Winter 2025
April 2, 2025, 4:00–5:00pm, EH3096
Speaker: Marcus Michelen (UIC)
Title: New lower bounds for sphere packings and independent sets via randomness
Abstract: We construct new lower bounds for sphere packings in high dimensions and for independent sets in graphs with not-too-large co-degrees. For dimension d, this achieves a sphere packing of density (1 + o(1)) d log d / 2^(d+1). In general dimension this provides the first asymptotically growing improvement for sphere packing lower bounds since Rogers' bound of c*d/2^d in 1947. The proof amounts to a random (very dense) discretization together with a new theorem on constructing independent sets on graphs with not-too-large co-degree. Both steps will be discussed and no knowledge of sphere packings will be assumed or required. Central to the analysis is the study of a random process on a graph. This is based on joint work with Marcelo Campos, Matthew Jenssen and Julian Sahasrabudhe.
March 26, 2025, 4:00–5:00pm, EH3096
Speaker: Reza Gheissari (Northwestern)
Title: Local geometry and spectral transitions in high-dimensional classification
Abstract: We study the spectral theory of a family of random matrices that are variants on empirical covariance matrices, with entries that can have correlations through projections of the data onto some O(1)-many directions. These matrices arise naturally when looking at the empirical Hessians of many high-dimensional statistical tasks at different points in parameter space, to probe the local geometry of their loss landscapes. We prove limits for the bulk distribution and any outlier eigenvalues, in a way that only depends on the point in parameter space through finitely many “summary statistics” of the parameter. This allows us to probe the evolution of the Hessian as one moves through parameter space via some training dynamics like stochastic gradient descent, establishing interesting phenomena like splitting and emergence of outliers over the course of training in basic classification tasks. Based on joint work with G. Ben Arous, J. Huang, and A. Jagannath.
March 12, 2025, 4:00–5:00pm, EH3096
Speaker: Dmitry Krachun (Princeton)
Title: A glimpse of universality in critical planar lattice models
Abstract: Many models of statistical mechanics are defined on a lattice, yet they describe behaviour of objects in our seemingly isotropic world. It is then natural to ask why, in the small mesh size limit, the directions of the lattice disappear. Physicists' answer to this question is partially given by the Universality hypothesis, which roughly speaking states that critical properties of a physical system do not depend on the lattice or fine properties of short-range interactions but only depend on the spatial dimension and the symmetry of the possible spins. Justifying the reasoning behind the universality hypothesis mathematically seems virtually impossible and so other ideas are needed for a rigorous derivation of universality even in the simplest of setups.
In this talk I will explain some ideas behind the recent result which proves rotational invariance of the FK-percolation model. In doing so, we will see how rotational invariance is related to universality among a certain one-dimensional family of planar lattices and how the latter can be proved using exact integrability of the six-vertex model using Bethe ansatz.
Based on joint works with Hugo Duminil-Copin, Karol Kozlowski, Ioan Manolescu, Mendes Oulamara, and Tatiana Tikhonovskaia.
February 26, 2025, 4:00–5:00pm, EH3096
Speaker: Konstantinos Kavvadias (MIT)
Title: Two-sided heat kernel bounds for \sqrt{8/3}-Liouville Brownian Motion
Abstract: Please click this link.
February 19, 2025, 4:00–5:00pm, EH3096
Speaker: Catherine Wolfram (Yale)
Title: The multinomial dimer model
Abstract: An N dimer cover of a graph is a collection of edges such that every vertex is contained in exactly N edges of the collection. The multinomial dimer model, introduced by Kenyon and Pohoata, studies a natural but non-uniform measure on N dimer covers. While the standard dimer model (N=1) is exactly solvable only in two dimensions (i.e. on planar graphs), in the N to infinity limit, the multinomial dimer model turns out to be exactly solvable even in three (or higher) dimensions. In this talk, I will define the model and discuss new results in two and three dimensions, including: explicit formulas for the free energy, a large deviation principle, Euler-Lagrange equations, and descriptions of limit shapes and some of their properties. This is joint work with Richard Kenyon.
