Previous Random Matrix & Probability Theory Seminars

02/26/2020 10:26 am - 10:27 am

Spring 2020:

Date Speaker Title/Abstract
2/26/2020 Louigi 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.

4/1/2020 Ian 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.

4/22/2020 Martin Gebert (UC Davis) This meeting will be taking place virtually on Zoom.

TitleLieb-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.

4/29/2020 Marcin Napiórkowski (University of Warsaw) This meeting will be taking place virtually on Zoom.

TitleFree 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.

5/6/2020 Antti Knowles (University of Geneva) TitleField 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.
5/13/2020 Sven Bachmann (University of British Columbia) TitleQuantized 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.

5/20/2020 Kristina 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.





Date Speaker Title/Abstract

*Friday, 10:00am*

Yash Deshpande (MIT) Title: Estimating low-rank matrices in noise: phase transitions from spin glass theory

Abstract: Estimating low-rank matrices from noisy observations is a common task in statistical and engineering applications. Following the seminal work of Johnstone, Baik, Ben-Arous and Peche, versions of this problem have been extensively studied using random matrix theory. In this talk, we will consider an alternative viewpoint based on tools from mean field spin glasses. We will present two examples that illustrate how these tools yield information beyond those from classical random matrix theory. The first example is the two-groups stochastic block model (SBM), where we will obtain a full information-theoretic understanding of the estimation phase transition. In the second example, we will augment the SBM with covariate information at nodes, and obtain results on the altered phase transition.

This is based on joint works with Emmanuel Abbe, Andrea Montanari, Elchanan Mossel and Subhabrata Sen.

10/3/2018 Ian Jauslin (IAS) Title: Liquid Crystals and the Heilmann-Lieb model

Abstract: In 1979, O.Heilmann and E.H. Lieb introduced an interacting dimer model with the goal of proving the emergence of a nematic liquid crystal phase in it. In such a phase, dimers spontaneously align, but there is no long range translational order. Heilmann and Lieb proved that dimers do, indeed, align, and conjectured that there is no translational order. I will discuss a recent proof of this conjecture. This is joint work with Elliott H. Lieb.

10/10/2018 Afonso Bandeira (NYU Title: Statistical estimation under group actions: The Sample Complexity of Multi-Reference Alignment

Abstract: Many problems in signal/image processing, and computer vision amount to estimating a signal, image, or tri-dimensional structure/scene from corrupted measurements. A particularly challenging form of measurement corruption are latent transformations of the underlying signal to be recovered. Many such transformations can be described as a group acting on the object to be recovered. Examples include the Simulatenous Localization and Mapping (SLaM) problem in Robotics and Computer Vision, where pictures of a scene are obtained from different positions and orientations; Cryo-Electron Microscopy (Cryo-EM) imaging where projections of a molecule density are taken from unknown rotations, and several others.

One fundamental example of this type of problems is Multi-Reference Alignment: Given a group acting in a space, the goal is to estimate an orbit of the group action from noisy samples. For example, in one of its simplest forms, one is tasked with estimating a signal from noisy cyclically shifted copies. We will show that the number of observations needed by any method has a surprising dependency on the signal-to-noise ratio (SNR), and algebraic properties of the underlying group action. Remarkably, in some important cases, this sample complexity is achieved with computationally efficient methods based on computing invariants under the group of transformations.



Thomas Chen (UT Austin) Title: Dynamics of a heavy quantum tracer particle in a Bose gas

Abstract: We consider the dynamics of a heavy quantum tracer particle coupled to a non-relativistic boson field in R^3. The pair interactions of the bosons are of mean-field type, with coupling strength proportional to 1/N where N is the expected particle number. Assuming that the mass of the tracer particle is proportional to N, we derive generalized Hartree equations in the limit where N tends to infinity. Moreover, we prove the global well-posedness of the associated Cauchy problem for sufficiently weak interaction potentials. This is joint work with Avy Soffer (Rutgers University).


