On minimal singular values of random matrices with correlated entries

Let $\mathbf X$ be a random matrix whose pairs of entries $X_{jk}$ and $X_{kj}$ are correlated and vectors $ (X_{jk},X_{kj})$, for $1\le j 0$ and $Q\ge 0$. Let $s_n(\mathbf X+\mathbf M_n)$ denote the least singular value of the matrix $\mathbf X+\mathbf M_n$. It is shown that there exist positive constants $A$ and $B$ depending on $K,Q,\rho$ only such that $$ \mathbb{P}(s_n(\mathbf X+\mathbf M_n)\le n^{-A})\le n^{-B}. $$ As an application of this result we prove the elliptic law for this class of matrices with non identically distributed correlated entries.