Schedule for: 16w5076 - Beta Ensembles: Universality, Integrability, and Asymptotics

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.

The traditional derivations of the joint eigenvalue densities focus on the symmetric eigenvalue factorization, but it has been long been clear that Laguerre deserves an SVD derivation, and Jacobi deserves a Generalized Singular Value or GSVD derivation. In the old days, the SVD was unfamiliar so it took longer to explain. Today the GSVD is less familiar, so it takes longer to explain. We carry out this derivation, and then turn to a "kind of" explanation (some would call it fictional, or "ghost" explanation) of what is happening for general beta ensembles. Joint work with Bernie Wang.

Quantum Painleve equations (QP) are Fokker-Planck (or non-stationary Schroedinger) equations in two independent variables (``time" and ``space")with diffusion-drift operators being quantized Painleve Hamiltonians. They are satisfied by certain eigenvalue probabilities of beta ensembles. E.g. QPII describes the soft edge limit of beta ensembles while QPIII does so for the hard edge. QP are relatively simple instances of (confluent) Belavin-Polyakov-Zamolodchikov (BPZ) equations of conformal field theory (CFT). The general multidimensional linear BPZ PDEs also naturally arise from beta-ensemble integrals. While CFT is known as quantum integrable theory, we show on the example of QPII how classical integrable structure can be extended to all values of beta. Using explicit Lax pair for even integer beta with QPII solution as eigenvector component (known before only for beta = 2 and 4), the case beta=6 is further studied. It turns out that again everything depends on the Hastings-McLeod solution of Painleve II (PII). The main result is a second order nonlinear ODE for the log-derivative of Tracy-Widom distribution for beta = 6, involving the PII function in the coefficients. Beyond even integers, the derived nonlinear integrable system associated with QPII possesses identically nice analytic properties for all beta, e.g. the Painleve property. Its general solutions are related by a Cole-Hopf transform with two linearly independent solutions of QPII.

Certain iterative numerical algorithms for computing eigenvalues have an unexpected connection to completely integrable Hamiltonian systems. Thus, the algorithm may be thought of as a particularly nice dynamical system on the space of symmetric matrices. A few years ago, we investigated the behavior of these dynamical systems on random matrices and found an intriguing form of universality for the fluctuations in "halting times". I'll present a tentative explanation for this universality and its connection to beta ensembles. This is joint work with several people: Percy Deift and Tom Trogdon (Courant), Enrique Pujals (IMPA) and Christian Pfrang (JP Morgan).

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 give a proof that the maximum of the centered log-potential of GUE is log N + o(log N) with high probability. This confirms the first term in a conjecture by Fyodorov and Simm. Moreover, we prove a general theorem about almost Gaussian fields that should be applicable to showing the law of large numbers for the log characteristic polynomial of beta-ensembles. We will also survey some extensions of this type of result: to other potentials, to other beta, and to higher degrees of precision. This is based on joint works with Gaultier Lambert and Ofer Zeitouni.

We present recent progess on the extremal values of (the logarithm of) the characteristic polynomial of a random unitary matrix whose spectrum is distributed according to the Circular Beta Ensemble. Using different techniques, it constitutes a follow-up to the work by Arguin, Belius, Bourgade on the one hand, and Paquette, Zeitouni on the other hand. They recently treated the CUE case, which corresponds to beta equal to 2

The beta-Jacobi ensembles complete the triad of ``classical" matrix ensembles (together with Hermite/Gaussian and Laguerre/Wishart). In the beta = 1,2,4 cases they have close relationships with the Haar orthogonal/unitary/symplectic ensembles. Although it is possible to find exact formulae for the distributions of the extreme eigenvalues of such ensembles, the are relatively opaque (involving multivariate hypergeometric functions) and not very practical. We will show a few cases where relatively simple asymptotics can be deduced from these formulae, and show a surprising numerical application.

