Diagonalizing Transition Matrices of Card Shuffles
2023-04-19 15:30 - 16:30 (America/New_York) | Science Center 232, Harvard Science Center, 1 Oxford Street, Cambridge MA 02138
Probability Seminar
Speaker: Evita Nestoridi (Stonybrook)

Title: Diagonalizing Transition Matrices of Card Shuffles
Abstract: In their seminal work, Diaconis and Shahshahani used representation theory of the symmetric group to diagonalize the transition matrix of random transpositions. More recently, Dieker and Saliola introduced another technique to diagonalize the random-to-random card shuffle. In this talk we will discuss connections between these techniques as well as application to card shuffling.

Boundary current fluctuations for the half space ASEP
2023-04-26 15:30 - 16:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Probability Seminar
Speaker: Jimmy He (MIT)
Title: Boundary current fluctuations for the half space ASEP
Abstract: The half space asymmetric simple exclusion process (ASEP) is an interacting particle system on the half line, with particles allowed to enter/exit at the boundary. I will discuss recent work on understanding fluctuations for the number of particles in the half space ASEP started with no particles, which exhibits the Baik-Rains phase transition between GSE, GOE, and Gaussian fluctuations as the boundary rates vary. As part of the proof, we find new distributional identities relating this system to two other models, the half space Hall-Littlewood process, and the free boundary Schur process, which allows exact formulas to be computed.

Random Neural Networks
2023-05-03 15:30 - 16:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Probability Seminar
Speaker: Boris Hanin (Princeton)

Title: Random Neural Networks
Abstract: Fully connected neural networks are described two by structural parameters: a depth L and a width N. In this talk, I will present results and open questions about the asymptotic analysis of such networks with random weights and biases in the regime where N (and potentially L) are large. The first set of results are for deep linear networks, which are simply products of L random matrices of size N x N. I’ll explain how the setting where the ratio L / N is fixed with both N and L large reveals a number of phenomena not present when only one of them is large. I will then state several results about non-linear networks in which this depth-to-width ratio L / N again plays a crucial role and gives an effective notion of depth for a random neural network.

How do the eigenvalues of a large non-Hermitian random matrix behave?
2023-05-11 13:30 - 14:30 | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Probability Seminar
Speaker: Giorgio Cipolloni (Princeton)
Title: How do the eigenvalues of a large non-Hermitian random matrix behave?
Abstract: We prove that the fluctuations of the eigenvalues converge to the Gaussian Free Field (GFF) on the unit disk. These fluctuations appear on a non-natural scale, due to strong correlations between the eigenvalues. Then, motivated by the long time behaviour of the ODE dot{u}=Xu, we give a precise estimate on the eigenvalue with the largest real part and on the spectral radius of X.
Location: Science Center Room 232

Light cones for open quantum systems
2023-09-06 16:00 - 17:00 | Science Center 232, Harvard Science Center, 1 Oxford Street, Cambridge MA 02138
Probability Seminar
Speaker: Marius Lemm, University of Tuebingen
Title: Light cones for open quantum systems

Abstract: We consider non-relativistic Markovian open quantum dynamics in continuous space. We show that, up to small probability tails, the supports of quantum states propagate with finite speed in any finite-energy subspace. More precisely, if the initial quantum state is localized in space, then any finite-energy part of the solution of the von Neumann-Lindblad equation is approximately localized inside an energy-dependent light cone. We also obtain an explicit upper bound on the slope of this light cone (i.e., on the maximal speed). The general method can be used to derive propagation bounds for a variety of other quantum systems including Lieb-Robinson bounds for lattice bosons. Based on joint works with S. Breteaux, J. Faupin, D.H. Ou Yang, I.M. Sigal, and J. Zhang.

Correlation decay for finite lattice gauge theories
2023-09-07 13:30 - 14:30 | Science Center 232, Harvard Science Center, 1 Oxford Street, Cambridge MA 02138
Probability Seminar
Speaker: Arka Adhikari (Stanford)
Title: Correlation decay for finite lattice gauge theories
Abstract: In the setting of lattice gauge theories with finite (possibly non-Abelian) gauge groups at weak coupling, we prove exponential decay of correlations for a wide class of gauge invariant functions, which in particular includes arbitrary functions of Wilson loop observables. Based on joint work with Sky Cao.

