# Student Probability Seminar

This semester, Spring 2017, we meet Mondays from 11.am to 12.pm in WWH 905. 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 background in probability. If you wish to speak please contact Reza Gheissari at reza [at] cims or Guillaume Dubach at dubach [at] cims.

## Spring 2017

### March 6th

Some problems of random cycles - and how to fix them.

Guillaume Dubach

Abstract: We shall consider the uniform distribution over the permutation groups and try to answer simple questions such as : how likely is it for two elements to belong to the same cycle ? How are fixed points distributed ? What is the typical size of a cycle ?... Some of our answers will be just as simple as the questions; whereas others will require to introduce usual tools and methods of random permutation theory, such as the celebrated Feller coupling, thanks to which we might even end up answering questions we didn't ask.

### February 27th

Critical slowdown of Swendsen--Wang dynamics.

Reza Gheissari

Abstract: Consider the $q$-state mean-field Potts model, a probability distribution $\pi$ on $\{1,...,q\}^{n}$ with $\pi(\sigma)\propto \exp(\beta H(\sigma))$ where $H(\sigma)$ counts the number of pairs of vertices in the same state. It is a canonical model of statistical physics generalizing the $q=2$ case (Ising Curie--Weiss). In this talk, we will introduce the Swendsen--Wang dynamics, introduced in the 1980's, as a Markov chain reversible w.r.t. $\pi$ that utilizes the Erdos--Renyi random graph $G(n,p)$ to make global changes and jump over the energy barriers in the Potts landscape. As a result, for some time, physicists believed the Swendsen--Wang algorithm to always be a fast (polynomial time) MCMC sampler of the Ising/Potts model. In 1999, Gore and Jerrum found that when $q\geq 3$, at the critical temperature $\beta_c(q)$, the dynamics is in fact slow: $t_{\mathrm {mix}}\geq \exp(c\sqrt n)$. We present that proof, then with some new random graph estimates, show that in fact, there, $t_{\mathrm{mix}}$ is truly exponential in $n$: $\log t_{\mathrm {mix}} \asymp n$. Joint work with E. Lubetzky and Y. Peres.

### February 13th

Interpretability of fixed points of recursive distributional equations.

Moumanti Podder

Abstract: Consider the property $A$ that there is a complete binary tree, as a subtree, starting at the root. We wish to find $P_{\lambda}(A) = P[T_{\lambda} \text{ satisfies } A]$, where $T_{\lambda}$ is the Galton-Watson process with $Poisson(\lambda)$ offspring distribution. This probability is given by a fixed point of the function (recursive distribution equation): $\Psi(x) = 1 - e^{-\lambda x} (1 + \lambda x)$. But for $\lambda$ bigger than a critical $\lambda_{0}$, this function has $3$ fixed points: the $0$ solution, the true probability $P_{\lambda}(A)$, and a third fixed point $q_{\lambda}$. We are interested in finding an interpretation for $q_{\lambda}$. In other words, we wish to find if there is any non-analytic reason why this fixed point appears.

## Fall 2016

### December 1st

A Driven Tagged Particle in the Symmetric Simple Exclusions in Z^3.

Zhe Wang

Abstract: We will consider the symmetric simple exclusions in Z^3, with a tagged particle driven by external forces. Starting from the Bernoulli measure initial measures with density \rho, the environment viewed from the tagged particles has is close to the Bernoulli measures. Particularly, the densities of large blocks would be close to \rho. We will see how to estimate this with applications of Feynman-Kac Formula, Dirichlet Forms, Generalized Duality and some random walk estimates. We may also talk about some open problems and similar results.

### November 17th

Universality of the large deviations for the roots of random polynomials.

Raphael Butez

Abstract: In this talk we will study the behavior of the complex roots of random polynomials P= a_0 = a_1 X+...+a_n X^n where the coefficients are i.i.d. random variables. We will see that when the coefficients are Gaussian random variables, the roots of P behave like a Coulomb gas (like electrons in a conductor). We will show how we can obtain global results on the zeros of P when the degree grows at infinity and we will answer partially to the question: how much do these results depend on the distribution of the coefficients?

### November 8th

Ergodic theory for SPDEs.

Liying Li

Abstract: In this talk I will talk about how to develop ergodic theory for SPDEs. On the one hand, SPDEs can be seen as Markov chains, so some Markov chain techniques, such as coupling, Lyapunov function, may be useful; on the other hand, the infinite dimensional nature of the problem brings extra difficulties. With some basic examples I will try to give a flavor of the related theory.

### November 3rd

Intermediate disorder limits for multi-layer random polymers.