*Room G02*

Tselil Schramm (Harvard/MIT) Title: (Nearly) Efficient Algorithms for the Graph Matching Problem in Correlated Random Graphs

Abstract: The Graph Matching problem is a robust version of the Graph Isomorphism problem: given two not-necessarily-isomorphic graphs, the goal is to find a permutation of the vertices which maximizes the number of common edges. We study a popular average-case variant; we deviate from the common heuristic strategy and give the first quasi-polynomial time algorithm, where previously only sub-exponential time algorithms were known.

Based on joint work with Boaz Barak, Chi-Ning Chou, Zhixian Lei, and Yueqi Sheng.




SC 507*

Lauren Williams (Harvard) Title: Introduction to the asymmetric simple exclusion process (from a combinatorialist’s point of view)

Abstract: The asymmetric simple exclusion process (ASEP) is a model of particles hopping on a one-dimensional lattice, subject to the condition that there is at most one particle per site. This model was introduced in 1970 by biologists (as a model for translation in protein synthesis) but has since been shown to display a rich mathematical structure. There are many variants of the model — e.g. the lattice could be a ring, or a line with open boundaries. One can also allow multiple species of particles with different “weights.” I will explain how one can give combinatorial formulas for the stationary distribution using various kinds of tableaux. I will also explain how the ASEP is related to interesting families of orthogonal polynomials, including Askey-Wilson polynomials, Koornwinder polynomials, and Macdonald polynomials.

11/7/2018 Willhelm Schlag (Yale) Title: on the Bourgain-Dyatlov fractal uncertainty principle

Abstract: We will present the Bourgain-Dyatlov theorem on the line, it’s connection with other uncertainty principles in harmonic analysis, and my recent partial progress with Rui Han on the problem of higher dimensions.

11/14/2018 David Gamarnik (MIT) Title: Two Algorithmic Hardness Results in Spin Glasses and Compressive Sensing.

Abstract: I will discuss two computational problems in the area of random combinatorial structures. The first one is the problem of computing the partition function of a Sherrington-Kirkpatrick spin glass model. While the the problem of computing the partition functions associated with arbitrary instances is known to belong to the #P complexity class, the complexity of the problem for random instances is open. We show that the problem of computing the partition function exactly (in an appropriate sense) for the case of instances involving Gaussian couplings is #P-hard on average. The proof uses Lipton’s trick of computation modulo large prime number, reduction of the average case to the worst case instances, and the near uniformity of the ”stretched” log-normal distribution.

In the second part we will discuss the problem of explicit construction of matrices satisfying the Restricted Isometry Property (RIP). This challenge arises in the field of compressive sensing. While random matrices are known to satisfy the RIP with high probability, the problem of explicit (deterministic) construction of RIP matrices eluded efforts and hits the so-called ”square root” barrier which I will discuss in the talk. Overcoming this barrier is an open problem explored widely in the literature. We essentially resolve this problem by showing that an explicit construction of RIP matrices implies an explicit construction of graphs satisfying a very strong form of Ramsey property, which has been open since the seminal work of Erdos in 1947.

11/28/2018 Sean O’ Rourke (UC Boulder) Title: Universality and least singular values of random matrix products

Abstract: We consider the product of m independent iid random matrices as m is fixed and the sizes of the matrices tend to infinity.  In the case when the factor matrices are drawn from the complex Ginibre ensemble, Akemann and Burda computed the limiting microscopic correlation functions.  In particular, away from the origin, they showed that the limiting correlation functions do not depend on m, the number of factor matrices. We show that this behavior is universal for products of iid random matrices under a moment matching hypothesis.  In addition, we establish universality results for the linear statistics for these product models, which show that the limiting variance does not depend on the number of factor matrices either. The proofs of these universality results require a near-optimal lower bound on the least singular value for these product ensembles.


*Room G02*

Omer Angel (UBC) Title: balanced excited random walks

Abstract: I will present results on the scaling limit and asymptotics of the balanced excited random walk and related processes. This is a walk the that moves vertically on the first visit to a vertex, and horizontally on every subsequent visit. We also analyze certain versions of “clairvoyant scheduling” of random walks.

