Columbia Student Probability Seminar

The seminar is organized by Hugo Falconet. If you want to be added on the mailing list and to participate, please send an email to hugo dot falconet at columbia dot edu and please use for subject : Probability Seminar.

Fall 2018

Date and location Speaker Title
September 20, 4.30 pm - 5.30 pm, Room 622 Hugo Falconet Mandelbrot's Percolation Process
I will give an introduction to the Mandelbrot's Percolation Process. After a brief introduction to the model, we will discuss connectivity properties following [CCD]) and [O]).
September 27, 4.30 pm - 5.30 pm, Room 622 Promit Ghosal Interplay between Liouville field theory and the regularity structures
In this talk, we discuss a recent paper by Christophe Garban on "Dynamical Liouville". In this paper, the author revisited the Gaussian multiplicative chaos (GMC) measure from the perspective of SPDE. This paper shows the existence of a stochastic process which is invariant under GMC measure. To prove its main result, the authors borrows some tools from the regularity structures. The plan of my talk is to give a very brief overview of the proof and point out some interesting directions at the end.
October 4, 4.30 pm - 5.30 pm, Room 622 Xuan Wu RSK and log gamma polymer model
The Robinson-Schensted-Knuth correspondence is a combinatorial bijection which associates to each matrix M with non-negative entries a pair (P,Q) of semi-standard Young tableaux of the same shape and it provides a natural framework for the study of last passage percolation and longest increasing sub-sequence problems. The geometric lifted RSK correspondence provides a similar framework for the study of log-gamma random polymers. First we will describe that using a geometric row insertion procedure, array z(n), the P-tableau analogue of the gRSK with input n*N weight matrix X, naturally has Markov property. To the direction of application to log-gamma polymer model, we will show it's equivalent to define the array z(n) via non-intersecting lattice paths and z_{N,1} now is the partition function for log-gamma polymer model when the entries of X are inverse gamma random variables. Applying an intertwining relation, the analogue shape z_{N,.}(n) also evolves marginally as a Markov process in it's own filtration. With the the transition kernel for z_{N,.}(n) explicitly computable due to the integrable structure behind, we will give the formula for the Laplace transformation of log-gamma partition function and moreover we will move to the discrete log-gamma line ensemble and explains how the random walk Gibbs property follows given the explicit expression of the transition kernel for z_{N,.}(n). [Reference].
October 11, 4.30 pm - 5.30 pm, Room 622 Xuan Wu RSK and log gamma polymer model (bis)
This week Xuan will continue and complete her previous talk.
October 18, 4.30 pm - 5.30 pm, Room 622 Shalin Parekh The Ising Model and Dynamical Phi^4
The 2d Ising model is a classical model of statistical mechanics which describes ferromagnets, and is one of the simplest models to exhibit a phase transition. For this model, one can introduce various types of Markovian dynamics (such as Glauber) which preserve the associated Gibbs measure. In this talk, we will cover a celebrated result of Mourrat and Weber, which roughly says that under an appropriate "weak scaling" towards the critical temperature, this Ising model with Glauber dynamics converges to an SPDE arising in quantum field theory, called the dynamical Phi^4 equation.
October 25, 4.30 pm - 5.30 pm, Room 622 Mark Rychnovsky Exactly solvable models
Integrable probability contains a rich hierarchy of exactly solvable models on which exact computations are possible. These allow us to analyze specific cases of phenomena which are difficult to treat in general. We will introduce several of the main models in the hierarchy and then discuss the main tools that allow for exact computations. We will discuss the coordinate bethe ansatz, duality, and integral formulas for q-Hahn TASEP and the cauchy identity for vertex models.
October 30, 4.30 pm - 5.30 pm, Room 528 Sayan Das Branching Random Walk with heavy-tailed Displacements.
Branching random walk (BRW) is an important model in the context of statistical physics and probability and has connections to tree indexed Random walks and Gaussian Free Field. In this talk, we will consider BRW with heavy-tailed displacements. Under appropriate assumptions, we will discuss the limiting behavior of the location of the rightmost particle and associated extremal point process of such Branching Random Walks ([Durrett83] , [BHR15]).
November 15-16, Courant Institute Northeast Probability Seminar Link to the conference website
November 22 No seminar Thanksgiving
November 27, 4.30 pm - 5.30 pm, Room 528 Yier Lin Vertex model and Gibbs state
I will introduce the higher spin vertex model defined in [CP16] and then describe how to use fusion to get the general horizontal spin. I will also derive Gibbs state of stochastic six vertex model, which appears in [Agg17].
December 4, 4.30 pm - 5.30 pm, Room 528 Mark Rychnovsky TBA
December 11, 4.30 pm - 5.30 pm, Room 528 Hugo Falconet TBA

