Random Matrix & Probability Theory Seminar

Beginning immediately, until at least December 31, all seminars will take place virtually, through Zoom.

In the 2020-2021 AY, the Random Matrix and Probability Theory Seminar will take place on select Wednesdays from 2:00 – 3:00pm virtually. This seminar is organized by Christian Brennecke (brennecke@math.harvard.edu ).

To learn how to attend this seminar, please fill out this form.

The schedule below will be updated as the details are confirmed.

Spring 2021:

3/31/2021Philippe Sosoe, Cornell UniversityTBA
4/7/2021Yue M. Lu, HarvardTBA
4/14/2021Jean-Christophe Mourrat, Courant Institute, NYUTBA

Fall 2020:

9/9/2020Yukun He (Zurich)Title: Single eigenvalue fluctuations of sparse Erdős–Rényi graphs

Abstract: I discuss the fluctuations of individual eigenvalues of the adjacency matrix of the Erdös-Rényi graph $G(N,p)$. I show that if $N^{-1}\ll p \ll N^{-2/3}, then all nontrivial eigenvalues away from 0 have asymptotically Gaussian fluctuations. These fluctuations are governed by a single random variable, which has the interpretation of the total degree of the graph. The main technical tool of the proof is a rigidity bound of accuracy $N^{-1-\varepsilon}p^{-1/2}$ for the extreme eigenvalues, which avoids the $(Np)^{-1}$-expansions from previous works. Joint work with Antti Knowles.
10/14/2020David Belius (University of Basel)Title: The TAP approach to mean field spin glasses

Abstract: The Thouless-Anderson-Palmer (TAP) approach to the Sherrington-Kirkpatrick mean field spin glass model was proposed in one of the earliest papers on this model. Since then it has complemented subsequently elaborated methods  in theoretical physics and mathematics, such as the replica method, which are largely orthogonal to the TAP approach. The TAP approach has the advantage of being interpretable as a variational principle optimizing an energy/entropy trade-off, as commonly encountered in statistical physics and large deviations theory, and potentially allowing for a more direct characterization of the Gibbs measure and its “pure states”. In this talk I will recall the TAP approach, and present preliminary steps towards a solution of mean field spin glass models entirely within a TAP framework.
10/28/2020Giuseppe Genovese (University of Basel)TitleNon-convex variational principles for the RS free energy of restricted Boltzmann machines

Abstract: From the viewpoint of spin glass theory, restricted Boltzmann machines represent a veritable challenge, as to the lack of convexity prevents us to use Guerra’s bounds. Therefore even the replica symmetric approximation for the free energy presents some challenges. I will present old and new results around the topic along with some open problems. 
11/4/2020Benjamin Landon (MIT)TitleFluctuations of the spherical Sherrington-Kirkpatrick model

Abstract:  The SSK model was introduced by Kosterlitz, Thouless and Jones as a simplification of the usual SK model with Ising spins. Fluctuations of its observables may be related to quantities from random matrix theory using integral representations.  In this informal talk we discuss some results on fluctuations of this model at critical temperature and with a magnetic field
3:00 – 4:00pm
Lucas Benigni (University of Chicago)TitleOptimal delocalization for generalized Wigner matrices

Abstract: We consider eigenvector statistics of large symmetric random matrices. When the matrix entries are sampled from independent Gaussian random variables, eigenvectors are uniformly distributed on the sphere and numerous properties can be computed exactly. In particular, we can bound their extremal coordinates with high probability. There has been an extensive amount of work on generalizing such a result, known as delocalization, to more general entry distributions. After giving a brief overview of the previous results going in this direction, we present an optimal delocalization result for matrices with sub-exponential entries for all eigenvectors. The proof is based on the dynamical method introduced by Erdos-Yau, an analysis of high moments of eigenvectors as well as new level repulsion estimates which will be presented during the talk. This is based on a joint work with P. Lopatto.
11/18/2020Simone Warzel (Technical University of Munich)TitleHierarchical quantum spin glasses

Abstract: Hierarchical spin glasses such as the generalised random energy model are known to faithfully model typical energy landscapes in the classical theory of mean-field spin glasses. Their built-in hierarchical structure is known to emerge spontaneously in the spin-glass phase of, e.g., the Sherrington-Kirkpatrick model. In this talk, I will review recent results on the effects of a transversal magnetic field on such hierarchical quantum spin glasses. 
In particular, I will present a formula of Parisi-type for their free energy which allows to make predictions about the phase diagram. 
12/2/2020Sabine Jansen (LMU Munich)Title: Thermodynamics of a hierarchical mixture of cubes

Abstract: The talk discusses a toy model for phase transitions in mixtures of incompressible droplets. The model consists of non-overlapping hypercubes of side-lengths 2^j, j\in \N_0. Cubes belong to an admissible set such that if two cubes overlap, then one cube is contained in the other, a picture reminiscent of Mandelbrot’s fractal percolation model. I will present exact formulas for the entropy and pressure, discuss phase transitions from a fluid phase with small cubes towards a condensed phase with a macroscopic cube, and briefly sketch some broader questions on renormalization and cluster expansions that motivate the model. Based on arXiv:1909.09546 (J. Stat. Phys. 179 (2020), 309-340).

For information on previous seminars, click here

The schedule will be updated as details are confirmed.

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