Probability and Mathematical Physics Seminar
This seminar covers a wide range of topics in pure and applied probability and in mathematical physics. Unless otherwise noted, the talks take place on Fridays, 11am12noon, in room WWH 512. See directions to the Courant Institute.
Thomas Leblé helps organizing, on behalf of the Probability/Mathematical physics group. To be kept informed, you can subscribe to the Probability Seminar Mailing list.
Seminar Organizer(s): Thomas Leblé
Upcoming Events

Friday, April 19, 201911AM, Warren Weaver Hall 512
TBA

Friday, April 26, 201910:30AM, Warren Weaver Hall 512
TBA
Alex Hening, Tufts University 
Friday, April 26, 201911:30AM, Warren Weaver Hall 512
A Scaling limit for the Cover Time of the Binary Tree
Oren Louidor, TechnionSynopsis:
We consider a continuous time random walk on the rooted binary tree of depth \(n\) with all transition rates equal to one and study its cover time, namely the time until all vertices of the tree have been visited. We prove that, normalized by \(2^{n+1} n\) and then centered by \((\log 2) n  \log n\), the cover time admits a weak limit as the depth of the tree tends to infinity. The limiting distribution is identified as that of a Gumbel random variable with rate one, shifted randomly by the logarithm of the sum of the limits of the derivative martingales associated with two negatively correlated discrete Gaussian free fields on the infinite version of the tree. The existence of the limit and its overall form were conjectured in the literature. Our approach is quite different from those taken in earlier works on this subject and relies in great part on a comparison with the extremal landscape of the discrete Gaussian free field on the tree. Joint work with Aser Cortines and Santiago Saglietti.

Friday, May 3, 201911AM, Warren Weaver Hall 512
Columbia/Courant joint seminar

Friday, May 10, 201911AM, Warren Weaver Hall 512
TBA
Past Events

Friday, April 12, 201911:35AM, Warren Weaver Hall 512
On microscopic derivation of a meancurvature flow
Sunder Sethuraman, University of ArizonaSynopsis:
We discuss a derivation of a continuum meancurvature flow as a scaling limit of a class of particle systems, more robust than previous methods. We consider zerorange + Glauber interacting particle systems, where the zerorange part moves particles while preserving particle numbers, and the Glauber part allows creation and annihilation of particles. When the two parts are simultaneously seen in certain (different) timescales, and the Glauber part is "bistable", a meancurvature flow can be captured directly as a limit of the mass empirical density.
Such a "direct" limit might be compared with a "twostage" approach: When the zerorange part is diffusively scaled but the Glauber part is not scaled, the hydrodynamic limit is a nonlinear AllenCahn reactiondiffusion PDE. It is wellknown in such PDEs, when the "bistable" reaction term is now scaled, that the limit of the solutions takes on stable values across an interface moving by a meancurvature flow. This is joint workinprogress with Tadahisa Funaki and Danielle Hilhorst.

Friday, April 12, 201910:30AM, Warren Weaver Hall 512
Eigenvectors of nonhermitian random matrices
Guillaume Dubach, CourantSynopsis:
Right and left eigenvectors of nonHermitian matrices form a biorthogonal system, to which one can associate homogeneous quantities known as overlaps. The matrix of overlaps is Hermitian and positivedefinite; it quantifies the stability of the spectrum, and characterizes the joint eigenvalues increments under Dysontype dynamics. These variables first appeared in the physics literature, when Chalker and Mehlig calculated their conditional expectation for complex Ginibre matrices (1998). For the same model, we extend their results by deriving the distribution of the overlaps and their correlations (joint work with P. Bourgade). Similar results are expected to hold in other integrable models, and some have been established for quaternionic Gaussian matrices.

Friday, April 5, 201911AM, Warren Weaver Hall 512
Some free boundary problems arising from branching Brownian motion with selection
Jim Nolen, Duke UniversitySynopsis:
I will explain some current work on a stochastic interacting particle system, branching Brownian motion with selection, and its hydrodynamic limit, which is a free boundary PDE problem. At each branch event in the branching Brownian motion, a particle is removed from the system according to a fitness function, so that the total number of particles, N, is preserved. It is interesting to understand how this selection process effects the evolution of the ensemble of particles. De Masi, Ferrari, Presutti, SopranoLoto recently showed that in one space dimension, when the leftmost particle is always selected, then as N grows the particle system converges to the solution of a certain parabolic free boundary problem which has traveling wave solutions  this scenario corresponds to a fitness function which is monotone. In joint work with Julien Berestycki, Éric Brunet, Sarah Penington, we study this problem in higher dimensions with a fitness function that has compact level sets. The hydrodynamic limit (large N limit) is also a parabolic free boundary problem, related to the parabolic obstacle problem. The solution of this PDE problem converges, in the large time limit, to an eigenfunction of the laplacian. With Erin Beckman, we also study the problem in 1d with nonmonotone fitness, which leads to a kind of pulsating traveling wave behavior and a metastability phenomenon depending on the fitness function.

