional-time analog of Brownian motion. Just as Brownian motion is the limit of the simple random walk (when time and space are appropriately scaled), the GFF is the limit of many incrementally varying random functions on d-dimensional grids. We present an overview of the GFF and some of the properties that are useful in light of recent connections between the GFF and the Schramm-Loewner evolution.
Will Perkins
Abstract: We give a few examples of boolean functions arising in probability theory, then introduce the basics of Fourier analysis on boolean functions. We'll talk about the influence of variables and present a couple of the most useful results in the area. No background necessary!
Pierre Nolin
Abstract Percolation is probably one of the simplest models of a random medium, but it nonetheless features the typical properties of most statistical mechanics systems, such as the existence of a phase transition and of critical exponents. After a general overview of this model, we will present techniques which were developed to describe two-dimensional percolation near its phase transition (mostly on the triangular lattice). This uses Smirnov's proof of conformal invariance in the scaling limit at criticality, results on the so-called Schramm-Loewner Evolution (SLE) obtained by Lawler, Schramm and Werner, and also Kesten's scaling relations that relate the near-critical regime to the critical regime itself.
Antonio Auffinger
Abstract: We study the landscape structure of high dimensional smooth Gaussian Fields and its connection to Random Matrix Theory. Through the talk, we will introduce some basic facts about large deviation principles in order to study the complexity of spin glass models. The talk will be self-contained, no background necessary!
Oren Louidor
Abstract Two vertices are said to be finitely connected if they belong to the same cluster and this cluster is finite. We derive sharp asymptotics for finite connection probabilities for supercritical Bernoulli bond percolation on Z2.
Ling Zhu
AbstractWe introduce the Wiener-Ito Chaos expansion and define the Skorohod integral and Malliavin Derivative. The Skorohod integral is a generalization of Ito integral and gives a different approach to defining the Malliavin Derivative. Finally, we state and prove the famous Clark-Ocone formula. If time permits, we some examples of Skorohod integrals and Malliavin Derivatives. All the materials can be found in the 2009 book, Malliavin Calculus for Levy Processes with Applications to Finance, by Oksendal et al.
Paul Bourgade
Abstract For large unitary matrices, the number of eigenvalues in distinct shrinking intervals satisfies a central limit theorem, whose covariance structure is related to some branching processes. In this talk, we present these random matrix results, following works of Diaconis, Evans and Wieand, and show the strict analogue for the zeros of L-functions.
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Sean Li
Abstract Given the family of maps from finite subsets of a metric space into Hilbert space, is it possible to extend each of these maps to the entire space so that the Lipschitz constants are bounded by the Lipschitz constant of the initial maps (and possibly some universal constant depending only on the metric space)? We give a sufficient characterization of the metric spaces for which such extensions are always possible and then relate it to the behavior of Markov chains on that space.
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Antonio Auffinger
Abstract We introduce First Passage Percolation model and its basic results. Then we focus on the structure of the optimal path, more precisely on the inequalities involving the longitudinal fluctuation exponent and the transversal fluctuation exponent.
Ivan Corwin
Abstract In this talk I will introduce some of the basic ideas and problems in random matrix theory. In particular I will study the eigenvalue distribution for symmetric matrices with Gaussian i.i.d. entries and, via a version of the moments method, prove that this distribution converges to the famous Wigner semi-circle. More general results will follow from the method of proof.
Michael Damron
Abstract: Invasion Percolation in 2D.
Will Perkins
Abstract I'll talk about Erdos-Renyi random graphs G(n,p) and the phase transition that occurs at p = 1/n. We'll use three branching process models to estimate the sizes of the largest connected components in the graph in five regions of the evolution from p < 1/n to p > 1/n and show how a giant component emerges.
Limit theorems for Random Variables with Slowly Varying Tails
Onur Gun
Abstract: We will cover some results about random variables with slowly varying tails, connecting them to dynamics of Spin Glasses Systems.
Break
Let's go to Columbia to the Probability Seminar!
Abstract
Brownian Excursion view on Riemann Theta
Dmytro Karabash
Abstract: The relation between probability laws of Brownian Excursion and Riemann Theta function will be established. Sketches of main results and various version of Riemann Hypothesis will be reviewed. The talk is based on: http://arxiv.org/abs/math/9912170
Continuum Random Tree
Yevhen Mohylevskyy
Abstract: We will study the scaling limits of a uniform random tree.
Antonio Auffinger and Ivan Corwin
Abstract: This talk will be based on Sourav Chatterjee's Ph.D. thesis. We will give a brief introduction on how to use Stein's method to get concentration inequalities. As an application, we will get concentration bounds for the magnetization on the Curie-Weiss model and concetration for the number of fixed points of a random permutation.
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