The seminar is organized by
Date and location  Speaker  Title  

September 20, 4.30 pm  5.30 pm, Room 622  Hugo Falconet  Mandelbrot's Percolation Process  
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 RobinsonSchenstedKnuth correspondence is a combinatorial bijection which associates to each matrix M with nonnegative entries a pair (P,Q) of semistandard Young tableaux of the same shape and it provides a natural framework for the study of last passage percolation and longest increasing subsequence problems. The geometric lifted RSK correspondence provides a similar framework for the study of loggamma random polymers. First we will describe that using a geometric row insertion procedure, array z(n), the Ptableau analogue of the gRSK with input n*N weight matrix X, naturally has Markov property. To the direction of application to loggamma polymer model, we will show it's equivalent to define the array z(n) via nonintersecting lattice paths and z_{N,1} now is the partition function for loggamma 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 loggamma partition function and moreover we will move to the discrete loggamma 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 qHahn TASEP and the cauchy identity for vertex models.


October 30, 4.30 pm  5.30 pm, Room 528  Sayan Das  Branching Random Walk with heavytailed 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 heavytailed 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 1516, 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  
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.

Date and location  Speaker  Title  

February 8, 3 pm  4 pm, Room 528  Yier Lin  KestenStigum theorem  
For Galton Watson process Z_n with mean m>1, it is well known Z_n / m^n \to Y for some nonnegative 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 GaltonWatson 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 KestenStigum theorem.


March 8, 5 pm  6 pm, Room 528  Shalin Parekh  CameronMartin Theory  
We will give a short introduction to the abstract theory of Gaussian measures on infinitedimensional 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  CameronMartin Theory (bis)  
We will continue to discuss the theory Gaussian measures on Banach spaces, specifically the CameronMartin space and absolute continuity of translation maps.


April 5, 5 pm  6 pm, Room 528  Xuan Wu  On the qTASEP model  
Let's look at the qTASEP 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 qTASEP model (bis)  
Xuan will continue her previous lecture.

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 TracyWidom 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 TracyWidom distribution (bis)  
We will use the steepest descent to show that the largest eigenvalue in GUE follows the TracyWidom 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 YangMills 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 betaensemble is connected to YangMills 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 brownianbridge 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.