Mihai Nica

Abstract: Random polymers are disordered systems made from a random walk in a disordered environemtn. The intermediate disorder regime is a scaling limit for disordered systems where the inverse temperature is critically scaled to zero as the size of the system grows to infinity. For a random polymer given by a single random walk, Alberts, Khanin and Quastel proved that under intermediate disorder scaling the polymer partition function converges to the solution to the stochastic heat equation with multiplicative white noise. In this talk, I consider polymers made up of multiple non-intersecting walkers and consider the same type of limit. The limiting object now is the multi-layer extension of the stochastic heat equation introduced by O'Connell and Warren. This result proves a conjecture about the KPZ line ensemble.

### October 25th

The Characteristic Polynomial of a Random Unitary Matrix.

Guillaume Dubach

Abstract: We will introduce the Circular Unitary Ensemble (CUE) of random unitary matrices distributed according to the Haar measure on Un(C), and compute the distribution of its characteristic polynomial thanks to an explicit decomposition of the ensemble. In particular, the moments of this characteristic polynomial can be computed ; they are strongly believed to have a link with the moments of the zeta function along its critical line, and we will give some intuitive (but striking) reasons to think it might be true.

### October 18th

Eigenvalues of random matrices: a ‘statistical physics’ approach

Thomas Leble

Abstract: Spectrum is arguably the central object of random matrix theory. I will explain how the random eigenvalues of some usual matrix ensembles can be seen as interacting particles with logarithmic repulsion, and how a ‘statistical physics’ approach can be used to derive results about the asymptotic behaviour of the eigenvalues at macroscopic and microscopic scales.

### October 11th

Gaussian comparison and its application to extremes

Lisa Hartung

Abstract: In this talk I will start by discussing Gaussian comparison. It allows to compare the expectation of functionals of Gaussian processes with each other. We will state and proof a Gaussian integration by parts formula and Kahane’s Theorem. We then apply this technique to compare the extremes of different Gaussian processes.

### October 4th

Static and dynamical phase transitions of the 2D Potts model

Reza Gheissari

Abstract: We introduce the two-dimensional q state Potts model, a generalization of the Ising model to q possible states. At critical temperature \beta_c(q), the model has a very rich phase transition that is continuous (second order) for q\leq 4 and believed to be discontinuous for q>4. We discuss the different phase transitions of the model. First we present recent results of Beffara and Duminil-Copin and Duminil-Copin, Sidoravicius, and Tassion identifying the critical temperature and the continuity of the phase transition for q\leq 4, including the expected conformal invariance. We then discuss how features of the static phase transition emerge in a parallel dynamical phase transition.

## Spring 2016

### Thursday, March 3th

The mixing time of Glauber dynamics on the critical 2D Ising model

Reza Gheissari

Abstract: Consider the Glauber dynamics with stationary distribution given by the Ising measure on $\mathbb Z^2$. The mixing time of the Markov chain (the time it takes to approach stationarity in $L_1$) is intimately related to the gap in the spectrum of its generator. We begin with some basic facts about the mixing time and spectral gap, then prove the block dynamics technique introduced by Martinelli, to recursively bound the spectral gap of a chain on a spin system. The rest of the talk will then provide a nice application of the technique to obtain a polynomial upper bound on the mixing time of the Ising model in 2D at its critical temperature (Lubetzky,Sly). If time permits we will discuss some related problems where the block dynamics could prove useful.

### Thursday, February 25th

Mere coincidences, or perhaps more.

Guillaume Dubach

Abstract: The story of the meeting at tea time between Freeman Dyson and Hugh Montgomery, and what happened that the latter would later recall as 'real serendipity', is now famous; but a good story never bores. We shall tell it again, sketch its mathematical background, and present a few other number-theoretic objects that exhibit amazing similarities with well-known results in Random Matrix theory : not only the zeros of the Zeta function, but also its moments on the critical line, and the number of points of some elliptic curves over finite fields.

### Thursday, February 18th

Last passage percolation and random matrices

Lucas Begnini

Abstract: We'll start with the description of last passage percolation (LPP) and some other interpretations of that model. We'll then look at the results given by Johansson that creates a link between LPP and random matrix theory : the similarity between the distribution of the LPP model and the largest eigenvalue of some random matrix ensembles. We'll see that by using random matrix theory tools such as orthogonal polynomials and its asymptotics, we can obtain the same type of central limit theorem with fluctuations given by the Tracy-Widom distribution. We'll then look at the combinatorial proof, using the Robinson-Schensted-Knuth correspondence and Schur polynomials, that gives the distribution of the LPP. Finally, If we have enough time, we'll look at some more recent results/improvements in last passage percolation theory.