Joint work with Mark Holmes and Alejandro Ramirez.


Science Center 530

Ramis Movassagh (IMB Research) Title: Generic Gaplessness, and Hamiltonian density of states from free probability theory

Abstract: Quantum many-body systems usually reside in their lowest energy states. This among other things, motives understanding the gap, which is generally an undecidable problem. Nevertheless, we prove that generically local quantum Hamiltonians are gapless in any dimension and on any graph with bounded maximum degree.

We then provide an applied and approximate answer to an old problem in pure mathematics. Suppose the eigenvalue distributions of two matrices M_1 and M_2 are known. What is the eigenvalue distribution of the sum M_1+M_2? This problem has a rich pure mathematics history dating back to H. Weyl (1912) with many applications in various fields. Free probability theory (FPT) answers this question under certain conditions. We will describe FPT and show examples of its powers for approximating physical quantities such as the density of states of the Anderson model, quantum spin chains, and gapped vs. gapless phases of some Floquet systems. These physical quantities are often hard to compute exactly (provably NP-hard). Nevertheless, using FPT and other ideas from random matrix theory excellent approximations can be obtained. Besides the applications presented, we believe the techniques will find new applications in fresh new contexts.

2/14/2019 Nike Sun (MIT) Title: Capacity lower bound for the Ising perceptron

Abstract: The perceptron is a toy model of a simple neural network that stores a collection of given patterns. Its analysis reduces to a simple problem in high-dimensional geometry, namely, understanding the intersection of the cube (or sphere) with a collection of random half-spaces. Despite the simplicity of this model, its high-dimensional asymptotics are not well understood. I will describe what is known and present recent results.

2/21/2019 Michael Loss (Georgia Tech) Title: Some results for functionals of Aharonov-Bohm type

Abstract: In this talk I present some variational problems of Aharonov-Bohm type, i.e., they include a  magnetic flux that is entirely concentrated at a point. This is maybe the simplest example of a  variational problems for systems, the wave function being necessarily complex. The functional is rotationally invariant and the issue to be discussed is whether the optimizer have this symmetry or whether it is broken.



Science Center 411

Ilya Kachkovskiy (Michigan State University) Title: Localization and delocalization for interacting 1D quasiperiodic particles.

Abstract: We consider a system of two interacting one-dimensional quasiperiodic particles as an operator on $\ell^2(\mathbb Z^2)$. The fact that particle frequencies are identical, implies a new effect compared to generic 2D potentials: the presence of large coupling localization depends on symmetries of the single-particle potential. If the potential has no cosine-type symmetries, then we are able to show large coupling localization at all energies, even if the interaction is not small (with some assumptions on its complexity). If symmetries are present, we can show localization away from finitely many energies, thus removing a fraction of spectrum from consideration. We also demonstrate that, in the symmetric case, delocalization can indeed happen if the interaction is strong, at the energies away from the bulk spectrum. The result is based on joint works with Jean Bourgain and Svetlana Jitomirskaya.



Science Center 232

Anna Vershynina (University of Houston) Title: How fast can entanglement be generated in quantum systems?

Abstract: We investigate the maximal rate at which entanglement can be generated in bipartite quantum systems. The goal is to upper bound this rate. All previous results in closed systems considered entanglement entropy as a measure of entanglement. I will present recent results, where entanglement measure can be chosen from a large class of measures. The result is derived from a general bound on the trace-norm of a commutator, and can, for example, be applied to bound the entanglement rate for Renyi and Tsallis entanglement entropies.


