Schedule

• September 14

  Oleg Zaboronski (Warwick)

  What is the probability that a large random matrix has no real eigenvalues?

  Abstract: In this talk I will discuss the calculation of the probability p(n) that a large real matrix with independent normal entries has no real eigenvalues. I will show how to guess the answer using the link between the law of real eigenvalues and a certain interacting particle system. I will then outline the steps of the proof of the result based on the determinantal expression for p(n).

• September 21

  Roland Bauerschmidt (Harvard)

  Local eigenvalue statistics for random regular graphs

  Abstract: The universality conjecture for random matrices states that the local spectral statistics are universal for broad classes of random matrices. It has been proved for random matrix ensembles that are invariant under large continuous symmetry groups (invariant ensembles) and for random matrices with i.i.d. entries (Wigner matrices). The adjacency matrix of a random regular graph is a random matrix with hard constraints and in neither of these classes. Nonetheless it has been conjectured and numerically confirmed that the eigenvalues of random regular graphs obey local random matrix statistics. We prove that the local semicircle law holds at the optimal scale and moreover that the local eigenvalue distribution in the bulk of the spectrum is indeed given by that of the GOE, both under suitable assumptions on the degrees of the graphs. This is joint work with J. Huang, A. Knowles, and H.-T. Yau.

• September 28

  Alexey Bufetov (MIT)

  Asymptotics of particle systems governed by Schur functions

  Abstract: We study the global limit behavior of probabilistic particle systems whose combinatorics is related to Schur functions. Our main examples are the decompositions of the tensor products of irreducible representations of classical Lie groups, the random lozenge tilings, the random domino tilings, and the decompositions of extreme characters of the infinite-dimensional unitary group. We prove the Law of Large Numbers and the Central Limit Theorem for all these systems. The talk is based on a joint work with V. Gorin.

• October 2

  Friday

  Charles River Lectures

• October 5

  Mariya Shcherbina (Kharkov)

  Fluctuations of eigenvalues of random matrices with independent or weakly dependent entries

  Abstract: We will discuss the method, which allows to prove automatically CLT for linear eigenvalue statistics (LES) of smooth test functions for almost all classical models of random matrix theory: deformed Wigner and sample covariance matrices, sparse matrices, diluted random matrices, matrices with heavy tails, etc. In particular, the application of the method allows us to prove CLT for LES of random band n-dimensional matrices, whose bandwidth b is assumed to grow with n in such a way that b/n \to 0, without any additional restrictions. Thus we remove the main technical restriction n >> b >> n^{1/2} of all the papers, where LES of band matrices was studied before.

• October 13

  Tuesday (due to Columbus Day) E17-136 (not usual room)

  Philippe Rigollet (MIT)

  Aggregation in high-dimensional statistics

  Abstract: Originally introduced as a tool for adaptation to smoothness in nonparametric statistics, aggregation has become a powerful tool in diverse areas of statistics and machine learning, primarily in high-dimensions, where it can be used to adapt to the underlying sparsity pattern of the problem. In this talk, I will give an overview of recent advances in the context of sparse linear regression where aggregation has been used successfully to derived the sharpest available oracle inequalities.

• October 19

  Nicholas Cook (UCLA)

  Random regular digraphs: singularity and spectrum

  Abstract: We consider two random matrix ensembles associated to large random regular digraphs: (1) the 0/1 adjacency matrix, and (2) the adjacency matrix with iid bounded edge weights. Motivated by universality conjectures, we show that the spectral distribution for the latter ensemble is asymptotically described by the circular law, assuming the graph has degree linear in the number of vertices. Towards establishing the same result for the unweighted adjacency matrix, we prove that it is invertible with high probability, even for sparse digraphs with degree growing only poly-logarithmically.

• October 26

  Peter Winkler (Dartmouth)

  Permutons

  Abstract: Permutons are doubly stochastic measures, i.e., probability measures on the unit square with uniform marginals. They function as limit structures for finite permutations, much as graphons do for graphs. As in the graph case, a variational principle holds, making it possible to count and describe certain classes of permutations. Unlike the graph case, most class-describing permutons are smooth and thus amenable to standard analytic techniques.

