Spectral statistics and dynamics of Lévy matrices

We study the spectral statistics and dynamics of a random matrix model where matrix elements are taken from power-law tailed distributions. Such distributions, labeled by a parameter mu, converge on the Lévy basin, giving the matrix model the label "Lévy matrix" [P. Cizeau and J. P. Bouchaud, Phys. Rev. E 50, 1810 (1994)]. Such matrices are interesting because their properties go beyond the Gaussian universality class and they model many physically relevant systems such as spin glasses with dipolar or Ruderman-Kittel-Kasuya-Yosida interactions, electronic systems with power-law decaying interactions, and the spectral behavior at the metal insulator transition. Regarding the density of states we extend previous work to reveal the sparse matrix limit as mu-->0. Furthermore, we find for 2 x 2 Lévy matrices that geometrical level repulsion is not affected by the distribution's broadness. Nevertheless, essential singularities particular to Lévy distributions for small arguments break geometrical repulsion and make it mu dependent. Level dynamics as a function of a symmetry breaking parameter gives new insight into the phases found by Cizeau and Bouchaud (CB). We map the phase diagram drawn qualitatively by CB by using the delta3 statistic. Finally we compute the conductance of each phase by using the Thouless formula, and find that the mixed phase separating conducting and insulating phases has a unique character.