Room G02

Xuwen Chen (University of Rochester) Title: The Derivation of the Energy-critical NLS from Quantum Many-body Dynamics

Abstract: We derive the 3D energy-critical quintic NLS from quantum many-body dynamics with 3-body interaction in the T^3 (periodic) setting. Due to the known complexity of the energy critical setting, previous progress was limited in comparison to the 2-body interaction case yielding energy subcritical cubic NLS. We develop methods to prove the convergence of the BBGKY hierarchy to the infinite Gross-Pitaevskii (GP) hierarchy, and separately, the uniqueness of large GP solutions. Since the trace estimate used in the previous proofs of convergence is the false sharp trace estimate in our setting, we instead introduce a new frequency interaction analysis and apply the finite dimensional quantum de Finetti theorem. For the large solution uniqueness argument, we discover the new HUFL (hierarchical uniform frequency localization) property for the GP hierarchy and use it to prove a new type of uniqueness theorem.

4/4/2019 Paul Bourgade (NYU) Title: Log-correlations and branching structures in analytic number theory

Abstract: Fyodorov, Hiary and Keating have predicted the size of local maxima of L-function along the critical axis, based on analogous random matrix statistics. I will explain this prediction in the context of the log-correlated universality class and branching structures. In particular I will explain why the Riemann zeta function exhibits log-correlations, and outline the proof for the leading order of the maximum in the Fyodorov, Hiary and Keating prediction. Joint work with Arguin, Belius, Radziwill and Soundararajan.




Room G02

Giulio Biroli (ENS Paris) Title: Large deviations for the largest eigenvalues and eigenvectors of spiked random matrices

Abstract: I consider matrices formed by a random $N\times N$ matrix drawn from the Gaussian Orthogonal Ensemble (or Gaussian Unitary Ensemble) plus a rank-one perturbation of strength $\theta$, and focus on the largest eigenvalue, $x$, and the component, $u$, of the corresponding eigenvector in the direction associated to the rank-one perturbation. I will show how to obtain the large deviation principle governing the atypical joint fluctuations of $x$ and $u$. Interestingly, for $\theta>1$, in large deviations characterized by a small value of $u$, i.e. $u<1-1/\theta$, the second-largest eigenvalue pops out from the Wigner semi-circle and the associated eigenvector orients in the direction corresponding to the rank-one perturbation. These results can be generalized to the Wishart Ensemble, and extended to the first $n$ eigenvalues and the associated eigenvectors.

Finally, I will discuss motivations and applications of these results to the study of the geometric properties of random high-dimensional functions—a topic that is currently attracting a lot of attention in physics and computer science.

4/11/2019 Rui Han (Georgia Tech) Title: Spectral gaps in graphene structures

Abstract: We present a full analysis of the spectrum of graphene in magnetic fields with constant flux through every hexagonal comb. In particular, we provide a rigorous foundation for self-similarity by showing that for irrational flux, the spectrum of graphene is a zero measure Cantor set. We also show that for vanishing flux, the spectral bands have nontrivial overlap, which proves the discrete Bethe-Sommerfeld conjecture for the graphene structure. This is based on joint works with S. Becker, J. Fillman and S. Jitomirskaya.

4/25/2019 Benjamin Fehrman (Oxford) Title:  Pathwise well-posedness of nonlinear diffusion equations with nonlinear, conservative noise

Abstract:  We present a pathwise well-posedness theory for stochastic porous media and fast diffusion equations driven by nonlinear, conservative noise.  Such equations arise in the theory of mean field games, approximate the Dean-Kawasaki equation in fluctuating fluid dynamics, describe the fluctuating hydrodynamics of the zero range process, and model the evolution of a thin film in the regime of negligible surface tension.  Motivated by the theory of stochastic viscosity solutions, we pass to the equation’s kinetic formulation, where the noise enters linearly and can be inverted using the theory of rough paths. The talk is based on joint work with Benjamin Gess.

4/30/2019 TBA TBA
5/2/2019 Jian Ding (UPenn) TBA


Date………… Name……………. Title/Abstract


Reza Gheissari (NYU) Dynamics of Critical 2D Potts ModelsAbstract: The Potts model is a generalization of the Ising model to $q\geq 3$ states with inverse temperature $\beta$. The Gibbs measure on $\mathbb Z^2$ has a sharp transition between a disordered regime when $\beta<\beta_c(q)$ and an ordered regime when $\beta>\beta_c(q)$. At $\beta=\beta_c(q)$, when $q\leq 4$, the phase transition is continuous while when $q>4$, the phase transition is discontinuous and the disordered and ordered phases coexist.