Spring 2018

Date and location Speaker Title
February 8, 3 pm - 4 pm, Room 528 Yier Lin Kesten-Stigum theorem
For Galton Watson process Z_n with mean m>1, it is well known Z_n / m^n \to Y for some non-negative random variable Y. The Kesten Stigum theorem gives equivalent condition on whether P(Y=0) = 1 or P(Y=0)=q, where q is the probability Z_n finally extincts. We will introduce the size biased Galton-Watson tree and prove this theorem.
February 15, 5 pm - 6 pm, Room 528 Maithreya Sitaraman The Upper Bound Lemma
The dual of an abelian group is just the thing we expect: the collection of characters of the group, and is therefore a group itself. When we move to the nonabelian case, we would like to extend this notion of duality in a way that allows us to take Fourier transforms.. and the way we do this is to call the set of irreducible representations of our group as it's dual. This, however, fails to be a group. In this talk, I will set up and prove Fourier inversion on the dual of a (possibly nonabelian) finite group, and then prove a key corollary of this setup: The Upper Bound Lemma (introduced by Diaconis and Shahshahani), which allows one to bound how quickly a random walk on a group converges to the uniform distribution. I will then discuss some applications of the Upper Bound Lemma to some concrete problems that are of interest to probabilists. Apart from concrete problems one can solve, the Upper Bound Lemma is interesting on a philosophical level, since it forms a connection between representation theory and probability theory.
February 22, 5 pm - 6 pm, Room 528 Yier Lin Kesten Stigum theorem (bis)
We introduce Galton Watson process with immigration (GWI), it turns out that for Galton Watson process, consider its behavior under a change of measure, it's almost a GWI. We will use this observation to prove the Kesten-Stigum theorem.
March 8, 5 pm - 6 pm, Room 528 Shalin Parekh Cameron-Martin Theory
We will give a short introduction to the abstract theory of Gaussian measures on infinite-dimensional vector spaces, with applications to the study of Brownian motion and the Gaussian free field.
March 22, 5 pm - 6 pm, Room 528 Shalin Parekh Cameron-Martin Theory (bis)
We will continue to discuss the theory Gaussian measures on Banach spaces, specifically the Cameron-Martin space and absolute continuity of translation maps.
April 5, 5 pm - 6 pm, Room 528 Xuan Wu On the q-TASEP model
Let's look at the q-TASEP model, which is continuous Markov chain with state space to be the location of particles and let's derive the exact moment integral formula with nested contours using Duality and Bethe ansatz. If time admits, we could talk about how the moment formula turn into a Fredholm determinant integral formula by deforming to contours.
April 12, 5 pm - 6 pm, Room 528 Xuan Wu On the q-TASEP model (bis)
Xuan will continue her previous lecture.

Fall 2017

Date and location Speaker Title
November 1, 5.30 pm - 6.30 pm, Room 528 Shalin Parekh Introduction to the stochastic heat equation
The stochastic heat equation with additive noise is one of the simplest yet most fundamental examples of a stochastic PDE. I will first motivate why this object is important, and talk about its invariance under scaling. Then I will give the basic solution theory and regularity properties for this equation, as well as its invariant measure and ergodicity properties. I will also briefly mention more interesting PDEs which arise as nonlinear perturbations of this equation, such as stochastic quantization equations and KPZ.
November 8, 5.30 pm - 6.30 pm, Room 528 Mark Rychnovosky GUE and Tracy-Widom distribution
We will introduce the Gaussian Unitary Ensemble and show that the distribution of its eigenvalues can be described by a Fredholm determinant. Using this expression, we will outline a steep descent argument to show that the largest eigenvalue follows the Tracy Widom distribution.
November 15, 5.30 pm - 6.30 pm, Room 528 Mark Rychnovosky GUE and Tracy-Widom distribution (bis)
We will use the steepest descent to show that the largest eigenvalue in GUE follows the Tracy-Widom distribution.
November 22, 5.30 pm - 6.30 pm, Room 528 Promit Ghosal Convergence of holonomy field to master field on the sphere
I will introduce Yang-Mills holonomy field on a compact Riemannian surface and state the result on the convergence of holonomy field to master field on the sphere. This will be followed by some characterization of the master field. Next, I will talk about how discrete beta-ensemble is connected to Yang-Mills on sphere and show the proof of one proposition (proposition 2.5 of [DN17]) where this connection is helpful.
November 29, 5.30 pm - 6.30 pm, Room 528 Hugo Falconet Ito's excursion theory and applications
I will give an introduction to Ito’s excursion theory. After a short detour with computations in the discrete setting, I will introduce local times for continuous diffusions and explain Ito's theorem : labelled by the local time at zero, the excursions away from zero form a Poisson point process. Then I will explain how to use it to perform computations and we will see for instance how to obtain the law of the maximum of the absolute value of the standard brownian-bridge between time zero and one using this theory, which is based on [PW]).
December 6, 5 pm - 6 pm, Room 622 Hugo Falconet Bernoulli Percolation
I will introduce the model of Bernoulli percolation and explain some of its classical properties together with the main ingredients of their proofs (following Percolation et modèle d'Ising by W.Werner). In particular we will see the existence of a phase transition, the uniqueness of the infinite component at supercriticality and exponential decay of the probability that zero is connected to distance n at subcriticality.