February 5, 2025, 4:00–5:00pm, EH3096
Speaker: Izak Oltman (Northwestern)
Title: Randomly Perturbed Berezin–Toeplitz Operators
Abstract: In this talk, I will prove a Weyl law for the spectrum of randomly perturbed Berezin–Toeplitz operators, generalizing a result of Martin Vogel from 2020 about quantizations of torii. I will first survey some results about randomly perturbed non–self-adjoint operators, then explain the construction of Berezin–Toeplitz operators (which are quantizations of smooth functions on compact Kähler manifolds), then discuss the main idea of the proof, which requires constructing an exotic calculus of Berezin–Toeplitz operators.
January 29, 2025, 4:00–5:00pm, EH3096
Speaker: Jinwoo Sung (UChicago)
Title: A quasi-invariant group action on SLE loops
Abstract: Conformal welding is an operation that encodes a large class of Jordan curves on the Riemann sphere, including the loop version of Schramm–Loewner evolution (SLE), in terms of circle homeomorphisms. In this talk, I will discuss a Cameron–Martin type quasi-invariance result for the SLE loop measure under the right group action by Weil–Petersson circle homeomorphisms on the welding homeomorphism. While this result was hinted at by Carfagnini and Wang's identification of the Onsager–Machlup action functional of the SLE loop measure with the Kähler potential of the unique right-invariant Kähler metric on the group of Weil–Petersson circle homeomorphisms (Loewner energy), the group structure of SLE welding has been little understood previously. Our proof is based on the characterization of the composition operator associated with Weil–Petersson circle homeomorphisms using Hilbert–Schmidt operators and the description of the SLE loop measure in terms of the welding of two independent quantum disks by Ang, Holden, and Sun. This is joint work with Shuo Fan (Tsinghua University and IHES).
January 22, 2025, 4:00–5:00pm, EH3096
Speaker: Eric Roon (Michigan State University)
Title: Toward a structure theory of disordered matrix product states
Abstract: In 1992, Fannes, Nachtergaele, and Werner classified translation invariant states on quantum spin chains and discovered that they admit a matrix product structure. Such matrix product states are simultaneously good approximations for general states, and natural candidates for ground states of specific local Hamiltonians. Following the observation by Vidal (2004) that matrix product states are ‘‘efficient,’’ the theory took root and is now an indispensable tool in many-body physics and quantum simulation. Recent work in this direction by Movassagh–Schenker (2022) and Nelson–R. (2024) adapted this structure to states generated by disordered matrix products. All such disordered matrix product states are translation co-variant. However both works above only had a ‘‘one-way’’ construction, not a classification. In this talk, I’ll report on some work in progress with Jeffrey Schenker where we successfully classify the translation co-variant states when the underlying probability space is a compact Hausdorff space.
January 15, 2025, 3:00–4:00pm, EH4448
Speaker: Yifan Chen (New York University)
Title: Convergence of Unadjusted Langevin in High Dimensions: Delocalization of Bias
Abstract: The unadjusted Langevin algorithm is commonly used to sample probability distributions in extremely high-dimensional settings. However, existing analyses of the algorithm for strongly log-concave distributions suggest that, as the dimension $d$ of the problem increases, the number of iterations required to ensure convergence within a desired error in the $W_2$ metric scales in proportion to $d$ or $\sqrt{d}$. In this work, we argue that, despite this poor scaling of the $W_2$ error for the full set of variables, the behavior for a \emph{small number} of variables can be significantly better: a number of iterations proportional to $K$, up to logarithmic terms in $d$, often suffices for the algorithm to converge to within a desired $W_2$ error for all $K$-marginals. We refer to this effect as \textit{delocalization of bias}. We show that the delocalization effect does not hold universally and prove its validity for Gaussian distributions and strongly log-concave distributions with certain sparse interactions. Our analysis relies on a novel $W_{2,\ell^\infty}$ metric to measure convergence. A key technical challenge we address is the lack of a one-step contraction property in this metric. Finally, we use asymptotic arguments to explore potential generalizations of the delocalization effect beyond the Gaussian and sparse interactions setting.
