Random Matrix & Probability Theory Seminar

Beginning immediately, until at least April 30, all seminars will take place virtually, through Zoom.

In the 2019-2020 AY, the Random Matrix and Probability Theory Seminar will take place on select Wednesdays from 2:00 – 3:00pm virtually. Please email the seminar organizers to obtain a link. The list below will reflect the dates of the scheduled talks. The list will be updated as the details are confirmed.

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

The schedule will be updated as details are confirmed.

Spring 2020:

2/26/2020Louigi Addario-Berry (McGill University) Title: Hipster random walks and their ilk 

Abstract: I will describe how certain recursive distributional equations can be solved by importing rigorous results on the convergence of approximation schemes for degenerate PDEs, from numerical analysis. This project is joint work with Luc Devroye, Hannah Cairns, Celine Kerriou, and Rivka Maclaine Mitchell. 
4/1/2020Ian Jauslin (Princeton)This meeting will be taking place virtually on Zoom.  

Title: A simplified approach to interacting Bose gases
Abstract: I will discuss some new results about an effective theory introduced by Lieb in 1963 to approximate the ground state energy of interacting Bosons at low density. In this regime, it agrees with the predictions of Bogolyubov. At high densities, Hartree theory provides a good approximation. In this talk, I will show that the ’63 effective theory is actually exact at both low and high densities, and numerically accurate to within a few percents in between, thus providing a new approach to the quantum many body problem that bridges the gap between low and high density.
4/22/2020Martin Gebert (UC Davis)This meeting will be taking place virtually on Zoom.  

Title: Lieb-Robinson bounds for a class of continuum many-body fermion systems

Abstract: We introduce a class of UV-regularized two-body interactions for
fermions in $\R^d$ and prove a Lieb-Robinson estimate for the dynamics
of this class of many-body systems. As a step towards this result, we
also prove a propagation bound of Lieb-Robinson type for continuum
one-particle Schr\“odinger operators. We apply the propagation bound to
prove the existence of a strongly continuous infinite-volume dynamics on
the CAR algebra.
4/29/2020Marcin Napiórkowski (University of Warsaw)This meeting will be taking place virtually on Zoom.  

Title: Free energy asymptotics of the quantum Heisenberg spin chain

Abstract: Spin wave theory suggests that low temperature properties of the Heisenberg model can be described in terms of noninteracting quasiparticles called magnons. In my talk I will review the basic concepts and predictions of spin wave approximation and report on recent rigorous results in that direction. Based on joint work with Robert Seiringer.
5/6/2020Antti Knowles (University of Geneva)Title: Field theory as a limit of interacting quantum Bose gases

Abstract: We prove that the grand canonical Gibbs state of an interacting quantum Bose gas converges to the Gibbs measure of a nonlinear Schrödinger equation in the mean-field limit, where the density of the gas becomes large and the interaction strength is proportional to the inverse density. Our results hold in dimensions d = 1,2,3. For d > 1 the Gibbs measure is supported on distributions of negative regularity and we have to renormalize the interaction. The proof is based on a functional integral representation of the grand canonical Gibbs state, in which convergence to the mean-field limit follows formally from an infinite-dimensional stationary phase argument for ill-defined non-Gaussian measures. We make this argument rigorous by introducing a white-noise-type auxiliary field, through which the functional integral is expressed in terms of propagators of heat equations driven by time-dependent periodic random potentials. Joint work with Jürg Fröhlich, Benjamin Schlein, and Vedran Sohinger.
5/13/2020Sven Bachmann (University of British Columbia)Title: Quantized quantum transport and Abelian anyons

Abstract: I’ll discuss recent developments in the study of quantized quantum transport, focussing on the quantum Hall effect. Beyond presenting an index taking rational values, and which is the Hall conductance in the adapted setting, I will explain how the index is intimately paired with the existence of quasi-particle excitations having non-trivial braiding properties.
5/20/2020Kristina Schubert (TU Dortmund) TitleFluctuation Results for General Ising Models — Block Spin Ising Models and Random Interactions

Abstract: Starting from the classical Curie-Weiss model in statistical mechanics, we will consider more general Ising models. On the one hand, we introduce a block structure, i.e. a model of spins in which the vertices are divided into a finite number of blocks and where pair interactions are given according to their blocks. The magnetization is then the vector of magnetizations within each block, and we are interested in its behaviour and in particular in its fluctuations. On the other hand, we consider Ising models on Erdős-Rényi random graphs. Here, I will also present results on the fluctuations of the magnetization.