Jack measures on partitions occur naturally in the study of continuum circular log-gases in generic background potentials V at arbitrary values \beta of Dyson’s inverse temperature. Our main result is a law of large numbers (LLN) and central limit theorem (CLT) for Jack measures in the macroscopic scaling limit, which corresponds to the large N limit in the log-gas. Precisely, the emergent limit shape and macroscopic fluctuations of profiles of these random Young diagrams are the push-forwards along V of the uniform measure on the circle (LLN) and of the restriction to the circle of a Gaussian free field on the upper half-plane (CLT), respectively. At \beta=2, this recovers Okounkov’s LLN for Schur measures (2003) and coincides with Breuer-Duits’ CLT for biorthogonal ensembles (2013). Our limit theorems follow from an all-order expansion (AOE) of joint cumulants of linear statistics, which has the same form as the all-order 1/N refined topological expansion for the log-gas on the line due to Chekhov-Eynard (2006) and Borot-Guionnet (2012). To prove our AOE, we rely on the Lax operator for the quantum Benjamin-Ono hierarchy with periodic profile V exhibited in collective field variables by Nazarov-Sklyanin (2013). The characterization of the limit laws as push-forwards follows from factorization formulas for resolvents of Toeplitz operators with symbol V due to Krein and Calderón-Spitzer-Widom (1958).

In this talk we'll recall how central limit theorems for the linear statistics of beta ensembles imply that certain natural objects of random matrix theory can be described asymptotically in terms of log-correlated Gaussian fields. We'll also briefly discuss a proof of such a central limit theorem for the circular beta ensemble.

We consider interacting particle systems on the real line which generalize beta-ensembles by allowing for different pair interactions. We show how these models can be represented as averages of beta-ensembles with random external fields. Particular emphasis is put on models resembling the beta=2-ensembles where we present detailed asymptotics in the bulk and at the edge. In the bulk, these asymptotics allow to consider empirical spacings of unfolded particles. At the edge, we describe the gradual transition from universal behavior in the regime of the Tracy-Widom law to non-universal behavior for large deviations of the largest particle.

We consider the 1d Schroedinger operator with random potential decaying of order \(\alpha\). The results are : (1) the fluctuation of density of states with different behavior depending on \(\alpha\), (2) the level statistics asymptotically obeys clock, \(Sine_{\beta}\), and Poisson processes for super-critical, critical, and sub-critical cases, respectively.

The eigenvalues of some random matrix models can be seen as particles with logarithmic interaction in dimension 1 or 2, and the first order of the interaction energy governs the macroscopic behaviour (convergence of the empirical measure). I will try to show how the « second order » allows one to study the microscopic behaviour of the particles/eigenvalues. We obtain a large deviation principle for the empirical fields (the microscopic analogue of the empirical measure, which describes the geometrical arrangement of the particles). In particular, it yields that Sine-beta processes minimize a certain explicit « free energy » functional. We also get bounds on the fluctuations of the counting measure at small scales, local laws, and a CLT for fluctuations in dimension 2. Joint work with S. Serfaty

We prove the local law and rigidity for two-dimensional beta ensembles with general beta and general confining potential. The local law, i.e. that the linear statistics of the beta ensemble are given by the equilibrium measure on mesoscopic scales, is achieved by a multiscale argument and the direct analysis of the equilibrium measure on the relevant scales. The suboptimal fluctuation estimate given by this argument is then improved to rigidity by using the loop equation.

The Gaussian Unitary/Orthogonal Ensembles (GUE/GOE) are some of the most studied Hermitian random matrix models. They may be generalized to a one parameter family of point processes indexed by beta called the beta-Hermite ensembles. When appropriately rescaled, the eigenvalues in the interior of the spectrum converge to a translation invariant limiting point process called the Sine_beta process. One expects the Sine_beta process to have a number of points that is roughly the length of the interval times a fixed constant (the density of the process). We find the asymptotic probability of two rare events. The first is a large deviation problem for the density of points in a large interval. The second is the asymptotic probability of overcrowding in a fixed interval.

In this talk, we will be interested in the behavior of beta (Hermite) ensembles in the microscopic regime. Thanks to the tridiagonal model discovered by Dumitriu and Edelman, Ramirez, Rider and Virag (for the edge) and Valko and Virag (for the bulk) were able to characterize the limiting point processes through a family of coupled diffusions. We use this approach to derive the large temperature limit i.e. the regime where the parameter beta tends to zero. Thanks to a careful analysis of the diffusions, we prove the convergence towards Poissonian point processes (joint works with Romain Allez).

