Theory for the conditioned spectral density of non-invariant random matrices

We develop a theoretical approach to compute the conditioned spectral density of $N \times N$ non-invariant random matrices in the limit $N \rightarrow \infty$. This large deviation observable, defined as the eigenvalue distribution conditioned to have a fixed fraction $k$ of eigenvalues smaller than $x \in \mathbb{R}$, provides the spectrum of random matrix samples that deviate atypically from the average behavior. We apply our theory to sparse random matrices and unveil strikingly new and generic properties, namely: (i) their conditioned spectral density has compact support; (ii) it does not experience any abrupt transition for $k$ around its typical value; (iii) its eigenvalues do not accumulate at $x$. Moreover, our work points towards other types of transitions in the conditioned spectral density for values of $k$ away from its typical value. These properties follow from the weak or absent eigenvalue repulsion in sparse ensembles and they are in sharp contrast to those displayed by classic or rotationally invariant random matrices. The exactness of our theoretical findings are confirmed through numerical diagonalization of finite random matrices.