Fall 2019:

Date Speaker Title/Abstract
9/11/2019 Subhabrata Sen Title: Sampling convergence for random graphs: graphexes and multigraphexes

Abstract: We will look at structural properties of large, sparse random graphs through the lens of sampling convergence (Borgs, Chayes, Cohn and Veitch ’17). Sam- pling convergence generalizes left convergence to sparse graphs, and describes the limit in terms of a graphex. We will introduce this framework and motivate the components of a graphex. Subsequently, we will discuss the graphex limit for several well-known sparse random (multi)graph models. This is based on joint work with Christian Borgs, Jennifer Chayes, and Souvik Dhara.

9/25/2019 Jeff Schenker (Michigan State)  Title: An ergodic theorem for homogeneously distributed quantum channels with applications to matrix product states  

Abstract: Quantum channels represent the most general physical evolution of a quantum system through unitary evolution and a measurement process. Mathematically, a quantum channel is a completely positive and trace preserving linear map on the space of $D\times D$ matrices. We consider ergodic sequences of channels, obtained by sampling channel valued maps along the trajectories of an ergodic dynamical system. The repeated composition of these maps along such a sequence could represent the result of repeated application of a given quantum channel subject to arbitrary correlated noise. It is physically natural to assume that such repeated compositions are eventually strictly positive, since this is true whenever any amount of decoherence is present in the quantum evolution. Under such an hypothesis, we obtain a general ergodic theorem showing that the composition of maps converges exponentially fast to a rank-one — “entanglement breaking’’ – channel. We apply this result to describe the thermodynamic limit of ergodic matrix product states and prove that correlations of observables in such states decay exponentially in the bulk. (Joint work with Ramis Movassagh)




Jian Ding (UPenn) Title: Distances associated with Liouville quantum gravity

Abstract: I will review some recent progresses on distances associated with Liouville quantum gravity, which is a random measure obtained from exponentiating a planar Gaussian free field.

The talk is based on works with Julien Dubédat, Alexander Dunlap, Hugo Falconet, Subhajit Goswami, Ewain Gwynne, Ofer Zeitouni and Fuxi Zhang in various combinations.

10/9/2019 Ruth Williams (UCSD) Title: Stability of a Fluid Model for Fair Bandwidth Sharing with General File Size Distributions

Abstract: Massoulie and Roberts introduced a stochastic model for a data communication network where file sizes are generally distributed and the network operates under a fair bandwidth sharing policy. It has been a standing problem to prove stability of this general model when the average load on the system is less than the network’s capacity. A crucial step in an approach to this problem is to prove stability of an associated measure-valued fluid model. We shall describe prior work on this question done under various strong assumptions and indicate how to prove stability of the fluid model under mild conditions.

This talk is based on joint work with Yingjia Fu.

10/11/2019  Cancelled  
10/16/2019  Wei-Kuo Chen (University of Minnesota)  Title: The generalized TAP free energy

Abstract: Spin glasses are disordered spin systems initially invented by theoretical physicists with the aim of understanding some strange magnetic properties of certain alloys. In particular, over the past decades, the study of the Sherrington-Kirkpatrick (SK) mean-field model via the replica method has received great attention. In this talk, I will discuss another approach to studying the SK model proposed by Thouless-Anderson-Palmer (TAP). I will explain how the generalized TAP correction appears naturally and give the corresponding generalized TAP representation for the free energy. Based on a joint work with D. Panchenko and E. Subag.

10/23/2019 Souvik Dhara (MIT) Title: A new universality class for critical percolation on networks with heavy-tailed degrees

Abstract: The talk concerns critical behavior of percolation on finite random networks with heavy-tailed degree distribution. In a seminal paper, Aldous (1997) identified the scaling limit for the component sizes in the critical window of phase transition for the Erdős-Rényi random graph. Subsequently, there has been a surge in the literature identifying two universality classes for the critical behavior depending on whether the asymptotic degree distribution has a finite or infinite third moment.

