Correlation functions in disordered systems

Breakdown of Random-Matrix Universality in Persistent Lotka-Volterra Communities

The eigenvalue spectrum of a random matrix often only depends on the first and second moments of its elements, but not on the specific distribution from which they are drawn. The validity of this universality principle is often assumed without proof in applications. In this Letter, we offer a pertinent counterexample in the context of the generalized Lotka-Volterra equations. Using dynamic mean-field theory, we derive the statistics of the interactions between species in an evolved ecological community. We then show that the full statistics of these interactions, beyond those of a Gaussian ensemble, are required to correctly predict the eigenvalue spectrum and therefore stability. Consequently, the universality principle fails in this system. We thus show that the eigenvalue spectra of random matrices can be used to deduce the stability of "feasible" ecological communities, but only if the emergent non-Gaussian statistics of the interactions between species are taken into account.

Eigenvalues of Random Matrices with Generalized Correlations: A Path Integral Approach

Random matrix theory allows one to deduce the eigenvalue spectrum of a large matrix given only statistical information about its elements. Such results provide insight into what factors contribute to the stability of complex dynamical systems. In this Letter, we study the eigenvalue spectrum of an ensemble of random matrices with correlations between any pair of elements. To this end, we introduce an analytical method that maps the resolvent of the random matrix onto the response functions of a linear dynamical system. The response functions are then evaluated using a path integral formalism, enabling us to make deductions about the eigenvalue spectrum. Our central result is a simple, closed-form expression for the leading eigenvalue of a large random matrix with generalized correlations. This formula demonstrates that correlations between matrix elements that are not diagonally opposite, which are often neglected, can have a significant impact on stability.

Universal Entanglement of Typical States in Constrained Systems

Constraints play an important role in the entanglement dynamics of many quantum systems. We develop a diagrammatic formalism to exactly evaluate the entanglement spectrum of random pure states in large constrained Hilbert spaces. The resulting spectra may be classified into universal "phases" depending on their singularities. The simplest class of local constraints reveals a nontrivial phase diagram with a Marchenko-Pastur phase which terminates in a critical point with new singularities. We propose a certain quantum defect chain as a microscopic realization of the critical point. The much studied Rydberg-blockaded or Fibonacci chain lies in the Marchenko-Pastur phase with a modified Page correction to the entanglement entropy. Our results predict the entanglement of infinite temperature eigenstates in thermalizing constrained Floquet spin chains, as we confirm numerically.

Modification of the Porter-Thomas Distribution by Rank-One Interaction

The Porter-Thomas (PT) distribution of resonance widths is one of the oldest and simplest applications of statistical ideas in nuclear physics. Previous experimental data confirmed it quite well, but recent and more careful investigations show clear deviations from this distribution. To explain these discrepancies, Volya, Weidenmüller, and Zelevinsky [Phys. Rev. Lett. 115, 052501 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.052501] argued that to get a realistic model of nuclear resonances is not enough to consider one of the standard random matrix ensembles which leads immediately to the PT distribution, but it is necessary to add a rank-one interaction which couples resonances to decay channels. The purpose of this Letter is to solve this model analytically and to find explicitly the modifications of the PT distribution due to such an interaction. Resulting formulas are simple, in good agreement with numerics, and could explain experimental results.

Eigenvalue spectra of large correlated random matrices

Using the diagrammatic method, we derive a set of self-consistent equations that describe eigenvalue distributions of large correlated asymmetric random matrices. The matrix elements can have different variances and be correlated with each other. The analytical results are confirmed by numerical simulations. The results have implications for the dynamics of neural and other biological networks where plasticity induces correlations in the connection strengths within the network. We find that the presence of correlations can have a major impact on network stability.

Parametric number covariance in quantum chaotic spectra

We study spectral parametric correlations in quantum chaotic systems and introduce the number covariance as a measure of such correlations. We derive analytic results for the classical random matrix ensembles using the binary correlation method and obtain compact expressions for the covariance. We illustrate the universality of this measure by presenting the spectral analysis of the quantum kicked rotors for the time-reversal invariant and time-reversal noninvariant cases. A local version of the parametric number variance introduced earlier is also investigated.

Universal covariance formula for linear statistics on random matrices

We derive an analytical formula for the covariance cov(A,B) of two smooth linear statistics A=[under ∑]ia(λ_{i}) and B=[under ∑]ib(λ_{i}) to leading order for N→∞, where {λ_{i}} are the N real eigenvalues of a general one-cut random-matrix model with Dyson index β. The formula, carrying the universal 1/β prefactor, depends on the random-matrix ensemble only through the edge points [λ_{-},λ_{+}] of the limiting spectral density. For A=B, we recover in some special cases the classical variance formulas by Beenakker and by Dyson and Mehta, clarifying the respective ranges of applicability. Some choices of a(x) and b(x) lead to a striking decorrelation of the corresponding linear statistics. We provide two applications-the joint statistics of conductance and shot noise in ideal chaotic cavities, and some new fluctuation relations for traces of powers of random matrices.

Spectral statistics of instantaneous normal modes in liquids and random matrices

We study the statistical properties of eigenvalues of the Hessian matrix H (matrix of second derivatives of the potential energy) for a classical atomic liquid, and compare these properties with predictions for random matrix models. The eigenvalue spectra (the instantaneous normal mode or INM spectra) are evaluated numerically for configurations generated by molecular dynamics simulations. We find that distribution of spacings between nearest-neighbor eigenvalues, s, obeys quite well the Wigner prediction s exp(-s(2)), with the agreement being better for higher densities at fixed temperature. The deviations display a correlation with the number of localized eigenstates (normal modes) in the liquid; there are fewer localized states at higher densities that we quantify by calculating the participation ratios of the normal modes. We confirm this observation by calculating the spacing distribution for parts of the INM spectra with high participation ratios, obtaining greater conformity with the Wigner form. We also calculate the spectral rigidity and find a substantial dependence on the density of the liquid.