Student Probability Seminar

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...

Fall 2009

Friday, September 18th

Incoming PhD student reception.

No seminar


Friday, September 25th

Traffic jams, polymer growth, and random matrices

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!!!


Friday, October 2nd

The Gaussian Free Field

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.


Friday, October 9th

Boolean Functions in Probability

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!


Friday, October 16th

Critical and near-critical percolation in two dimensions (and a little bit of SLE(6))

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.


Friday, October 23th

Random Morse functions and the Gaussian Orthogonal Ensemble

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!


Friday, October 30th

Finite connections for supercritical Bernoulli bond percolation in 2D.

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.


Friday, November 6th

Elementary Malliavin Calculus

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.


Friday, November 13th

Mesoscopic fluctuations of the zeta zeros

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.


Friday, November 20th

Northeast Probability Seminar

No meeting

Abstract No meeting


Friday, November 27th

Thanksgiving!!!!

No meeting

Abstract No meeting


Friday, December 4th

Markov Type 2

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.


Friday, December 11th

TBA

TBA

Abstract TBA


    Past semesters

      Fall 2008

      Friday, September 19th

      Some exponents for First Passage Percolation.

      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.


      Friday, September 26th

      Wigner's Semi-Circle Law via the moments method.

      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.


      Friday, October 3rd

      Invasion Percolation in 2D.

      Michael Damron

      Abstract: Invasion Percolation in 2D.


      Friday, October 10th

      Phase Trasitions on Erdos-Renyi Graphs.

      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.


      Friday, October 17th

      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.


      Friday, October 24th

      Break

      Let's go to Columbia to the Probability Seminar!

      Abstract


      Friday, October 31st

      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


      Friday, November 7th

      Continuum Random Tree

      Yevhen Mohylevskyy

      Abstract: We will study the scaling limits of a uniform random tree.


      Friday, November 14th

      Stein's Method for Concentration Inequalities and a simple application to Spin Glass Systems and Random Permutations.

      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.

      Friday, November 21st

      TBA

      TBA


      Friday, November 28th

      TBA

      TBA