In this talk, we will present a completely new universality class that arises in the context of degrees having infinite second moment. Specifically, the scaling limit of the rescaled component sizes is different from the general description of multiplicative coalescent given by Aldous and Limic (1998). Moreover, the study of critical behavior in this regime exhibits several surprising features that have never been observed in any other universality classes so far.

This is based on joint works with Shankar Bhamidi, Remco van der Hofstad, Johan van Leeuwaarden.

10/30/2019 Aram Harrow (MIT)  Title: Random quantum circuits, phase transitions and complexity

Abstract: Random unitary dynamics are a toy model for chaotic quantum dynamics and also have applications to quantum information theory and computing. Recently, random quantum circuits were the basis of Google’s announcement of “quantum computational supremacy,” meaning performing a task on a programmable quantum computer that would difficult or infeasible for any classical computer. Google’s approach is based on the conjecture that random circuits are as hard to classical computers to simulate as a worst-case quantum computation would be. I will describe evidence in favor of this conjecture for deep random circuits and against this conjecture for shallow random circuits. (Deep/shallow refers to the number of time steps of the quantum circuit.) For deep random circuits in Euclidean geometries, we show that quantum dynamics match the first few moments of the Haar measure after roughly the amount of time needed for a signal to propagate from one side of the system to the other. In non-Euclidean geometries, such as the Schwarzschild metric in the vicinity of a black hole, this turns out not to be always true. I will also explain how shallow quantum circuits are easier to simulate when the gates are randomly chosen than in the worst case. This uses a simulation algorithm based on tensor contraction which is analyzed in terms of an associated stat mech model.

This is based on joint work with Saeed Mehraban (1809.06957) and with John Napp, Rolando La Placa, Alex Dalzell and Fernando Brandao (to appear).

11/6/2019 Bruno Nachtergaele (UC Davis)  TitleThe transmission time and local integrals of motion for disordered spin chains

Abstract:  We investigate the relationship between zero-velocity Lieb-Robinson bounds and the existence of local integrals of motion (LIOMs) for disordered quantum spin chains. We also study the effect of dilute random perturbations on the dynamics of many-body localized spin chains. Using a notion of transmission time for propagation in quantum lattice systems we demonstrate slow propagation by proving a lower bound for the transmission time. This result can be interpreted as a robustness property of slow transport in one dimension. (Joint work with Jake Reschke)

11/13/2019 Gourab Ray (University of Victoria)  Title: Logarithmic variance of height function of square-iceAbstract: A homomorphism height function on a finite graph is a integer-valued function on the set of vertices constrained to have adjacent vertices take adjacent integer values. We consider the uniform distribution over all such functions defined on a finite subgraph of Z^2 with predetermined values at some fixed boundary vertices. This model is equivalent to the height function of the six-vertex model with a = b = c = 1, i.e. to the height function of square-ice. Our main result is that in a subgraph of Z^2 with zero boundary conditions, the variance grows logarithmically in the distance to the boundary. This establishes a strong form of roughness of the planar uniform homomorphisms.

Joint work with: Hugo Duminil Copin, Matan Harel, Benoit Laslier and Aran Raoufi.

11/20/2019 Vishesh Jain (MIT) Title: A combinatorial approach to the quantitative invertibility of random matrices.

Abstract: Abstract: Let $s_n(M_n)$ denote the smallest singular value of an $n\times n$ random matrix $M_n$. We will discuss a novel combinatorial approach (in particular, not using either inverse Littlewood–Offord theory or net arguments) for obtaining upper bounds on the probability that $s_n(M_n)$ is smaller than $\eta \geq 0$ for quite general random matrix models. Such estimates are a fundamental part of the non-asymptotic theory of random matrices and have applications to the strong circular law, numerical linear algebra etc. In several cases of interest, our approach provides stronger bounds than those obtained by Tao and Vu using inverse Littlewood–Offord theory.



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