### Thursday, February 11th

First order sentences on G(n,p)​, zero-one laws, almost sure and complete theories

Moumanti Podder

Abstract: We start with G(n, p)​, at first with a constant p​, and then examine the probability of the presence of some property, especially the limit as n​ goes to ∞​. We discuss the Fagin-GKLT theorem which shows that for constant p​, any first order property will hold almost surely or almost never in the above set-up. We informally discuss the intuition and the consquence of a well known result: when p(n) = n-α​, G(n, p)​ satisfies the Zero-One law if and only if α​ is irrational. Time permitting, we define almost sure and complete theories, and countable models. We look at examples of such theories for G(n, p)​ where p = p(n)​ varies over different ranges, especially in the case of very sparse random graphs.

## Fall 2015

### Thursday, September 24th

The Parisi Variational Problem: A New Approach

Aukosh Jagannath

Abstract: The Parisi Variational Problem is a challenging non-local, strictly convex variational problem over the space of probability measures whose analysis is of great interest to the study of mean field spin glasses. In this talk, I will present a new, conceptually simple approach to the study of this problem using techniques from PDEs, stochastic optimal control, and convex optimization. We will begin with a new characterization of the minimizers of this problem whose origin lies in the first order optimality conditions for this functional. As a demonstration of the power of this new approach, we will study a prediction of de Almeida and Thouless regarding the validity of the 1 atomic anzatz. We will generalize their conjecture to all mixed p-spin glasses and prove that their condition is correct in the entire temperature-external field plane except for a compact set whose phase is unknown at this level of generality. A key element of this analysis is a new class of estimates regarding gaussian integrals in the large noise limit called Dispersive Estimates of Gaussians’’ . This is joint work with Ian Tobasco (NYU).

### Thursday, October 1st

The Random Cluster Model and Applications

Reza Gheissari

Abstract: We motivate and introduce the random cluster model, a class of models that unify percolation, Ising, Potts and other statistical mechanics models. We define random cluster percolation with parameters p and q and define its coupling with Ising and Potts models. We then discuss its many phase transitions and critical properties and how its monotonicity properties can help solve problems in other models that don't exhibit monotonicity. Finally we discuss some open problems.

### Thursday, October 8th

Optimal Strategy in "Guess Who?": Beyond Binary Search

Mihai Nica

Abstract: "Guess Who?" is a popular two player game where players ask "Yes" or "No" questions to search for their opponent's secret identity from a pool of possible candidates. We model this as a simple stochastic game under the assumption that the opponent's secret identity is uniformly distributed. Using this model, we explicitly find the optimal strategy and prove it is optimal for both players and for all possible candidate pool sizes. Contrary to popular belief, performing a binary search is not the optimal strategy for both players. Instead, the optimal strategy for the player who trails is to make certain bold plays in an attempt catch up. We also find an interesting log-periodic behavior that naturally arises in the landscape of this game.

### Thursday, October 15th

The Lovasz Local Lemma and a couple of its magical applications

Moumanti Podder

Abstract: The Lovasz Local Lemma, which is an extremely powerful tool in probabilistic combinatorics, is really a generalization of the case of mutual independence of events to the scenario where there is rare dependencies among them. It helps to show that in certain cases, none of the finitely many "bad events" occur with positive probability. And it has some stunning uses, for example, providing sufficient conditions for the 2-colourability of hypergraphs, and exploring the properties of colourings of the real line. I hope to be able to give a concise proof of the lemma, and discuss two such nice applications.

### Thursday, October 22th

2D Gibbsian point processes: Preservation and breaking of symmetries for the hard core potential

Alexisz Gaal

Abstract: A mathematically rigorous model of crystallization using the Gibbs distribution haven't been constructed yet. The phase transitions of freezing and melting seem out of reach for mathematicians. We will look at a Poisson point process in a bounded sets of R^2 with constant intensity z>0 conditioned on the event that two Poisson points have distance greater than 1. I will quote a result to extend this process to the whole plane and we will look at properties of such extensions. Existence (for large class of such processes) and translational invariance were shown in 2011 and 2008. Breaking of the rotational symmetry is indicated in a few toy models. My goal is to discuss one of these toy models. Breaking of the rotational symmetry is believed to model the phase transition from liquid to solid crystal.

### Thursday, October 29th

Around Ulam's Problem

Guillaume Dubach

Abstract: This talk will recall and sketch famous results about the longest increasing subsequence of a random (uniformly chosen) permutation. For this purpose, we shall learn and play a solitaire game, give a probabilistic proof of two elegant hook-formulas, and explain the Robinson-Schensted algorithm, among other things. These tools will give us two different approaches of the same problem, and even allow us to draw some conclusions.

