# — CUNY Probability Seminar, Spring 2018

The CUNY Probability Seminar is typically held on Tuesdays from 4:15 to 5:15 pm at the CUNY Graduate Center Math Department. The exact dates, times and locations are mentioned below. If you want to talk at the seminar, or want to be added to/removed from the seminar mailing list, then please contact the Seminar Coordinator

Jack Hanson (http://jhanson.ccny.cuny.edu/)

Tuesday, February 6, 2018, 4:15 PM, Rm. 3307:

Speaker: Joshua Rosenberg, UPenn

Title: Invasion Percolation on Galton-Watson Trees

Abstract: This talk will focus on invasion percolation on Galton-Watson trees.  On almost every Galton-Watson tree, the invasion cluster almost surely contains only one infinite path.  This means that for almost every Galton-Watson tree, invasion percolation induces a probability measure on infinite paths from the root.  I will discuss our proof demonstrating that under certain conditions of the progeny distribution, this measure is absolutely continuous with respect to the limit uniform measure.  This confirms that invasion percolation, an efficient self-tuning algorithm, may be used to sample approximately from the limit uniform distribution.  Time permitting, I will also discuss a related result we achieved which involved proving a limit law for the forward maximal weights along the backbone of the invasion cluster.  This is based on joint work with Marcus Michelen and Robin Pemantle.

Tuesday, February 13, 4:15 PM, Rm. 3307:

Speaker:  Jay Rosen, CUNY (College of Staten Island)

Title: Tightness for the Cover Time of compact two dimensional manifolds

Abstract: Let $\CC^\ast_{\ep,M}$ denote the cover
time of the two dimensional compact manifold $M$ by
a Wiener sausage of radius $\ep$. We prove that
$$\sqrt{\CC^{\ast}_{\ep,M} } -\sqrt{\frac{2A_{M}}{\pi}}$$\log \ep^{-1}-\frac14\log\log \ep^{-1}$$$$ is tight, where $A_{M}$ denotes the Riemannian area of $M$.

Joint work with David Belius and Ofer Zeitouni

Tuesday, February 27, 4:15 PM, Rm. 3307:

Speaker: Shirshendu Chatterjee, CUNY (City College)

Title: TBA

Abstract:TBA

Tuesday, March 6, 4:15 PM, Rm. 3307:

Speaker: Yuri Bakhtin, NYU

Title: Invariant densities for systems with random switchings

Abstract: In piecewise deterministic Markov processes, the system solves an ODE with the right-hand side that switches randomly between vector fields from a fixed family. I will talk about smoothness and singularities of invariant distributions for such systems. The most recent progress is inspired by the Malliavin calculus approach to hypoellipticity in diffusion processes. Joint work with Tobias Hurth, Jonathan Mattingly, Sean Lawley.

Tuesday, March 20, 4:15 PM, Rm. 3307:

Speaker: Alexander Moll, HCM

Title: Random Permutations and the Quantum Linear Wave Equation

Abstract: The quantum linear wave equation – also known as the free massless scalar bosonic quantum field – is a geometric quantization Q with respect to the Peierls bracket of the classical linear wave equation for u(x,t) for x on any Riemannian manifold (X,g).  We first define from probabilistic first principles (without using path integrals) the Fock space of a Sobolev space, the quantization Q, phonon number states, mixed thermal states, and pure coherent states.  Next, we describe the distributions of observables in coherent states as fractional Gaussian fields and describe the h^{1/2} correction in the semi-classical limit at fixed time by specializing a general result from [A. Moll 2017].  Finally, we specialize to the quantum linear wave equation in (1+1) dimensions, where we encounter simultaneously two objects of recent interest in extreme value theory:

(I) the number of k-cycles in a random permutation of size d with law weighted by arbitrary multiplicative cycle weights possibly depending on d

(II) regularized exponentials of log-correlated fractional Gaussian fields in 1D

and relate their asymptotics to the semi-classical analysis of coherent states.