Large deviations for the 3D dimer model
2023-09-27 15:30 - 16:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Probability Seminar
Speaker: Catherine Wolfram (MIT)
Title: Large deviations for the 3D dimer model
Abstract: A dimer tiling of Z^d is a collection of edges such that every vertex is covered exactly once. In 2000, Cohn, Kenyon, and Propp showed that 2D dimer tilings satisfy a large deviations principle. In joint work with Nishant Chandgotia and Scott Sheffield, we prove an analogous large deviations principle for dimers in 3D. A lot of the results for dimers in two dimensions use tools and exact formulas (e.g. the height function representation of a tiling or the Kasteleyn determinant formula) that are specific to dimension 2. In this talk, I will try to give some intuition for why three dimensions is different from two, explain how to formulate the large deviations principle in 3D, show simulations, and explain some of the ways that we use a smaller set of tools (e.g. Hall’s matching theorem or a double dimer swapping operation) in our arguments.

Geometry of the doubly periodic Aztec dimer model
2023-10-18 15:30 - 16:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Probability Seminar
Speaker: Tomas Berggren (MIT)
Title: Geometry of the doubly periodic Aztec dimer model
Abstract: Random dimer models (or equivalently tiling models) have been a subject of extensive research in mathematics and physics for several decades. In this talk, we will discuss the doubly periodic Aztec diamond dimer model of growing size, with arbitrary periodicity and only mild conditions on the edge weights. In this limit, we see three types of macroscopic regions — known as rough, smooth and frozen regions. We will discuss how the geometry of the arctic curves, the boundary of these regions, can be described in terms of an associated amoeba and an action function. In particular, we determine the number of frozen and smooth regions and the number of cusps on the arctic curves. We will also discuss the convergence of local fluctuations to the appropriate translation-invariant Gibbs measures. Joint work with Alexei Borodin.

Tail estimates for stationary KPZ models
2023-10-25 15:30 - 16:30 | Virtual
Probability Seminar
Speaker: Benjamin Landon (University of Toronto)
Title: Tail estimates for stationary KPZ models
Abstract: The limiting distributions of the KPZ universality class exhibit tail exponents of 3/2 and 3. In this talk we will review recent work studying the upper tail exponent 3/2 in the moderate deviations regime of several KPZ models at finite size, including the stochastic six vertex model, the ASEP and a class of non-integrable interacting diffusions.
Joint work with Christian Noack and Phil Sosoe.

Universality of max-margin classifiers
2023-11-01 15:30 - 16:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Probability Seminar
Speaker: Youngtak Sohn (MIT)
Title: Universality of max-margin classifiers
Abstract: Many modern learning methods, such as deep neural networks, are so complex that they perfectly fit the training data. Despite this, they generalize well to the unseen data. Motivated by this phenomenon, we consider high-dimensional binary classification with linearly separable data. First, we consider Gaussian covariates and characterize linear classification problems for which the minimum norm interpolating prediction rule, namely the max-margin classification, has near-optimal prediction accuracy. Then, we discuss universality of max-margin classification. In particular, we characterize the prediction accuracy of the non-linear random features model, a two-layer neural network with random first layer weights. The spectrum of the kernel random matrices plays a crucial role in the analysis. Finally, we consider the wide-network limit, where the number of neurons tends to infinity, and show how non-linear max-margin classification with random features collapse to a linear classifier with a soft-margin objective.
Joint work with Andrea Montanari, Feng Ruan, Jun Yan, and Basil Saeed.

Solving spin systems, the Babylonian way
2023-11-02 15:30 - 16:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Probability Seminar
Speaker: Nicola Kistler (Johann Wolfgang Goethe-Universität Frankfurt am Main)
Title: Solving spin systems, the Babylonian way
Abstract: The replica method, together with Parisi’s symmetry breaking mechanism, is an extremely powerful tool to compute the limiting free energy of virtually any mean field disordered system. Unfortunately, the tool is dramatically flawed from a mathematical point of view. I will discuss a truly elementary procedure which allows to rigorously implement two (out of three) steps of the replica method, and conclude with some remarks on the relation between this new point of view and old work by Mezard and Virasoro on the microstructure of ultrametricity, the latter being the fundamental yet unjustified Ansatz in the celebrated Parisi solution. We are still far from a clear understanding of the issues, but quite astonishingly, evidence is mounting that Parisi’s ultrametricity assumption, the onset of scales and the universal hierarchical self-organisation of random systems in the infinite volume limit, is intimately linked to hidden geometrical properties of large random matrices which satisfy rules reminiscent of the popular SUDOKU game.

