Universality of the Tearing Phase in Matrix Models

The spontaneous symmetry breaking associated to the tearing of a random surface, where large dynamical holes fill the surface, was recently analized obtaining a non-universal critical exponent on a border phase. Here the issue of universality is explained by an independent analysis. The one hole sector of the model is useful to manifest the origin of the (limited) non-universal behaviour, that is the existence of two inequivalent critical points.Comment: 9 pages, 1 figure non include

Real symmetric random matrices and replicas

Various ensembles of random matrices with independent entries are analyzed by the replica formalism in the large-N limit. A result on the Laplacian random matrix with Wigner-rescaling is generalized to arbitrary probability distribution.Comment: 17 page

Unifying model for random matrix theory in arbitrary space dimensions.

A sparse random block matrix model suggested by the Hessian matrix used in the study of elastic vibrational modes of amorphous solids is presented and analyzed. By evaluating some moments, benchmarked against numerics, differences in the eigenvalue spectrum of this model in different limits of space dimension d, and for arbitrary values of the lattice coordination number Z, are shown and discussed. As a function of these two parameters (and their ratio Z/d), the most studied models in random matrix theory (Erdos-Renyi graphs, effective medium, and replicas) can be reproduced in the various limits of block dimensionality d. Remarkably, the Marchenko-Pastur spectral density (which is recovered by replica calculations for the Laplacian matrix) is reproduced exactly in the limit of infinite size of the blocks, or d→∞, which clarifies the physical meaning of space dimension in these models. We feel that the approximate results for d=3 provided by our method may have many potential applications in the future, from the vibrational spectrum of glasses and elastic networks to wave localization, disordered conductors, random resistor networks, and random walks

Real symmetric random matrices and paths counting

Exact evaluation of $$ is here performed for real symmetric matrices $S$ of arbitrary order $n$, up to some integer $p$, where the matrix entries are independent identically distributed random variables, with an arbitrary probability distribution. These expectations are polynomials in the moments of the matrix entries ; they provide useful information on the spectral density of the ensemble in the large $n$ limit. They also are a straightforward tool to examine a variety of rescalings of the entries in the large $n$ limit.Comment: 23 pages, 10 figures, revised pape

Probability density of determinants of random matrices

In this brief paper the probability density of a random real, complex and quaternion determinant is rederived using singular values. The behaviour of suitably rescaled random determinants is studied in the limit of infinite order of the matrices

Yang-Mills Integrals

Two results are presented for reduced Yang-Mills integrals with different symmetry groups and dimensions: the first is a compact integral representation in terms of the relevant variables of the integral, the second is a method to analytically evaluate the integrals in cases of low order. This is exhibited by evaluating a Yang-Mills integral over real symmetric matrices of order 3.Comment: LaTeX, 10 pages, references added and minimal change

Sparse block-structured random matrices: universality

We study ensembles of sparse block-structured random matrices generated from the adjacency matrix of a Erdös–Renyi random graph with N vertices of average degree Z , inserting a real symmetric d  ×  d random block at each non-vanishing entry. We consider several ensembles of random block matrices with rank r  <  d and with maximal rank, r  =  d . The spectral moments of the sparse block-structured random matrix are evaluated for $N \to \infty$ , d finite or infinite, and several probability distributions for the blocks (e.g. fixed trace, bounded trace and Gaussian). Because of the concentration of the probability measure in the $d \to \infty$ limit, the spectral moments are independent of the probability measure of the blocks (with mild assumptions of isotropy, smoothness and sub-Gaussian tails). The effective medium approximation is the limiting spectral density of the sparse block-structured random ensembles with finite rank. Analogous classes of universality hold for the Laplacian sparse block-structured ensemble. The same limiting distributions are obtained using random regular graphs instead of Erdös–Renyi graphs

Patterns of synchronization in the hydrodynamic coupling of active colloids

International audienceA system of active colloidal particles driven by harmonic potentials to oscillate about the vertices of a regular polygon, with hydrodynamic coupling between all particles, is described by a piecewise linear model which exhibits various patterns of synchronization. Analytical solutions are obtained for this class of dynamical systems. Depending only on the number of particles, the synchronization occurs into states in which nearest neighbors oscillate in in-phase, antiphase, or phase-locked (time-shifted) trajectories