Schedule for: 19w5176 - From Many Body Problems to Random Matrices

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

We consider systems of N trapped bosons interacting through a repulsive potential with scattering length of the order 1/N (Gross-Pitaevskii regime). We determine the low-energy spectrum of the Hamilton operator in the limit of large N. Our results confirm the predictions of Bogoliubov theory. This talk is based on joint works with C. Boccato, C. Brennecke and S. Cenatiempo.

In 1957 Lee, Huang, and Yang (LHY) predicted a universal expression for a two-term asymptotic formula for the ground state energy of a dilute Bose gas. The formula is universal in the sense that the two terms depend on the interaction potential only through its scattering length. In 2009 Yau and Yin proved an upper bound of the LHY form for a fairly large class of potentials. I will discuss recent joint work with Fournais complementing this by a corresponding lower bound establishing the LHY universality formula.

I will illustrate some recent results about hydrodynamic limit in Euler scaling for one dimensional chain of oscillators: - in the harmonic case with random masses, Anderson localization allows to obtain Euler equation in the hyperbolic scaling limit, while temperature profile does not evolve in any time scale (with F. Huveneers and C. Bernardin). - If the chain is in contact with a Langevin heat bath conserving momentum and volume (isothermal evolution), we prove convergence to $L^2$-valued weak entropic thermodynamic solutions of the non-linear wave equation, even in presence of boundary tension. (with S. Marchesani).

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Kitaev's quantum double models provide a rich class of examples of two-dimensional lattice systems with topological order in the ground states and a spectrum described by anyonic elementary excitations. The infinite volume ground states of the abelian quantum double models come in a number of equivalence classes called superselection sectors. We prove that the superselection structure remains unchanged under uniformly small perturbations of the Hamiltonians. (joint work with Matthew Cha and Pieter Naaijkens)

Geometric matrix Brownian motion is the solution (in $N\times N$ matrices) to the stochastic differential equation $dG_t = G_t dZ_t$, $G_0 = I$, where $Z_t$ is a Ginibre Brownian motion (all independent complex Brownian motion entries). It can also be described as the standard Brownian motion on the Lie group $\mathrm{GL}(N,\mathbb{C})$. For $N>2$, with probability $1$ it is not a normal matrix for any $t>0$. Over the last 5 years, we have made progress in understanding its asymptotic moments and fluctuations, but the non-normality (and lack of explicit symmetry) has made understanding its large-$N$ limit empirical eigenvalue distribution quite challenging. The tools around the circular law are now rich and provide a (log) potential course of action to understand the eigenvalues. There are two sides to this problem in general, both quite difficult: proving that the empirical law of eigenvalues converges (which amounts to certain tightness conditions on singular values), and computing what it converges {\em to}. In the case of the geometric matrix Brownian motion, the question of convergence is still a work in progress; but in recent joint work with Bruce Driver and Brian Hall, we have explicitly calculated the limit empirical eigenvalue distribution. It has an analytic density with a nice polar decomposition, supported on a region that resembles a lima bean for small $t>0$, then folds over and becomes a topological annulus when $t>4$. Our methods blend stochastic analysis, complex analysis, and PDE, and approach the log potential in a new way that we hope will be useful in a wider context.

The Harish-Chandra/Itzykson-Zuber integral and its additive counterpart, the Brezin-Gross-Witten integral, play an important role in random matrix theory. I will present recent work which proves a longstanding conjecture on the large dimension asymptotic behavior of these special functions.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We consider a Hamiltonian given by the Laplacian plus a Bernoulli potential on the two dimensional lattice. We prove that, for energies sufficiently close to the edge of the spectrum, the resolvent on a large square is likely to decay exponentially. This implies almost sure Anderson localization for energies sufficiently close to the edge of the spectrum. Our proof follows the program of Bourgain-Kenig, using a new unique continuation result inspired by a Liouville theorem of Buhovsky-Logunov-Malinnikova-Sodin. This is based on joint work with Charles Smart.

Quantum spin chains provide some of the mathematically most accessible examples of quantum many-body systems. However, even these toy models pose considerable analytical and numerical challenges, due to the fact that the number of degrees of freedom involved grows exponentially fast with the system’s size. We will discuss the recent progress in establishing many body localization at the bottom of the spectrum of a disordered XXZ chain. In particular, we will introduce a new approach to many body localization that works beyond the droplet phase (joint work in progress with A. Klein).

We consider the macroscopic disordered system of free lattice fermions with the one-body Hamiltonian, which is the Schrödinger operator with i.i.d. potential in d>1. Assuming that the fractional moment criteria for the Anderson localization is satisfied, we prove Central Limit Theorem for the large block entanglement entropy.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Random band matrices (RBM) are natural intermediate models to study eigenvalue statistics and quantum propagation in disordered systems, since they interpolate between mean-field type Wigner matrices and random Schrodinger operators. In particular, RBM can be used to model the Anderson metal-insulator phase transition (crossover) even in 1d. In this talk we will discuss some recent progress in application of the supersymmetric method (SUSY) and transfer matrix approach to the analysis of local spectral characteristics of some specific types of 1d RBM.