Friday, March 29, 201911AM, Warren Weaver Hall 512
Free probability for random band matrices
Benson Au, UC San DiegoSynopsis:
Free probability provides a unifying framework for studying random multimatrix models in the large \(N\) limit. Typically, the purview of these techniques is limited to invariant or meanfield ensembles. Nevertheless, we show that random band matrices fit quite naturally in this framework. Our considerations extend to the infinitesimal level, where finer results can be stated for the \(\frac{1}{N}\) correction. As an application, we consider the question of outliers for localized perturbations of our model.

Friday, March 8, 201911AM, Warren Weaver Hall 512
TBA
Jonathan Mattingly, Duke 
Friday, March 1, 201911AM, Warren Weaver Hall 512
Stochastic Ricci Flow on surfaces
Julien Dubédat, Columbia UniversitySynopsis:
The Ricci flow on a surface is an intrinsic evolution of the metric converging to a constant curvature metric within the conformal class. It can be seen as an (infinitedimensional) gradient flow. We introduce a natural 'Langevinization' of that flow, thus constructing an SPDE with invariant measure expressed in terms of Liouville Conformal Field Theory.
Joint work with Hao Shen (Wisconsin).

Friday, February 22, 201911AM, Warren Weaver Hall 512
New coupling techniques for exponential ergodicity of SPDEs in the hypoelliptic and effectively elliptic settings
Oleg Butkovsky, TU BerlinSynopsis:
We will present new coupling techniques for analyzing ergodicity of nonlinear stochastic PDEs with additive forcing. These methods complement the HairerMattingly approach (2006, 2011). In the first part of the talk, we demonstrate how a generalized coupling approach can be used to study ergodicity for a broad class of nonlinear SPDEs, including 2D stochastic NavierStokes equations. This extends the results of [N. GlattHoltz, J. Mattingly, G. Richards, 2017]. The second part of the talk is devoted to SPDEs that satisfy comparison principle (e.g., stochastic reactiondiffusion equation). Using a new version of the coupling method, we establish exponential ergodicity of such SPDEs in the hypoelliptic setting and show how the corresponding HairerMattingly results can be refined.
O. Butkovsky, A. Kulik, M. Scheutzow (2018). Generalized couplings and ergodic rates for SPDEs and other Markov models. arXiv:1806.00395; to appear in "The Annals of Applied Probability".
(Joint work with Alexey Kulik and Michael Scheutzow)

Friday, February 15, 201911AM, Warren Weaver Hall 512
On the spectrum of the hierarchical Schrödinger – type operators
Stanislav Molchanov, UNC Charlotte and HSE MoscowSynopsis:
The hierarchical Laplacian was initially introduced in the works of N. Bogolubov and his school (V. Vladimirov, I. Volovich, E. Zelinov) as an essential object in the \(p\)–adic analysis. Similar ideas were developed by F. Dyson in his famous paper on the phase transitions in \(1D\) Ising model with the long range potentials.
We define Dyson–Vladimirov hierarchical Laplacian \(\Delta\) as the nonlocal operator in \(L^2 (\mathbb{R}, dx)\) associated the Dyson metric on \(\mathbb{R}\). Such Laplacian has many features of the classical fractals (renorm group etc.).
The talk will present the elements of the spectral theory of the hierarchical Hamiltonian \(H = \Delta + V(x)\). The theory includes the standard results (on the essential selfadjointness, negative spectrum etc.) for the deterministic operators and the results in the spirit of the Anderson localization for the class of the random Schrödinger operators.

Friday, February 8, 201911AM, Warren Weaver Hall 512
Products of Many Random Matrices and Gradients in Deep Neural Networks
Boris Hanin, Texas A&M / Facebook AISynopsis:
Neural networks have experienced a renaissance in recent years, finding success in tasks from machine vision (e.g. selfdriving cars) to natural language processing (e.g. Alexa or Siri) and reinforcement learning (e.g. AlphaGo). A mathematical theory of how and why they work is only in the very starting stages.
The purpose of this talk is to address an important numerical stability issue for neural networks, known as the exploding and vanishing gradient problem. I will explain what this problem is and how it turns precisely into a problem of studying products of many matrices in the regime where both the sizes and the number of matrices tend to infinity together. I will present some joint work with Mihai Nica on the behavior of matrices in this regime.

Friday, February 1, 201911AM, Warren Weaver Hall 512
Restricted critical exponents in highdimensional percolation
Jack Hanson, CUNY  City CollegeSynopsis:
Critical percolation is fairly wellunderstood on \(Z^d\) for \(d > 11\). Exact values of many critical exponents are rigorously known: for instance, the “onearm” probability that the origin is connected by an open path to distance \(r\) scales as \(r^{2}\). However, most existing methods rely heavily on the symmetries of the lattice, so they do not extend to fractional spaces. We will discuss progress on these questions in the highdimensional upper halfspace (and within cubes), including the result that the halfspace onearm probability scales as \(r^{3}\).