The Limiting Spectral Distribution for Sparse Elliptic Random Matrices

Abstract

This paper studies sparse elliptic random matrix models which generalize both the classical elliptic ensembles and sparse i.i.d. matrix models by incorporating correlated entries and a tunable sparsity parameter $p_n$. Each $n\times n$ matrix $X_n$ is formed by entry-wise multiplication of an elliptic random matrix by an elliptic matrix of Bernoulli($p_n$) variables, where $np_n\to\infty$, allowing for interpolation between dense and sparse regimes. The main result establishes that under appropriate normalization, the empirical spectral measures of these matrices converge weakly in probability to the uniform measure on a rotated ellipsoid in the complex plane as the dimension $n$ tends to infinity. Interestingly, the shape of the limiting ellipsoid depends not just on the mirrored entry-wise correlation structure, but also non-trivially on the sparsity limit $p=\lim\limits_{n\to\infty}p_n\in[0,1]$. The main result generalizes and recovers many classical results in sparse and dense regimes for elliptic and i.i.d. random matrix models.