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We study the distribution of the least singular value associated to anensemble of sparse random matrices. Our motivating example is the ensemble of$N\times N$ matrices whose entries are chosen independently from a Bernoullidistribution with parameter $p$. These matrices represent the adjacencymatrices of random Erd\H{o}s--R\'enyi digraphs and are sparse when $p\ll 1$. Weprove that in the regime $pN\gg 1$, the distribution of the least singularvalue is universal in the sense that it is independent of $p$ and equal to thedistribution of the least singular value of a Gaussian matrix ensemble. We alsoprove the universality of the joint distribution of multiple small singularvalues. Our methods extend to matrix ensembles whose entries are chosen fromarbitrary distributions that may be correlated, complex valued, and haveunequal variances.

Universality of the least singular value for sparse random matrices