Disordered ensembles of random matrices

Abstract

It is shown that the families of generalized matrix ensembles recently considered which give rise to an orthogonal invariant stable Lévy ensemble can be generated by the simple procedure of dividing Gaussian matrices by a random variable. The nonergodicity of this kind of disordered ensembles is investigated. It is shown that the same procedure applied to random graphs gives rise to a family that interpolates between the Erdös-Renyi and the scale free models.

Data-adaptive unfolding of nonergodic spectra: Two-Body Random Ensemble

“…For ξ ≥ 4, there is a further low approach to the ergodic limit, in correspondence with the results of Ref. [15] for the fluctuation measure Σ 2 , obtained after a traditional ensemble unfolding. The expected value of γ = 1 for the ergodic limit is never obtained because of the small ensemble dimensions used in the present calculations N = M = 50, where the power law can be followed only over a very limited range of less than one order of magnitude.…”

“…To fix the ideas, let H G (σ) be a random matrix from GOE (with Dyson index β = 1) of dimension N × N with matrix elements chosen independently from the Gaussian distribution N (µ, σ) with N (0, 1) for the diagonal elements, and N (0, 1/ √ 2) for the nondiagonal elements [22]. A new, so-called disordered random-matrix ensemble H(σ, ξ) can be introduced by imposing an external source of randomness ξ to the fluctuations of the Gaussian matrix [15,16],…”

Heavy-tailed chiral random matrix theory

“…Alternatively one can also consider heavy-tailed matrix ensembles with rotational invariance, at the expense of losing statistical independence of matrix elements. Both directions have been actively pursued [25][26][27][28][29][30][31][32][33][34][35], revealing a plethora of exotic behaviors not seen in Gaussian RMTs.…”

Properties of sparse random matrices over finite fields

“…Random matrices are ubiquitous in many branches of the natural sciences and mathematics ranging from biology to computer science, nuclear physics and quantum chaos. Activity in the area has been recently boosted by the application of techniques originated in the statistical mechanics of disordered systems [1,2,3,4,5,6]. These techniques have been used to analyse ensemble properties, including variational techniques and the replica method, have proved to be valuable tools in several of these approaches.…”