On statistics of bi-orthogonal eigenvectors in real and complex Ginibre ensembles: combining partial Schur decomposition with supersymmetry

We suggest a method of studying the joint probability density (JPD) of an eigenvalue and the associated 'non-orthogonality overlap factor' (also known as the 'eigenvalue condition number') of the left and right eigenvectors for non-selfadjoint Gaussian random matrices of size $N\times N$. First we derive the general finite $N$ expression for the JPD of a real eigenvalue $\lambda$ and the associated non-orthogonality factor in the real Ginibre ensemble, and then analyze its 'bulk' and 'edge' scaling limits. The ensuing distribution is maximally heavy-tailed, so that all integer moments beyond normalization are divergent. A similar calculation for a complex eigenvalue $z$ and the associated non-orthogonality factor in the complex Ginibre ensemble is presented as well and yields a distribution with the finite first moment. Its 'bulk' scaling limit yields a distribution whose first moment reproduces the well-known result of Chalker and Mehlig \cite{ChalkerMehlig1998}, and we provide the 'edge' scaling distribution for this case as well. Our method involves evaluating the ensemble average of products and ratios of integer and half-integer powers of characteristic polynomials for Ginibre matrices, which we perform in the framework of a supersymmetry approach. Our paper complements recent studies by Bourgade and Dubach \cite{BourgadeDubach}.Comment: published versio

A spin glass model for reconstructing nonlinearly encrypted signals corrupted by noise

An encryption of a signal ${\bf s}\in\mathbb{R^N}$ is a random mapping ${\bf s}\mapsto \textbf{y}=(y_1,\ldots,y_M)^T\in \mathbb{R}^M$ which can be corrupted by an additive noise. Given the Encryption Redundancy Parameter (ERP) $\mu=M/N\ge 1$, the signal strength parameter $R=\sqrt{\sum_i s_i^2/N}$, and the ('bare') noise-to-signal ratio (NSR) $\gamma\ge 0$, we consider the problem of reconstructing ${\bf s}$ from its corrupted image by a Least Square Scheme for a certain class of random Gaussian mappings. The problem is equivalent to finding the configuration of minimal energy in a certain version of spherical spin glass model, with squared Gaussian-distributed random potential. We use the Parisi replica symmetry breaking scheme to evaluate the mean overlap $p_{\infty}\in [0,1]$ between the original signal and its recovered image (known as 'estimator') as $N\to \infty$, which is a measure of the quality of the signal reconstruction. We explicitly analyze the general case of linear-quadratic family of random mappings and discuss the full $p_{\infty} (\gamma)$ curve. When nonlinearity exceeds a certain threshold but redundancy is not yet too big, the replica symmetric solution is necessarily broken in some interval of NSR. We show that encryptions with a nonvanishing linear component permit reconstructions with $p_{\infty}>0$ for any $\mu>1$ and any $\gamma<\infty$, with $p_{\infty}\sim \gamma^{-1/2}$ as $\gamma\to \infty$. In contrast, for the case of purely quadratic nonlinearity, for any ERP $\mu>1$ there exists a threshold NSR value $\gamma_c(\mu)$ such that $p_{\infty}=0$ for $\gamma>\gamma_c(\mu)$ making the reconstruction impossible. The behaviour close to the threshold is given by $p_{\infty}\sim (\gamma_c-\gamma)^{3/4}$ and is controlled by the replica symmetry breaking mechanism.Comment: 33 pages, 5 figure

Spectra of Random Matrices Close to Unitary and Scattering Theory for Discrete-Time Systems

We analyze statistical properties of complex eigenvalues of random matrices $\hat{A}$ close to unitary. Such matrices appear naturally when considering quantized chaotic maps within a general theory of open linear stationary systems with discrete time. Deviation from unitarity are characterized by rank $M$ and eigenvalues $T_i, i=1,...,M$ of the matrix $\hat{T}=\hat{{\bf 1}}-\hat{A}^{\dagger}\hat{A}$. For the case M=1 we solve the problem completely by deriving the joint probability density of eigenvalues and calculating all $n-$ point correlation functions. For a general case we present the correlation function of secular determinants.Comment: 4 pages, latex, no figures, a few misprints are correcte

High-Dimensional Random Fields and Random Matrix Theory

Our goal is to discuss in detail the calculation of the mean number of stationary points and minima for random isotropic Gaussian fields on a sphere as well as for stationary Gaussian random fields in a background parabolic confinement. After developing the general formalism based on the high-dimensional Kac-Rice formulae we combine it with the Random Matrix Theory (RMT) techniques to perform analysis of the random energy landscape of $p-$spin spherical spinglasses and a related glass model, both displaying a zero-temperature one-step replica symmetry breaking glass transition as a function of control parameters (e.g. a magnetic field or curvature of the confining potential). A particular emphasis of the presented analysis is on understanding in detail the picture of "topology trivialization" (in the sense of drastic reduction of the number of stationary points) of the landscape which takes place in the vicinity of the zero-temperature glass transition in both models. We will reveal the important role of the GOE "edge scaling" spectral region and the Tracy-Widom distribution of the maximal eigenvalue of GOE matrices for providing an accurate quantitative description of the universal features of the topology trivialization scenario.Comment: 40 pages; 2 figures; In this version the original lecture notes are converted to an article format, new Eqs. (82)-(85) and Appendix about anisotropic fields added, noticed misprints corrected, references updated. references update

The Spectral Autocorrelation Function in Weakly Open Chaotic Systems: Indirect Photodissociation of Molecules

We derive the statistical limit of the spectral autocorrelation function and of the survival probability for the indirect photodissociation of molecules in the regime of non-overlapping resonances. The results are derived in the framework of random matrix theory, and hold more generally for any chaotic quantum system that is weakly coupled to the continuum. The "correlation hole" that characterizes the spectral autocorrelation in the bound molecule diminishes as the typical average total width of a resonance increases.Comment: 13 pages, 1 Postscript figure included, RevTe

On Random Matrix Averages Involving Half-Integer Powers of GOE Characteristic Polynomials

Correlation functions involving products and ratios of half-integer powers of characteristic polynomials of random matrices from the Gaussian Orthogonal Ensemble (GOE) frequently arise in applications of Random Matrix Theory (RMT) to physics of quantum chaotic systems, and beyond. We provide an explicit evaluation of the large-$N$ limits of a few non-trivial objects of that sort within a variant of the supersymmetry formalism, and via a related but different method. As one of the applications we derive the distribution of an off-diagonal entry $K_{ab}$ of the resolvent (or Wigner $K$-matrix) of GOE matrices which, among other things, is of relevance for experiments on chaotic wave scattering in electromagnetic resonators.Comment: 25 pages (2 figures); published version (conclusion added, minor changes