We will discuss recent progress, joint with E. Lubetzky, in analyzing the time to equilibrium (mixing time) of natural Markov chains (e.g., heat bath/Metropolis) for the 2D Potts model, where the mixing time on an $n \times n$ torus should transition from $O(\log n)$ at high temperatures to $\exp(c_\beta n)$ at low temperatures, via a critical slowdown at $\beta_c(q)$ that is polynomial in $n$ when $q \leq 4$ and exponential in $n$ when $q>4$.



Mustazee Rahman (MIT) On shocks in the TASEPAbstract: The TASEP particle system runs into traffic jams when the particle density to the left is smaller than the density to the right. Macroscopically, the particle density solves Burgers’ equation and traffic jams correspond to its shocks. I will describe work with Jeremy Quastel on a specialization of the TASEP shock whereby we identify the microscopic fluctuations around the shock by using exact formulas for the correlation functions of TASEP and its KPZ scaling limit. The resulting laws are related to universal laws of random matrix theory.

For the curious, here is a video of the shock forming in Burgers’ equation:

4-20-20182:00-3:00pm Carlo Lucibello(Microsoft Research NE) The Random Perceptron Problem: thresholds, phase transitions, and geometryAbstract: The perceptron is the simplest feedforward neural network model, the building block of the deep architectures used in modern machine learning practice. In this talk, I will review some old and new results, mostly focusing on the case of binary weights and random examples. Despite its simplicity, this model provides an extremely rich phenomenology: as the number of examples per synapses is increased, the system undergoes different phase transitions, which can be directly linked to solvers’ performances and to information theoretic bounds. A geometrical analysis of the solution space shows how two different types of solutions, akin to wide and sharp minima, have different generalization capabilities when presented with new examples.  Solutions in dense clusters generalize remarkably better,  partially closing the gap with Bayesian optimal estimators.  Most of the results I will present were first obtained using non rigorous techniques from spin glass theory and many of them haven’t been rigorously established yet,  although some big steps forward have been taken in recent years.
4-20-20183:00-4:00pm Yash Despande(MIT) Phase transitions in estimating low-rank matricesAbstract: Low-rank perturbations of Wigner matrices have been extensively studied in random matrix theory; much information about the corresponding spectral phase transition can be gleaned using these tools. In this talk, I will consider an alternative viewpoint based on tools from spin glass theory, and two examples that illustrate how these they yield information beyond traditional spectral tools. The first example is the stochastic block model,where we obtain a full information-theoretic picture of estimation. The second example demonstrates how side information alters the spectral threshold. It involves a new phase transition that interpolates between the Wigner and Wishart ensembles.
Date Name Title/Abstract
9-27-17 Herbert Spohn, Technische Universität München Hydrodynamics of integrable classical and quantum systems

Abstract:  In the cold atoms community there is great interest in developing Euler-type hydrodynamics for one-dimensional integrable quantum systems, in particular with application to domain wall initial states.  I provide some mathematical physics background and also compare with integrable classical systems.


*12:00-1:00pm, Science Center 232*

 Madhu Sudan, Harvard SEAS General Strong Polarization

A recent discovery (circa 2008) in information theory called Polar Coding has led to a remarkable construction of error-correcting codes and decoding algorithms, resolving one of the fundamental algorithmic challenges in the field. The underlying phenomenon studies the “polarization” of a “bounded” martingale. A bounded martingale, X_0,…,X_t,…  is one where X_t in [0,1]. This martingale is said to polarize if Pr[lim_{t\to infty} X_t \in {0,1}] = 1. The questions of interest to the results in coding are the rate of convergence and proximity: Specifically, given epsilon and tau > 0 what is the smallest t after which it is the case that Pr[X_t in (tau,1-tau)] < epsilon? For the main theorem, it was crucial that t <= min{O(log(1/epsilon)), o(log(1/tau))}. We say that a martingale polarizes strongly if it satisfies this requirement. We give a simple local criterion on the evolution of the martingale that suffices for strong polarization. A consequence to coding theory is that a broad class of constructions of polar codes can be used to resolve the afore-mentioned algorithmic challenge.

