On the Largest and the Smallest Singular Value of Sparse Rectangular Random Matrices

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 $N{p}_N\sim {\log }^{\alpha }N$ for some $\alpha >1$, and assume that the moments of the matrix elements satisfy the condition $\mathbf{E}\mathrm{\mid }{X}_{jk}{\mathrm{\mid }}^{4+\delta }\le C<\mathrm{\infty }$. We assume also that the entries of matrices we consider are truncated at the level $(N{p}_N{)}^{\frac{1}{2}-\mathrm{\varkappa }}$ with $\mathrm{\varkappa }:=\frac{\delta }{2(4+\delta )}$.