On the largest and the smallest singular value of sparse rectangular random matrices

Abstract

We derive estimates for the largest and smallest singular values of sparse rectangular $N\times n$ random matrices, assuming ${lim}_{N,n\to \mathrm{\infty }}\frac{n}{N}=y\in (0,1)$. We consider a model with sparsity parameter $p_N$ such that $N{p}_N\sim {log}^{\mathit{\alpha }}N$ for some $\mathit{\alpha }>1$, and assume that the moments of the matrix elements satisfy the condition $\mathbb{E}|X_{jk}{|}^{4+\mathit{\delta }}\le C<\mathrm{\infty }$. We assume also that the entries of matrices we consider are truncated at the level ${(N{p}_N)}^{\frac{1}{2}-\mathit{\varkappa }}$ with $\mathit{\varkappa }:=\frac{\mathit{\delta }}{2(4+\mathit{\delta })}$.