In this talk I will introduce the concepts of polarization and strong polarization.  Depending on the audience interest I can explain why this concept is useful to construct codes and decoding algorithms, or explain the local criteria that help establish strong polarization (and the proof of why it does so).

Based on joint work with Jaroslaw Blasiok, Venkatesan Guruswami, Preetum Nakkiran, and Atri Rudra.



Subhabrata Sen (Microsoft and MIT)

Noga Alon,(Tel Aviv University)

Subhabrata Sen, “Partitioning sparse random graphs: connections with mean-field spin glasses”

Abstract: The study of graph-partition problems such as Maxcut, max-bisection and min-bisection have a long and rich history in combinatorics and theoretical computer science. A recent line of work studies these problems on sparse random graphs, via a connection with mean field spin glasses. In this talk, we will look at this general direction, and derive sharp comparison inequalities between cut-sizes on sparse Erd\ ̋{o}s-R\'{e}nyi and random regular graphs.

Based on joint work with Aukosh Jagannath.

Noga Alon, “Random Cayley Graphs”

Abstract: The study of random Cayley graphs of finite groups is related to the  investigation of Expanders and to problems in Combinatorial Number Theory and in Information Theory. I will discuss this topic, describing the motivation and focusing on the question of estimating the chromatic number of a random Cayley graph of a given  group with a prescribed number of generators.  Several intriguing questions that remain open will be mentioned as well.



Kay Kirkpatrick (Illinois)


Wei-Ming Wang (CNRS)

Kay Kirkpatrick, Quantum groups, Free Araki-Woods Factors, and a Calculus for Moments

 Abstract: We will discuss a central limit theorem for quantum groups: that the joint distributions with respect to the Haar state of the generators of free orthogonal quantum groups converge to free families of generalized circular elements in the large (quantum) dimension limit. We also discuss a connection to free Araki-Woods factors, and cases where we have surprisingly good rates of convergence. This is joint work with Michael Brannan. Time permitting, we’ll mention another quantum central limit theorem for Bose-Einstein condensation and work in progress.

Wei-Min Wang, Quasi-periodic solutions to nonlinear PDE’s

Abstract: We present a new approach to the existence of time quasi-periodic solutions to nonlinear PDE’s. It is based on the method of Anderson localization, harmonic analysis and algebraic analysis. This can be viewed as an infinite dimensional analogue of a Lagrangian approach to KAM theory, as suggested by J. Moser.

11-8-17 Elchanan Mossel Optimal Gaussian Partitions.

Abstract: How should we partition the Gaussian space into k parts in a way that minimizes Gaussian surface area, maximize correlation or simulate a specific distribution.

The problem of Gaussian partitions was studied since the 70s first as a generalization of the isoperimetric problem in the context of the heat equation. It found a renewed interest in context of the double bubble theorem proven in geometric measure theory and due to connection to problems in theoretical computer science and social choice theory.

I will survey the little we know about this problem and the major open problems in the area.


*12pm SC 232*

Zhe Wang (NYU) A Driven Tagged Particle in One-dimensional Simple Exclusion Process

Abstract: We study the long-time behavior of a driven tagged particle in a one-dimensional non-nearest- neighbor simple exclusion process.  We will discuss two scenarios when the tagged particle has a speed. Particularly, for the ASEP, the tagged particle can have a positive speed even when it has a drift with negative mean; for the SSEP with removals, we can compute the speed explicitly. We will characterize some nontrivial invariant measures of the environment process by using coupling arguments and color schemes.