Fall 2024
December 4, 2024 (Wednesday), 4:00–5:00pm, EH1096
Speaker: Theo McKenzie (Stanford)
Title: The spectral edge of constant degree Erdős-Rényi graphs
Abstract: Understanding the spectrum and eigenvectors of the adjacency matrix of random graphs is a fundamental problem with broad applications in computer science and statistical physics. A widely studied model is the Erdős-Rényi graph, where edges are included independently with a fixed probability. In this talk, we show that for Erdős-Rényi graphs with constant expected degree, the most positive and most negative eigenvalues are completely localized, in that eigenvector entries decay away from individual, high degree vertices, and eigenvalues are almost completely determined by the geometry surrounding these high degree vertices. This resolves a question of Alice Guionnet.
This talk is based on joint work with Ella Hiesmayr.
November 20 , 2024 (Wednesday), 3:00–4:00pm Eastern, EH 3096
Speaker: Jorge Garza Vargas (Caltech)
Title: A new approach to strong convergence of random matrices
Abstract: Friedman's celebrated 2004 result states that, as the number of vertices goes to infinity, random d-regular graphs are (with high probability) nearly optimal expanders, meaning that the top non-trivial eigenvalue of their (random) adjacency matrix converges in probability to 2 sqrt(d-1). Since expanders are of great interest in mathematics and computer science, Friedman's paper (which is ~100 pages long) has attracted a lot of attention in the last two decades and more efficient proofs of his result (which yield vast generalizations) have been found. However, all the approaches to Friedman's theorem and its extensions relied on very delicate and sophisticated combinatorial considerations, making it hard to apply those ideas to other settings of interest.
In this talk I will discuss a fundamentally new (analytic) approach to Friedman's theorem which yields an elementary proof that can be written in just a few pages. Our approach also allows us to establish strong convergence (i.e. sharp norm estimates) for much more general models of tuples of random matrices (random regular graphs corresponding to the particular case of adding independent random permutations). These results can be used to show that certain infinite objects admit very strong finite dimensional approximations, which has important implications in operator algebras, spectral geometry, and differential geometry.
This is joint work with Chi-Fang Chen, Joel Tropp, and Ramon van Handel.
November 7, 2024, 4:00–5:00pm Eastern, EH4088
Speaker: Tomas Berggren (KTH Stockholm)
Title: Crystallization of the Aztec diamond
Abstract: Random planar dimer models (or equivalently tiling models) have been a subject of extensive research in mathematics and physics for several decades. Recent progress has been made in understanding the behavior of planar dimer models with periodic edge weights, particularly models on the subgraph of the square lattice known as the Aztec diamond graph. In this talk, I will discuss the effect of a temperature parameter in the doubly periodic Aztec diamond dimer model in the zero temperature limit. In this limit, the Aztec diamond undergoes crystallization: The limit shape converges to a piecewise linear function we call the tropical limit shape, and the local fluctuations are governed by the Gibbs measures with the slope dictated by the tropical limit shape for low enough temperature.
The tropical limit shape and the tropical arctic curve (consisting of ridges of the crystal) are described in terms of a tropical curve and a tropical action function on that curve, which are the tropical analogs of the spectral curve and the action function that describe the finite-temperature models. The tropical curve is explicit in terms of the edge weights, and the tropical action function is a solution to Kirchhoff's problem on the tropical curve.
Based on joint work with Alexei Borodin.
October 31, 2024, 4:00–5:00pm Eastern, EH4088
Speaker: Guilherme Silva (Universidade de São Paulo)
Title: Tail estimates of the stochastic six-vertex model: A tale from A^2
Abstract: Long story short: our goal is to describe tail decay and moderate deviation estimates for the distribution of the height function of the stochastic six-vertex model, with narrow wedge initial condition.
Now the long version. This whole tale of the tail started in Ann Arbor, many many years ago. Explaining the embarrassment of our failure to find such estimates for so many years is the main reason for our talk. As we had to learn for our task, such tail estimates are intimately connected with conditional thinning ensembles from random matrix theory, and nonlocal versions of the so-called Painlevé equations. The unraveling of such connections was a parallel task that we had to overcome, together with many other colleagues, and led to interesting phenomena that we hadn't anticipated with our original question. During our talk, we plan to go through such developments as well.
The talk is mostly based on joint works with Promit Ghosal (University of Chicago), but partially also in works in preparation with Tom Claeys (UC Louvain), Leslie Molag (Universidad Carlos III) and Lun Zhang (Fudan University).