On the real spectrum of a product of Gaussian matrices

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Schur function expansion in non-Hermitian ensembles and averages of characteristic polynomials

We study $k$-point correlators of characteristic polynomials in non-Hermitian ensembles of random matrices, focusing on the real, complex and quaternion $N \times N$ Ginibre ensembles. Our approach is based on the technique of character expansions, which expresses the correlator as a sum over partitions involving Schur functions. We show how to re-sum the expansions in terms of representations which interchange the roles of $N$ and $k$. We also provide a probabilistic interpretation of the character expansion analogous to the Schur measure, linking the correlators to the distribution of the top row in certain Young diagrams. In more specific examples we evaluate these expressions explicitly in terms of $k \times k$ determinants or Pfaffians. We show that our approach extends to other ensembles, such as truncations of random unitary matrices.Comment: 35 page

Secular Coefficients and the Holomorphic Multiplicative Chaos

We study the secular coefficients of $N \times N$ random unitary matrices $U_{N}$ drawn from the Circular $\beta$-Ensemble, which are defined as the coefficients of $\{z^n\}$ in the characteristic polynomial $\det(1-zU_{N}^{*})$. When $\beta > 4$ we obtain a new class of limiting distributions that arise when both $n$ and $N$ tend to infinity simultaneously. We solve an open problem of Diaconis and Gamburd by showing that for $\beta=2$, the middle coefficient tends to zero as $N \to \infty$. We show how the theory of Gaussian multiplicative chaos (GMC) plays a prominent role in these problems and in the explicit description of the obtained limiting distributions. We extend the remarkable magic square formula of Diaconis and Gamburd for the moments of secular coefficients to all $\beta>0$ and analyse the asymptotic behaviour of the moments. We obtain estimates on the order of magnitude of the secular coefficients for all $\beta > 0,$ and these estimates are sharp when $\beta \geq 2$. These insights motivated us to introduce a new stochastic object associated with the secular coefficients, which we call Holomorphic Multiplicative Chaos (HMC). Viewing the HMC as a random distribution, we prove a sharp result about its regularity in an appropriate Sobolev space. Our proofs expose and exploit several novel connections with other areas, including random permutations, Tauberian theorems and combinatorics

Moments of random matrices and hypergeometric orthogonal polynomials

We establish a new connection between moments of $n \times n$ random matrices $X_n$ and hypergeometric orthogonal polynomials. Specifically, we consider moments \mathbb{E}\Tr X_n^{-s} as a function of the complex variable $s \in \mathbb{C}$, whose analytic structure we describe completely. We discover several remarkable features, including a reflection symmetry (or functional equation), zeros on a critical line in the complex plane, and orthogonality relations. An application of the theory resolves part of an integrality conjecture of Cunden \textit{et al.}~[F. D. Cunden, F. Mezzadri, N. J. Simm and P. Vivo, J. Math. Phys. 57 (2016)] on the time-delay matrix of chaotic cavities. In each of the classical ensembles of random matrix theory (Gaussian, Laguerre, Jacobi) we characterise the moments in terms of the Askey scheme of hypergeometric orthogonal polynomials. We also calculate the leading order $n\to\infty$ asymptotics of the moments and discuss their symmetries and zeroes. We discuss aspects of these phenomena beyond the random matrix setting, including the Mellin transform of products and Wronskians of pairs of classical orthogonal polynomials. When the random matrix model has orthogonal or symplectic symmetry, we obtain a new duality formula relating their moments to hypergeometric orthogonal polynomials

Large-N expansion for the time-delay matrix of ballistic chaotic cavities

We consider the 1/N-expansion of the moments of the proper delay times for a ballistic chaotic cavity supporting N scattering channels. In the random matrix approach, these moments correspond to traces of negative powers of Wishart matrices. For systems with and without broken time reversal symmetry (Dyson indices β=1 and β=2) we obtain a recursion relation, which efficiently generates the coefficients of the 1/N-expansion of the moments. The integrality of these coefficients and their possible diagrammatic interpretation is discussed

Correlators for the Wigner–Smith time-delay matrix of chaotic cavities

We study the Wigner–Smith time-delay matrix Q of a ballistic quantum dot supporting N scattering channels. We compute the v-point correlators of the power traces Tr Qk for arbitrary v>1 at leading order for large N using techniques from the random matrix theory approach to quantum chromodynamics. We conjecture that the cumulants of the Tr Qkʼs are integer-valued at leading order in N and include a MATHEMATICA code that computes their generating functions recursively