Fitting ellipsoids to random points
2023-11-08 15:30 - 16:30 | Virtual
Probability Seminar
Speaker: Antoine Maillard (ETH Zürich)
Title: Fitting ellipsoids to random points
Abstract: We consider the problem of exactly fitting an ellipsoid (centered at 0) to n standard Gaussian random vectors in dimension d, for very large n and d. This problem has connections to questions in statistical learning and theoretical computer science, and is conjectured to undergo a sharp transition: with high probability, it has a solution if n < d^2/4, while it is not satisfiable if n > d^2/4. In this talk we will discuss the origin of this conjecture, and highlight some recent progress, in three different directions:

A proof that the problem is feasible for n < d^2 / C, for some (large) constant C, significantly improving over previously-known bounds.
A non-rigorous characterization of the conjecture, as well as significant generalizations, using analytical methods of statistical physics.
A rigorous proof of a satisfiability transition exactly at n = d^2 / 4 in a slightly relaxed version of the problem, the first rigorous result characterizing the expected phase transition in ellipsoid fitting. The proof is inspired by the non-rigorous characterization discussed above.

This talk is based on the three manuscripts: arXiv:2307.01181, arXiv:2310.01169, arXiv:2310.05787, which are joint works with A. Bandeira, Tim Kunisky, Shahar Mendelson and Elliot Paquette.

Thresholds
2023-11-15 15:30 - 16:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Probability Seminar
Speaker: Jinyoung Park (NYU)
Title: Thresholds
Abstract: For a finite set X, a family F of subsets of X is said to be increasing if any set A that contains B in F is also in F. The p-biased product measure of F increases as p increases from 0 to 1, and often exhibits a drastic change around a specific value, which is called a “threshold.” Thresholds of increasing families have been of great historical interest and a central focus of the study of random discrete structures (e.g. random graphs and hypergraphs), with estimation of thresholds for specific properties the subject of some of the most challenging work in the area. In 2006, Jeff Kahn and Gil Kalai conjectured that a natural (and often easy to calculate) lower bound q(F) (which we refer to as the “expectation-threshold”) for the threshold is in fact never far from its actual value. A positive answer to this conjecture enables one to narrow down the location of thresholds for any increasing properties in a tiny window. In particular, this easily implies several previously very difficult results in probabilistic combinatorics such as thresholds for perfect hypergraph matchings (Johansson–Kahn–Vu) and bounded-degree spanning trees (Montgomery). I will present recent progress on this topic. Based on joint work with Keith Frankston, Jeff Kahn, Bhargav Narayanan, and Huy Tuan Pham.

A random matrix model towards the quantum chaos transition conjecture
2023-11-28 12:00 - 13:00 | MA
Probability Seminar
Speaker: Jun Yin (UCLA)
Title: A random matrix model towards the quantum chaos transition conjecture
Abstract: The Quantum Chaos Conjecture has long fascinated researchers, postulating a critical spectrum phase transition that separates integrable systems from chaotic systems in quantum mechanics. In the realm of integrable systems, eigenvectors remain localized, and local eigenvalue statistics follow the Poisson distribution. Conversely, chaotic systems exhibit delocalized eigenvectors, with local eigenvalue statistics mirroring the Sine kernel distribution, akin to the standard random matrix ensembles GOE/GUE.
This talk delves into the heart of the Quantum Chaos Conjecture, presenting a novel approach through the lens of random matrix models. By utilizing these models, we aim to provide a clear and intuitive demonstration of the same phenomenon, shedding light on the intricacies of this long-standing conjecture.

A Gaussian convexity for logarithmic moment generating function
2023-11-30 16:00 - 17:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Probability Seminar
Speaker: Wei-Kuo Chen (University of Minnesota)
Title: A Gaussian convexity for logarithmic moment generating function
Abstract: Convex functions of Gaussian vectors are prominent objectives in many fields of mathematical studies. In this talk, I will establish a new convexity for the logarithmic moment generating function for this object and draw two consequences. The first leads to the Paouris-Valettas small deviation inequality that arises from the study of convex geometry. The second provides a quantitative bound for the Dotsenko-Franz-Mezard conjecture in the Sherrington-Kirkpatrick mean-field spin glass model, which states that the logarithmic anneal partition function of negative replica is asymptotically equal to the free energy.