Tuesday, March 27, 4:15 PM, Rm. 3307:

Speaker: Li-Cheng Tsai, Columbia

Title: KPZ equation limit for the six vertex model

Abstract: The Six Vertex (6V) model, initially introduced as a model for ice, is an integrable model for tiling in two dimensions. In this talk we consider symmetric and stochastic 6V models, and show that, under certain scaling into the ferroelectric/disordered phase critical point, fluctuations described by the Kardar–Parisi–Zhang (KPZ) equation arises.

Our approach utilizes the one- and two-point Markov duality enjoyed by the stochastic 6V model. One-point duality gives the (so-called) microscopic Hopf–Cole transform, and thereby exposes the connection to KPZ equation. On the other hand, two-point duality provides exact, analyzable formulas that help to establish certain self-averaging, which is the key step in the proof given the aforementioned transform.
Joint work with Ivan Corwin, Promit Ghosal, and Hao Shen

Tuesday, April 10, 4:15 PM, Rm. 3307:

Speaker: Arjun Krishnan, U. of Rochester

Title: TBA

Abstract: TBA

Tuesday, April 17, 4:15 PM, Rm. 3307:

Speaker: Philippe Sosoe, Cornell University

Title: Dispersive equations with random initial data

Abstract: Beginning the 1980s, there has been interest in considering certain classical nonlinear equations such as nonlinear Schroedinger, Korteweg de Vries and wave equations, with random initial data. I will explain the motivation for this setting, describe some of the results obtained by using probabilistic methods for dispersive nonlinear equations, and finish by describing some recent and ongoing work  by myself and collaborators on the subject.

Tuesday April 24, 4:15 PM, Rm. 3307

Speaker: Guillaume Dubach

Title: Eigenvectors of non-hermitian random matrices

Abstract: Eigenvectors of non-hermitian matrices are non-orthogonal, and their distance to a unitary basis can be quantified through the matrix of overlaps. These variables quantify the stability of the spectrum, and characterize the joint eigenvalues increments under Dyson-type dynamics. They first appeared in the physics literature; well known work by Chalker and Mehlig calculated the expectation of these overlaps for complex Ginibre matrices. For the same model, we extend their results by deriving the distribution of the overlaps and their correlations. (Joint work with P. Bourgade)

Tuesday May 1, 4:15 PM, Rm. 3307

Speaker: Matthew Junge, Duke U.

Title: Chase-Escape

Abstract:  Imagine barnacles and mussels spreading across the surface of a rock. Barnacles move to adjacent unfilled spots. Mussels too, but they can only attach to barnacles. Barnacles with a mussel on top no longer spread. What conditions on the rock geometry (i.e. graph) and spreading rates (i.e. exponential clocks) ensure that barnacles can survive? Chase-escape can be formalized in terms of competing Richardson growth models; one on top of the other. New, tantalizing open problems will be presented. Joint work with Rick Durrett and Si Tang.​

Tuesday May 8, 4:15 PM, Rm. 3307

Title: Levy driven Ornstein-Uhlenbeck type processes and intermittency

Abstract: We consider finite variance Levy driven Ornstein-Uhlenbeck processes. Although these processes provide a rich class of stationary models, their dependence structure is rather limited from a modeling perspective. But superpositions of such Ornstein-Uhlenbeck processes introduced by Barndorff-Nielsen, provide far more flexibility and can exhibit both short-range dependence and long-range dependence. They have found many applications, especially in finance where they are used in models for stochastic volatility. The asymptotic behavior of such  processes, integrated and normalized, can be, however, unusual. The cumulants and moments turn out to have an unexpected rate of growth, akin to a physical phenomenon called intermittency. Self-similarity of the limit and intermittency, however, are typically not compatible. We will try to shed some light on this unusual situation.

Tuesday May 15, 4:15 PM, Rm. 3307

Speaker: Sixian Jin, Fordham U.

Title: TBA

Abstract: TBA