Bose–Einstein-like condensation of deformed random matrix: a replica approach

Abstract

In this work, we investigate a symmetric deformed random matrix, which is obtained by perturbing the diagonal elements of the Wigner matrix. The eigenvector of the minimal eigenvalue of the deformed random matrix tends to condensate at a single site. In certain types of perturbations and in the limit of the large components, this condensation becomes a sharp phase transition, the mechanism of which can be identified with the Bose–Einstein condensation in a mathematical level. We study this Bose–Einstein like condensation phenomenon by means of the replica method. We first derive a formula to calculate the minimal eigenvalue and the statistical properties of . Then, we apply the formula for two solvable cases: when the distribution of the perturbation has the double peak, and when it has a continuous distribution. For the double peak, we find that at the transition point, the participation ratio changes discontinuously from a finite value to zero. On the contrary, in the case of a continuous distribution, the participation ratio goes to zero either continuously or discontinuously, depending on the distribution.