Columbia Student Probability Seminar

The seminar is now organized by Sayan Das.

Fall 2019

Date and location Speaker Title Abstract
September 17, 3pm - 4pm, Room 528 Shalin Parekh Integer Partitions and Large Deviations
I will talk about fun combinatorial problems related to large integer partitions, and I will show how these problems can be solved using variational methods and large deviations theory.
September 24, 3pm - 4pm, Room 528 Maithreya Sitaraman Asymptotics of Plancherel measures via free probability
The (rescaled) Plancherel measure converges to the delta measure at a curve. This curve can be identified with the curve whose transition measure is the semicircle measure (the central object in free probability), and the convergence can be viewed as a consequence of the free central limit theorem.
October 1, 3pm - 4pm, Room 528 Hugo Falconet Introduction to Brownian geometry
I will give an introduction to the Brownian geometry developed by Le Gall (and many others). The talk is based on the Takagi lectures notes, which is a very nice survey of the theory.
October 8, 3pm - 4pm, Room 528 Mark Rychnovosky Introduction to Random Graphs
I will introduce the random graph models we are interested in, state some properties without proof, and then go into some detail on local weak convergence of these graph models.
October 15, 3pm - 4pm, Room 528 Sayan Das Largest Component in Erdös–Rényi Random Graph
We will talk about the size of connected components in Erdös–Rényi Random Graph G(n,d/n). As noted in the previous lecture, G(n,d/n) is closely connected to the Galton Watson Trees. Like in the Galton Watson Trees, we will see G(n,d/n) has a related phase transition depending on d<1 or d>1.
October 29, 3pm - 4pm, Room 528 Weitao Zhu Diameter and distances on random graphs
We use branching process results to study the typical distance between two points on the giant component of Erdos-Renyi graphs. We also discuss similar types of results for random d-regular graphs and random walks on random d-regular graphs.
November 12, 3pm - 4pm, Room 528 Hugo Falconet Liouville Brownian motion
The Liouville Brownian motion was constructed independently by Berestycki and Garban, Rhodes, Vargas. In this talk, I will explain how to construct it and describe some of its properties.
November 19, 3pm - 4pm, Room 528 Shalin Parekh Contact process on random graphs
I’ll talk about the contact process on trees and random graphs, presenting results by Liggett, Pemantle and Lalley-Su.
November 26, 3pm - 4pm, Room 528 Sayan Das The Reconstruction Problem for Spin Systems on Trees
In the setting of spin systems on rooted deterministic trees, we introduce the reconstruction problem. We show how a given Gibbs measure can be identified as extremal depending on the solvability of the corresponding reconstruction problem. If time permits, we will also discuss the applicability of this result to the Ising model on the d-ary tree.

Spring 2019: reading seminar on First-Passage Percolation

Main reference: 50 years of first passage percolation. University Lecture Series of the AMS. arXiv. (by A. Auffinger, M. Damron and J. Hanson)

Date and location Speaker Title
February 5, 3 pm - 4 pm, Room 528 Hugo Falconet Introduction to First-Passage percolation
February 12, 3 pm - 4 pm, Room 528 Shalin Parekh Limit shape theorem (bis) and Gromov-Hausdorff convergence
February 19, 3 pm - 4 pm, Room 528 Hugo Falconet Log improvement to variance bounds
February 26, 3 pm - 4 pm, Room 528 Shalin Parekh Brownian meanders in directed polymers
March 5, 3 pm - 4 pm, Room 528 Sayan Das Tail estimates
March 12, 3 pm - 4 pm, Room 528 Yier Lin Large deviations
March 26, 3 pm - 4 pm, Room 528 Xuan Wu Study of geodesics
April 2, 3 pm - 4 pm, Room 528 Mark Rychnovosky Busemann functions
April 9, 3 pm - 4 pm, Room 528 Maithreya Sitaraman Large dimensional limit shapes of FPP
April 30, 3 pm - 4 pm, Room 528 Romain Panis Hyperbolic Gaussian Analytic Functions and determinantal point processes

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 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 Random Matrices over finite fields
I will give a brief overview of what is known about random matrices over finite fields. I will also briefly describe a few of the main tools used to study them.

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.