Statistics of the eigenvalues of Tsallis matrices

Abstract

The statistics of the eigenvalues of symmetric random matrices, composed by real and statistically independent elements following the distribution that maximizes Tsallis's entropy, is carried numerically in the limit of large matrices. For entropic indexes in the interval −∞<q< 5 3 , by using a convenient rescale of variables, it is possible to show that such matrices fall in the same class of the Gaussian orthogonal ensemble (GOE). For the entropic index 5 3 ⩽q<3 , the density of eigenvalues and the distribution of level spacings do not seem to follow a simple rescale of variables involving different values of q, and exhibit a behavior very distinct from the GOE: both quantities present long tails for 5 3 ⩽q⩽2 , and such long tails die out when q>2. The density of eigenvalues appears to be symmetric around zero, exhibiting a peak at the origin that becomes steeper for increasing values of q, approaching a delta function at the origin when q→3. The distribution of level spacings displays a form that resembles the well-known Wigner's surmise (augmented by a long tail) for q slightly greater than 5 3 , but gets deformed for increasing values of q, approaching an exponential decay for q>2. For q close to 3, our results resemble those of very sparse random matrices (characterized by many zero matrix elements).