Random Matrix & Probability Theory Seminar

Beginning immediately, until at least April 30, all seminars will take place virtually, through Zoom. Links to connect can be found in the schedule below once they are created.

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.

The schedule will be updated as details are confirmed.

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)

10/3/2019

Thursday

4:30pm

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.