Subcritical multiplicative chaos for regularized counting statistics from random matrix theory

For an N×N random unitary matrix U_N, we consider the random field defined by counting the number of eigenvalues of U_N in a mesoscopic arc of the unit circle, regularized at an N-dependent scale Ɛ_N>0. We prove that the renormalized exponential of this field converges as N → ∞ to a Gaussian multiplicative chaos measure in the whole subcritical phase. In addition, we show that the moments of the total mass converge to a Selberg-like integral and by taking a further limit as the size of the arc diverges, we establish part of the conjectures in [55]. By an analogous construction, we prove that the multiplicative chaos measure coming from the sine process has the same distribution, which strongly suggests that this limiting object should be universal. The proofs are based on the asymptotic analysis of certain Toeplitz or Fredholm determinants using the Borodin-Okounkov formula or a Riemann-Hilbert problem for integrable operators. Our approach to the L¹-phase is based on a generalization of the construction in Berestycki [5] to random fields which are only asymptotically Gaussian. In particular, our method could have applications to other random fields coming from either random matrix theory or a different context

The Sloan Digital Sky Survey Reverberation Mapping Project : sample characterization

Y.S. acknowledges support from an Alfred P. Sloan Research Fellowship and NSF grant AST-1715579. P.H. acknowledges support from the Natural Sciences and Engineering Research Council of Canada (NSERC), funding reference number 2017-05983. W.N.B. acknowledges support from NSF grant AST-1516784. C.J.G., W.N.B., and D.P.S. acknowledge support from NSF grant AST-1517113. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science.We present a detailed characterization of the 849 broad-line quasars from the Sloan Digital Sky Survey Reverberation Mapping (SDSS-RM) project. Our quasar sample covers a redshift range of 0.1 < z < 4.5 and is flux-limited to i PSF < 21.7 without any other cuts on quasar properties. The main sample characterization includes: (1) spectral measurements of the continuum and broad emission lines for individual objects from the coadded first-season spectroscopy in 2014, (2) identification of broad and narrow absorption lines in the spectra, and (3) optical variability properties for continuum and broad lines from multi-epoch spectroscopy. We provide improved systemic redshift estimates for all quasars and demonstrate the effects of the signal-to-noise ratio on the spectral measurements. We compile measured properties for all 849 quasars along with supplemental multi-wavelength data for subsets of our sample from other surveys. The SDSS-RM sample probes a diverse range in quasar properties and shows well-detected continuum and broad-line variability for many objects from first-season monitoring data. The compiled properties serve as the benchmark for follow-up work based on SDSS-RM data. The spectral fitting tools are made public along with this work.Publisher PDFPeer reviewe

Neuroimaging the consciousness of self: Review, and conceptual-methodological framework

We review neuroimaging research investigating self-referential processing (SRP), that is, how we respond to stimuli that reference ourselves, prefaced by a lexical-thematic analysis of words indicative of “self-feelings”. We consider SRP as occurring verbally (V-SRP) and non-verbally (NV-SRP), both in the controlled, “top-down” form of introspective and interoceptive tasks, respectively, as well as in the “bottom-up” spontaneous or automatic form of “mind wandering” and “body wandering” that occurs during resting state. Our review leads us to outline a conceptual and methodological framework for future SRP research that we briefly apply toward understanding certain psychological and neurological disorders symptomatically associated with abnormal SRP. Our discussion is partly guided by William James’ original writings on the consciousness of self

Fluctuations and correlations for products of real asymmetric random matrices

We study the real eigenvalue statistics of products of independent real Ginibre random matrices. These are matrices all of whose entries are real i.i.d. standard Gaussian random variables. For such product ensembles, we demonstrate the asymptotic normality of suitably normalised linear statistics of the real eigenvalues and compute the limiting variance explicitly in both global and mesoscopic regimes. A key part of our proof establishes uniform decorrelation estimates for the related Pfaffian point process, thereby allowing us to exploit weak dependence of the real eigenvalues to give simple and quick proofs of the central limit theorems under quite general conditions. We also establish the universality of these point processes. We compute the asymptotic limit of all correlation functions of the real eigenvalues in the bulk, origin and spectral edge regimes. By a suitable strengthening of the convergence at the edge, we also obtain the limiting fluctuations of the largest real eigenvalue. Near the origin we find new limiting distributions characterising the smallest positive real eigenvalue