Large deviation theory for diluted Wishart 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 diluted Wishart 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 $\mathcal{I}_N(x)$ smaller than $x\in\mathbb{R}^{+}$, from which all cumulants of $\mathcal{I}_N(x)$ and the rate function $\Psi_{x}(k)$ controlling its large deviation probability $\text{Prob}[\mathcal{I}_N(x)=kN] \asymp e^{-N\Psi_{x}(k)}$ follow. Explicit results for the mean value and the variance of $\mathcal{I}_N(x)$, its rate function, and its third cumulant are discussed and thoroughly compared to numerical diagonalization, showing a very good agreement. The present work establishes the theoretical framework put forward in a recent letter [Phys. Rev. Lett. 117, 104101] as an exact and compelling approach to deal with eigenvalue fluctuations of sparse random matrices.