Random matrix ensembles from nonextensive entropy

Abstract

The classical Gaussian ensembles of random matrices can be constructed by maximizing Boltzmann-Gibbs-Shannon's entropy, SBGS = − dH[P (H)] ln[P (H)], with suitable constraints. Here we construct and analyze random-matrix ensembles arising from the generalized entropyq /(q − 1) (thus S1 = SBGS). The resulting ensembles are characterized by a parameter q measuring the degree of nonextensivity of the entropic form. Making q → 1 recovers the Gaussian ensembles. If q = 1, the joint probability distributions P (H) cannot be factorized, i.e., the matrix elements of H are correlated. In the limit of large matrices two different regimes are observed. When q < 1, P (H) has compact support, and the fluctuations tend asymptotically to those of the Gaussian ensembles. Anomalies appear for q > 1: Both P (H) and the marginal distributions P (Hij) show power-law tails. Numerical analyses reveal that the nearest-neighbor spacing distribution is also long-tailed (not Wigner-Dyson) and, after proper scaling, very close to the result for the 2×2 case -a generalization of Wigner's surmise. We discuss connections of these "nonextensive" ensembles with other non-Gaussian ones, like the so-called Lévy ensembles and those arising from soft-confinement.

Power law deformation of Wishart–Laguerre ensembles of random matrices

“…(2.2). A similar model was introduced earlier generalising the (non-chiral) Wigner-Dyson ensembles for a Gaussian potential [27,29,28], and a similar interplay between the deformation parameter γ and the matrix size N was observed.…”

“…The same trick was used for the generalisation of the Gaussian Wigner-Dyson ensembles introduced previously in [27,28,29]. In fact, a similar technique was employed much earlier in [32] when solving the fixed and restricted trace ensembles by writing them as integral transforms of the Wigner-Dyson ensembles.…”

“…[32,28]. In particular, the large-N behaviour of any spectral property cannot be taken for fixed γ, and a prescription about the way both quantities should approach infinity needs to be given, respecting the inequality eq.…”

“…However, the analysis in [25] was restricted to the macroscopic spectral density, whereas many more interesting results, and ultimately a complete solution of the model can be obtained. This goal is achieved by exploiting techniques and results introduced in previous works, where the WD class was generalised using non-standard entropy maximisation [27,28] and super-statistical approaches [29,30].…”

“…We rederive a generalisation of the semi-circle and MP spectral density, see refs. [27,28,29] and [25] respectively. As a new result we compute the average position of an eigenvalue and the position of a pseudo edge.…”