  I'll show how both rigorous and empirical methods can help understand permutations with given pattern densities---an objective whose surface has so far only been scratched.

  Joint work with Rick Kenyon (Brown), Dan Kral (Warwick) and Charles Radin (Texas).

• November 2

  Xin Sun (MIT)

  Bi-Laplacian Gaussian field and uniform spanning forests

  Abstract: In this talk, I will first review Gaussian free field in $\R^d$ and its generalization called fractional Gaussian field, which includes log correlated field and bi-Laplacian Gaussian field as examples. Fractional Gaussian field arises naturally as scaling limits of spin models, i.e. Ising model and phi^4 model, at or above their critical dimension for the mean field behavior. We describe a simple spin model from uniform spanning forests in $\Z^d$ whose critical dimension is 4 and prove that the scaling limit is the bi-Laplacian Gaussain field for $d\ge 4$. At dimension 4, there is a $log n$ correction for the spin-spin correlation and the bi-Laplacian Gaussian field is a log correlated field. Based on a joint work with Greg Lawler and Wei Wu and a survey with Asad Lodhia, Sam Watson and Scott Sheffield.

• November 9

  Shirshendu Ganguly (UW)

  Competitive erosion is conformally invariant

  Abstract: We study a graph-theoretic model of interface dynamics called competitive erosion. Each vertex of the graph is occupied by a particle, which can be either red or blue. New red and blue particles are emitted alternately from their respective sources and perform random walk. On encountering a particle of the opposite color they remove it and occupy its position. This is a finite, competitive version of the celebrated Internal DLA growth model first analyzed by Lawler, Bramson and Griffeath in 1992.

  We establish conformal invariance of competitive erosion on discretizations of smooth, simply connected planar domains. This is done by showing that at stationarity, with high probability the red and the blue regions are separated by an orthogonal circular arc on the disc and more generally by a hyperbolic geodesic. The proof relies on convergence of solutions of the discrete Poisson problem with Neumann boundary conditions to their continuous counterparts and robust IDLA estimates.

  (Joint work with Yuval Peres, available at arXiv:1503.06989).

• November 16

  Hoi Nguyen (Ohio)

  Anti concentration of random walks and eigenvalue repulsion of random matrices

  Abstract: I will address recent characterization results on random walks (in both abelian and non-abelian groups) which stick to a small region unusually long. As an application, we show a Wegner-type estimate for the number of eigenvalues inside an extremely small interval for Wigner matrices of discrete type.

• November 23

  Guillaume Barraquand (Columbia)

  Random walks in Beta random environment

  Abstract: We consider a model of random walks in space-time random environment, with Beta-distributed transition probabilities. This model is exactly solvable, in the sense that the law of the (finite time) position of the walker can be completely characterized by Fredholm determinantal formulas. This enables to prove a limit theorem towards the Tracy-Widom distribution for the second order corrections to the large deviation principle satisfied by the walker, thus extending the scope of KPZ universality to RWRE. We will also discuss a few similar results about degenerations of the model: a first passage percolation model which is the "zero-temperature" limit, and a certain diffusive limit which leads to well-studied stochastic flows. (Work in collaboration with Ivan Corwin).

• November 30

  Damien Gayet (Institut Fourier)

  Universal components of random nodal hypersurfaces

  Abstract: Let S be a compact hypersurface of R^n, and M be a compact Riemannian manifold of dimension n. I will explain that if we take at random a linear combination of eigenfunctions of the Laplacian on M with eigenvalues less than L, then, in average, at least c_S Vol(M) L^{n/2} diffeomorphic copies of S appear in the vanishing locus of this sum, as L grows to infinity, where c_S is a positive constant depending only on S. This is a joint work with Jean-Yves Welschinger.

• December 7

  Eyal Lubetzky (Courant)

  Effect of initial conditions on mixing for spin systems

  Abstract: Recently, the "information percolation" framework was introduced as a way to obtain sharp estimates on mixing for spin systems at high temperatures, and in particular, to establish cutoff for the Ising model in three dimensions up to criticality from a worst starting state. I will describe how this method can be used to understand the effect of different initial states on the mixing time, both random (''warm start'') and deterministic. Joint work with Allan Sly.