Large-deviation theory for dilutedWishart random matrices

Abstract

Wishart random matrices with a sparse or diluted structure are ubiquitous in the processing of large datasets, with applications in physics, biology, and economy. In this work, we develop a theory for the eigenvalue fluctuations of dilutedWishart random matrices based on the replica approach of disordered systems.We derive an analytical expression for the cumulant generating function of the number of eigenvalues IN(x) smaller than x ∈ R+, from which all cumulants of IN(x) and the rate function x (k) controlling its large-deviation probability Prob[IN(x) = kN] e −N x (k) follow. Explicit results for themean value and the variance of IN(x), its rate function, and its third cumulant are discussed and thoroughly compared to numerical diagonalization, showing very good agreement. The presentwork establishes the theoretical framework put forward in a recent letter [Phys. Rev. Lett. 117, 104101 (2016)] as an exact and compelling approach to deal with eigenvalue fluctuations of sparse random matrices.