The \(L\)-site XXZ spin-1/2 chain is an exactly solvable quantum model of one dimensional condensed matter physics. As such is provides an excellent playground for testing the various manifestations of universality in one dimensional quantum models. For such model, universality manifests itself in the \(L \rightarrow + \infty \) limit and concerns various observables such as the structure of the low-lying spectrum of the model or the large-distance asymptotic behavior of its correlation functions. The XXZ chain is solvable within the Bethe Ansatz approach; its eigenvectors are parametrized by solutions to the Bethe equations, a set of high degree algebraic equations in a number of variables blowing up with \(L\). The whole description of the thermodynamic limit \(L \rightarrow +\infty\) of the model's observables is based on a conjecture due to Hulten in 1938 which stipulates that the roots of the equation associated with the model's ground are real and form a dense distribution in the \(L \rightarrow +\infty\) limit. Generalizations of Hulten's conjecture exist for excited states as well. Variants of Hulten's conjecture adjoined to other techniques allow one to establish various universality results for the chain. In 2009 Dorlas and Samsonov managed to prove the conjecture for some values of the model's coupling constants where it is possible to build the proof on certain convexity arguments. In this talk, I will present the main ideas of the method that I developed so as to prove condensation properties of Bethe roots corresponding to certain classes of solutions to the Bethe equations. The method works independently of the value taken by the coupling constants and appears to be generalisable to many other Bethe Ansatz solvable models. However, first, I shall discuss in more details the nature and history of the problem as well as its connections to universality results. I will also review how the proof of Hulten's conjecture completes the proof of certain multiple integral representations for the correlations functions. These multiple integrals bare certain structural similarities with those of \(\beta\)-ensembles.

In the theory of orthogonal polynomials, some sum rules are remarkable relationships between a functional defined on a subset of all probability measures involving the reverse Kullback-Leibler divergence with respect to a particular distribution and recursion coefficients related to the orthogonal polynomial construction. I will give a short historical introduction, from Szegö until Killip and Simon and descendants. These last authors proved in 2003 a quite surprising sum rule for measures dominating the semicircular distribution on [-2,2]. This sum rule includes a contribution of the atomic part of the measure away from [-2,2]. It is possible to recover this sum rule and to establish new ones by using large deviations of spectral measures in beta-ensembles. These formulas include a contribution of ouliers. The method is robust enough to allow extensions to matrix-valued measures and to measures on the unit circle. (joint work with F. Gamboa and J. Nagel)

We talk about stochastic dynamics whose (unlabeled) equilibrium states are point processes appearing in random matrix theory. These dynamics are called interacting Brownian motions (IBMs). We give various examples of IBMs related to random matrices. For example, sine, Airy, Bessel IBMs in one space dimension, and Ginibre IBM and stochastic dynamics related to zero points of planer Gaussian analytic functions (GAF) in two space dimensions. We construct these except GAF as a pathwise unique strong solution of an infinite dimensional stochastic differential equation (ISDE). Our method is analytic, and based on stochastic analysis. We present a sequence of general theorems to solve ISDEs. We establish a new formulation of solutions of ISDEs in terms of tail sigma-fields of labeled path spaces consisting of trajectories of infinitely many particles. These formulations are equivalent to the original notions of solutions of ISDEs, and more feasible to treat in infinite dimensions. If beta-ensembles satisfy some moment bounds of n-particle approximations, then we can immediately obtain stochastic dynamics using the general theory. When beta=2 and d=1, there exists another construction of stochastic dynamics based on space-time correlation functions, called the algebraic construction. Our method yields the same stochastic dynamics obtained by the algebraic construction. Analytic method gives qualitative information. If we would have a time, we will talk about the following as examples. (1) Girsanov-like formula for tagged particles of IBMs: The resulting IBMs are then locally same as Brownian motions. (2) Dynamical rigidity: Random point fields appearing in random matrix theory have various rigidities. We talk their dynamical counter parts and prove that their global behavior is very rigid and different from Brownian motions. We explain how geometric rigidity yields the dynamical rigidity for interacting Brownian motions in infinite dimensions arising from random matrix theory. This talk is based on the joint work with Hideki Tanemura in Chiba University.

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.