Daniel Sussman (BU) Multiple Network Inference: From Joint Embeddings to Graph Matching

Abstract: Statistical theory, computational methods, and empirical evidence abound for the study of individual networks. However, extending these ideas to the multiple-network framework remains a relatively under-explored area. Individuals today interact with each other through numerous modalities including online social networks, telecommunications, face-to-face interactions, financial transactions, and the sharing and distribution of goods and services. Individually these networks may hide important activities that are only revealed when the networks are studied jointly. In this talk, we’ll explore statistical and computational methods to study multiple networks, including a tool to borrow strength across networks via joint embeddings and a tool to confront the challenges of entity resolution across networks via graph matching.




 Yue M. Lu


Asymptotic Methods for High-Dimensional Inference: Precise Analysis, Fundamental Limits, and Optimal Designs
Abstract: Extracting meaningful information from the large datasets being compiled by our society presents challenges and opportunities to signal and information processing research. On the one hand, many classical methods, and the assumptions they are based on, are simply not designed to handle the explosive growth of the dimensionality of the modern datasets. On the other hand, the increasing dimensionality offers many benefits: in particular, the very high-dimensional settings allow one to apply powerful asymptotic methods from probability theory and statistical physics to obtain precise characterizations that would otherwise be too complicated in moderate dimensions. I will mention recent work on exploiting such blessings of dimensionality via sharp asymptotic methods. In particular, I will show (1) the exact characterization of a widely-used spectral method for nonconvex signal recoveries; (2) the fundamental limits of solving the phase retrieval problem via linear programming; and (3) how to use scaling and mean-field limits to analyze nonconvex optimization algorithms for high-dimensional inference and learning. In these problems, asymptotic methods not only clarify some of the fascinating phenomena that emerge with high-dimensional data, they also lead to optimal designs that significantly outperform commonly used heuristic choices.
11-29-17 David Gamarink (MIT) (Arguably) Hard on Average Constraint Satisfaction Problems

Abstract: Many combinatorial optimization problems defined on random instances such as random graphs, exhibit an apparent gap between the optimal value, which can be estimated by non-constructive means, and the best values achievable by fast (polynomial time) algorithms. Through a combined effort of mathematicians, computer scientists and statistical physicists, it became apparent that a potential and in some cases a provable obstruction for designing algorithms bridging this gap is an intricate geometry of nearly optimal solutions, in particular the presence of chaos and a certain Overlap Gap Property (OGP), which we will introduce in this talk. We will demonstrate how for many such problems, the onset of the OGP phase transition indeed nearly coincides with algorithmically hard regimes. Our examples will include the problem of finding a largest independent set of a graph, finding a largest cut in a random hypergrah, random NAE-K-SAT problem, the problem of finding a largest submatrix of a random matrix, and a high-dimensional sparse linear regression problem in statistics.

Joint work with Wei-Kuo Chen, Quan Li, Dmitry Panchenko,  Mustazee Rahman, Madhu Sudan and Ilias Zadik.



Philippe Rigollet (MIT)

2-3 pm


Ankur Moitra (MIT)

3-4 pm

Philippe Rigollet (MIT), Exact Recovery in the Ising Block Model 

Abstract: Over the past fifteen years, the problem of learning Ising models from independent samples has been of significant interest in the statistics, machine learning, and statistical physics communities. Much of the effort has been directed towards finding algorithms with low computational cost for various restricted classes of models, primarily in the case where the interaction graph is sparse. In parallel, stochastic blockmodels have played a more and more preponderant role in community detection and clustering as an average case model for the minimum bisection model. In this talk, we introduce a new model, called Ising blockmodel for the community structure in an Ising model. It imposes a block structure on the interactions of a dense Ising model and can be viewed as a structured perturbation of the celebrated Curie-Weiss model. We show that interesting phase transitions arise in this model and leverage this probabilistic analysis to develop an algorithm based on semidefinite programming that recovers exactly the community structure when the sample size is large enough. We also prove that exact recovery of the block structure is actually impossible with fewer samples.

This is joint work with Quentin Berthet (University of Cambridge) and Piyush Srivastava (Tata Institute).