### Thursday, November 11th

Global solution for Burgers equation and its connection with geodesics in first-passage percolation

Liying Li

Abstract: In this talk I will explain how to construct skew-invariant global solution for 1D Burgers equation with Poisson random forcing. After writing down the variational formula for the solution, we can make use of several results about geodesics in poissonian point field. In particular, infinite geodesics exist and each of them has an asymptotic slope, and also every two geodesics with the same asymptotic slope almost surely coalesce. The key step is a concentration inequality for action which handles the non-compactness of our domain and also gives an upper bound for the fluctuation exponent. In the end I might talk about some similar results for positive viscosity.

### Thursday, Dec 10th

Rigorous derivation of Brownian motion from deterministic hard spheres

Ryan Denlinger

Abstract: The problem we consider is N identical hard spheres colliding in a periodic box. We will confine our attention to the Boltzmann-Grad scaling; this is a low-density limit which should imply the mutual independence of all particles, asymptotically as N tends to infinity, under very weak conditions on the initial data. The main difficulty, however, is that we want to consider the full deterministic evolution of all N particles, and deduce independence from first principles. We will describe a proof of asymptotic independence in the case of a very small perturbation of equilibrium; in particular, we only perturb the initial distribution of one "tagged" particle, or a finite number of particles. The diffusive limit was proven in this case, and the resulting evolution is a Brownian motion for the tagged particle. All results discussed in this talk are from Bodineau, Gallagher, Saint-Raymond (Invent. math. 2015).

## Fall 2014

### Friday, September 19th

Moment proof of the CLT and the semi-circle law!

Mihai Nica

Abstract: We will prove the CLT for i.i.d. random varialbes and the semi-circle law using the method of moments!

### Friday, September 26th

Janson's Inequalities

Moumanti Podder

Abstract: These inequalities provide exponential upper bounds for the probability that none of a set of bad events occurs. When the events are mutually independent, the probability that none of them occurs is equal to the product of the probabilities that they do not occur individually. When these events are "mostly" independent, Janson's inequalities show us that the upper bound in that case is nearly the same as in the independent setup.

### Friday, October 3rd

Courant Instructor Day

TBA

Abstract:TBA

### Friday, October 10th

Columbia-NYU Probability Seminar

### Friday, October 17th

Hydrodynamic Limit Theory

Insuk Seo

Abstract: The hydrodynamic limit theory provide a natural way to understand very huge interacting particle system via the scaling limit PDE. This theory has been developed during the end of the last century mainly by S.R.S. Varadhan and many coworkers. I introduce the main idea of this theory with some toy models including interacting diffusion, simple exclusion process and zero range process. Special backgrounds about this topic will not be assumed.

### Friday, October 24th

Lyapunov exponents and the multiplicative ergodic theorem

Alex Blumenthal

Abstract: The Multiplicative Ergodic Theorem (MET) can be thought of as a noncommutative version of the strong law of large numbers for products of matrices drawn from some distribution. The MET is a critical ingredient in modern studies of smooth ergodic theory. I will describe the information the MET provides, analyze some simple examples, give motivation for its applications in dynamical systems, and present the main step in the proof of the MET.

### Friday, October 31rd

SLE and Conformal Invariance

Reza Gheissari

Abstract: Motivated by the scaling limits of a wide range of statistical mechanics models including LERW, percolation, GFF and Ising model, Schramm in 2000 introduced Stochastic/Schramm Loewner Evolution (SLE_\kappa). We motivate and introduce chordal SLE_\kappa and some interesting properties of its dependence on parameter /kappa. We then give flavors for the proofs of convergence of critical percolation and Ising interfaces to SLE_\kappa due to Smirnov (2010).

### Friday, November 6th

A simplified Version of Wiener Sausage

Zhe Wang

Abstract: We are interested in the large time behavior of simple random walk in Z^d, especially the number of sites visited by it. The asymptotic behavior of the number could be analyzed via the behavior of its moment generating function. A rate function will provided as an application of Large Deviation Theory. Some inequalities and typical results for LDP will be introduced. No special background is assumed.

### Friday, November 14th

Self Avoiding Walks

Alex Rozinov

Abstract: A big open problem in probability is proving that self avoiding walks (SAW) are conformally invariant. In this talk we will discuss recent progress on SAWs. In particular we will cover Smirnov's and Dominil-Copin's incredible proof that the connective constant for SAWs is \sqrt(2+\sqrt(2)). We will give an overview of what it means to be conformally invariant and time-pertaining, discuss other problems in this area. The talk requires no prior knowledge and will not dwell much on Schramm-Loewner evolution (SLE) theory.

### Friday, November 14th

Northeast Probability Seminar

No Seminar

Abstract: None

Previous Years: Spring 2013 2008-2009