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We prove a central limit theorem for the real part of the logarithm of the characteristic polynomial of random Jacobi matrices. Our results cover the G β E models for β > 0 . [ABSTRACT FROM AUTHOR]

We study in this paper lower bounds for the generalization error of models derived from multi-layer neural networks, in the regime where the size of the layers is commensurate with the number of samples in the training data. We derive explicit generalization lower bounds for general biased estimators, in the cases of two-layered networks. For linear activation function, the bound is asymptotically tight. In the nonlinear case, we provide a comparison of our bounds with an empirical study of the stochastic gradient descent algorithm. In addition, we derive bounds for unbiased estimators, which show that the latter have unacceptable performance for truly nonlinear networks. The analysis uses elements from the theory of large random matrices. [ABSTRACT FROM AUTHOR]

Let PN be a uniform random N × N permutation matrix and let χN(z) = det(zIN − PN) denote its characteristic polynomial. We prove a law of large numbers for the maximum modulus of χN on the unit circle, specifically, sup∣z∣=1∣χNz∣=Nx0+o1with probability tending to 1 as N → ∞, for a numerical constant x0 ≈ 0.652. The main idea of the proof is to uncover a logarithmic correlation structure for the distribution of (the logarithm of) χN, viewed as a random field on the circle, and to adapt a well‐known second‐moment argument for the maximum of the branching random walk. Unlike the well‐studied CUE field in which PN is replaced with a Haar unitary, the distribution of χN(e2πit) is sensitive to Diophantine properties of the point t. To deal with this we borrow tools from the Hardy‐Littlewood circle method. © 2020 Wiley Periodicals LLC [ABSTRACT FROM AUTHOR]

We present estimates on the small singular values of a class of matrices with independent Gaussian entries and inhomogeneous variance profile, satisfying a broad-connectedness condition. Using these estimates and concentration of measure for the spectrum of Gaussian matrices with independent entries, we prove that for a large class of graphs satisfying an appropriate expansion property, the Barvinok-Godsil-Gutman estimator for the permanent achieves sub-exponential errors with high probability. [ABSTRACT FROM AUTHOR]

We consider the regularization of matrices MN in Jordan form by additive Gaussian noise N-γ GN, where GN is a matrix of i.i.d. standard Gaussians and γ > 1/2 so that the operator norm of the additive noise tends to 0 with N. Under mild conditions on the structure of MN, we evaluate the limit of the empirical measure of eigenvalues of MN + N-γ GN and show that it depends on γ , in contrast with the case of a single Jordan block. [ABSTRACT FROM AUTHOR]

We discuss regularization by noise of the spectrum of large random non-normal matrices. Under suitable conditions, we show that the regularization of a sequence of matrices that converges in *-moments to a regular element a by the addition of a polynomially vanishing Gaussian Ginibre matrix forces the empirical measure of eigenvalues to converge to the Brown measure of a. [ABSTRACT FROM AUTHOR]

In this paper, we apply the theory of random Schrödinger operators to the analysis of multiusers communication channels similar to the Wyner model, which are characterized by short-range intercell interference. With H the channel transfer matrix, HH^\dagger is a narrow band matrix, a fact that does not permit the use of classical random matrices theory. On the other hand, HH^\dagger is in many aspects similar to a random Schrödinger operator. We relate the per-cell sum-rate capacity of the channel to the integrated density of states of a random Schrödinger operator; the latter is then related to the top Lyapunov exponent of a random sequence of matrices via a version of the Thouless formula. We also derive several bounds on the limiting per-cell sum-rate capacity, some based on the theory of random Schrödinger operators, and some derived from information theoretical considerations. Finally, we get explicit results in the high-signal-to-noise ratio (SNR) regime for some particular cases. [ABSTRACT FROM PUBLISHER]

In this paper we study the spectrum of certain large random Hermitian Jacobi matrices. These matrices are known to describe certain communication setups. In particular, we are interested in an uplink cellular channel which models mobile users experiencing a soft-handoff situation under joint multicell decoding. Considering rather general fading statistics we provide a closed-form expression for the per-cell sum—rate of this channel in high signal-to-noise ratio (SNR), when an intra-cell time-division multiple-access (TDMA) protocol is employed. Since the matrices of interest are tridiagonal, their eigenvectors can be considered as sequences with second-order linear recurrence. Therefore, the problem is reduced to the study of the exponential growth of products of two-by-two matrices. For the case where K users are simultaneously active in each cell, we obtain a series of lower and upper bound on the high-SNR power offset of the per-cell sum-rate, which are considerably tighter than previously known bounds. [ABSTRACT FROM AUTHOR]

We correct an omission in Guionnet and Zeitouni (J. Funct. Anal. 188 (2002) 461) and improve the main result there to a full-large deviations trajectorial result. [Copyright &y& Elsevier]

Consider the spherical integral I(β)N(DN, EN)≔∫ exp{N tr(UDNU*EN)} dmβN(U), where mβN denote the Haar measure on the orthogonal group ON when β=1 and on the unitary group UN when β=2, and DN, EN are diagonal real matrices whose spectral measures converge to μD, μE. In this paper we prove the existence and represent as solution to a variational problem the limit I(β)(μD, μE)≔lim N−2 log I(β)N(DN, EN). This limit appears in so-called “matrix models” but also in the evaluation of large deviations of the spectral measure of generalized Wishart matrices. Our technique is based on stochastic calculus, large deviations, and elements from free probability. [Copyright &y& Elsevier]