Ankur Moitra (MIT), A New Approach to Approximate Counting and Sampling 

Abstract: Over the past sixty years, many remarkable connections have been made between statistical physics, probability, analysis and theoretical computer science through the study of approximate counting. While tight phase transitions are known for many problems with pairwise constraints, much less is known about problems with higher-order constraints.
Here we introduce a new approach for approximately counting and sampling in bounded degree systems. Our main result is an algorithm to approximately count the number of solutions to a CNF formula where the degree is exponential in the number of variables per clause. Our algorithm extends straightforwardly to approximate sampling, which shows that under Lovasz Local Lemma-like conditions, it is possible to generate a satisfying assignment approximately uniformly at random. In our setting, the solution space is not even connected and we introduce alternatives to the usual Markov chain paradigm.

12-14-17 TBD
Date Name Title
09-16-2015 Scott Aaronson, MIT BosonSampling and the Permanents of Gaussian Matrices
09-23-2015 Xin Sun, MIT Almost sure multi fractal spectrum of SLE
09-28-2015 Li-Cheng Tsai, Stanford KPZ equation limit of interacting particle systems
09-30-2015 Kyle Luh, Yale Random Matrices: l1 Concentration and Dictionary Learning with Few Samples
10-07-2015 Martin Zirnbauer, Cologne/Simons Center Bott periodicity and the “Periodic Table” of topological insulators and superconductors
10-14-2015 Benjamin Schweinhart, Harvard CMSA Universality Conjectures for Curvature Flow on Graphs
10-21-2015 Nicholas Cook, UCLA
10-28-2015 Vu-Lan Nguyen, Université Paris Diderot Variants of geometric RSK, geometric PNG and the multipoint distribution of the log-gamma polymer
11-04-2015 Vadim Gorin, MIT Largest eigenvalues in random matrix beta-ensembles: structures of the limit.
11-18-2015 Louis-Pierre Arguin, CUNY The maximum of the characteristic polynomial of random unitary matrices 
11-19-2015 Nicholas Zygouras, Univ. of Warwick From disorder relevance to the 2d Stochastic Heat Equation
11-25-2015 Thanksgiving No seminar
12-02-2015 Eero Saksman (Helsinki) The uniqueness of Gaussian multiplicative chaos revisited
12-04-2015 Guillaume Barraquand, Columbia Random walks in Beta random environment
01-27-2016 Louigi Addario-Berry, McGill Slowdown of the front for branching Brownian motion with decay of mass
02-03-2016 Antti Knowles, ETH Zurich An optimal rotational invariant estimator for general covariance matrices
02-10-2016 No Seminar this week
02-17-2016 Florent Bekerman, MIT Transport Methods and Universality for beta-matrix models
02-24-2016 Aukosh Jagannath, Courant Institute The Parisi variational problem
03-02-2016 No Seminar this week Two next week
03-09-2016 Adam Marcus, Princeton Polynomials and (finite) free probability
03-11-2016 Hao Shen, Columbia The Sine-Gordon stochastic PDE and regularity structures
03-16-2016 Spring Recess
03-23-2016 Zeev Rudnick, Tel-Aviv and IAS Quantum chaos, eigenvalue statistics and the Fibonacci sequence
03-30-2016 Nikolai Makarov, Caltech Random normal matrices with hard edge spectrum
04-06-2016 Timo Seppalainen, Wisconsin Variational formulas and Busemann functions for random paths in a random medium
04-11-2016 (Science Center 530) Milton D. Jara, IMPA Around the strong KPZ universality conjecture
04-20-2016 Mark Rudelson, Michigan Delocalization of eigenvectors of random matrices
04-27-2016 Marek Biskup, UCLA Local limit theory for extreme values of 2D Discrete Gaussian Free Field
05-04-2016 No Talk
05-11-2016 (2:30-3:30pm, Room G10) Laure Saint-Raymond, École Normale Supérieure Fluctuating dynamics for a 2D rarified gas of hard disks
06-01-2016 Jun Yin, University of Wisconsin Generalized Circular Law
06-08-2016 Paul Bourgade, NYU Extremes of random matrices and log-correlated fields