Much of the theory of machine learning is concerned with the optimization of non-convex functions. Convex relaxations and their associated heirarchies (sum-of-squares, lassere) provide a systematic approach for approximately optimizing non-convex functions. Recent breakthroughs in robust statistics have produced the first polynomial time (efficient) algorithms for computing the robust mean of a high dimensional gaussian. Building on these developments, we construct a framework for robust learning via convex relaxations yielding the first polynomial time algorithm for robust regression when the overwhelming majority of the dataset is comprised of outliers.

In this talk, based on recent joint works with Nick Cook and with Sohom Bhattacharya, I will discuss recent developments in the emerging theory of nonlinear large deviations, focusing on sharp upper tails for counts of a fixed subgraph in large sparse random graphs, such as Erdos–Renyi or uniformly d-regular. Time permitting, I will explain our approach via quantitative versions of the regularity and counting lemmas suitable for the study of sparse random graphs in the large deviations regime.

I review recent results on the extremal eigenvalues of the adjacency matrix A of the Erdos-Renyi graph G(N,p). If p is large then, after a suitable rescaling, A behaves like a Wigner matrix and its extremal eigenvalues converge to the edges -2, +2 of the asymptotic support of the eigenvalue distribution. If p is small, this is no longer true. We analyse the behaviour of the extremal eigenvalues for small p, and in particular describe this transition around a critical p. The proof is based on a tridiagonal representation of A and on a detailed analysis of the geometry of the neighbourhood of the large degree vertices. An important ingredient in our estimates is a matrix inequality obtained via the associated nonbacktracking matrix and an Ihara-Bass formula. Joint work with Johannes Alt and Raphael Ducatez.

The speaker will describe various universality results in numerical computation with random data. The talk will provide an overview of prior results, and also some recent developments. This is joint work with various authors, but mostly with Tom Trogdon.

I will present a multiscale generalisation of the Bakry--Emery criterion for a measure to satisfy a Log-Sobolev inequality. It relies on the control of an associated PDE well known in renormalisation theory: the Polchinski equation. It implies the usual Bakry--Emery criterion, but we show that it remains effective for measures which are far from log-concave. Indeed, we prove that the massive continuum Sine-Gordon model on $R^2$ with $\beta < 6\pi$ satisfies asymptotically optimal Log-Sobolev inequalities for Glauber and Kawasaki dynamics. These dynamics can be seen as singular SPDEs recently constructed via regularity structures, but our results are independent of this theory. This is joint work with T. Bodineau.

We consider uniformly random lozenge tilings of essentially arbitrary domains and show that the local statistics of this model around any point in the liquid region of its limit shape are given by the infinite-volume, translation-invariant, extremal Gibbs measure of the appropriate slope. In this talk, we outline a proof of this result, which proceeds by locally coupling a uniformly random lozenge tiling with a model of Bernoulli random walks conditioned to never intersect. Central to implementing this procedure is to establish a local law for the random tiling, which states that the associated height function is approximately linear on any mesoscopic scale.

We study the two-dimensional stochastic sine-Gordon equation (SSG) in the hyperbolic setting. In particular, by introducing a suitable time-dependent renormalization, we prove local well- posedness of SSG for any value of a parameter \beta > 0 in the nonlinearity. This is in contrast to the parabolic case studied by Hairer and Shen (2016) and Chandra-Hairer-Shen (2018), where the parameter is restricted to the subcritical range \beta^2 < 8*pi. This is joint work with Tadahiro Oh, Tristan Robert and Yuzhao Wang.

In the last decade, Wigner-Dyson-Mehta (WDM) conjecture has been proven for very general random matrix ensembles in the bulk and at the edge of the self consistent density of states (scDos). Recently we proved universality at the cusp of the scDos completing the last remaining case of the WDM conjecture. About universality for non-Hermitian matrices much less is known. The only available result is the proof by Tao and Vu assuming (non-optimal) four moments matching with Ginibre ensemble. In a very recent work we proved universality at the circular edge of any non-Hermitian matrix X with entries i.i.d. real or complex centred random variables without any moment condition.

We consider the fluctuations of the overlap between two replicas in the 2-spin spherical SK model in the low temperature phase. We show that the fluctuations are of order $N^{-1/3}$ and are given by a simple, explicit function of the eigenvalues of a GOE matrix. We show that this quantity converges and describe its limit in terms of quantities from random matrix theory. Joint work with P. Sosoe.

It is meant to increase interaction. One can, for example, ask the connection and difference between two different talks.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.