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 $1\le j<k\le n$, are mutually independent. Assume that the diagonal entries are independent from off-diagonal entries as well. We assume that $\mathbb{E} X_{jk}=0$, $\mathbb{E} X_{jk}^2=1$, for any $j,k=1,\ldots,n$ and $\mathbb{E} X_{jk}X_{kj}=\rho$ for $1\le j<k\le n$. Let $\mathbf M_n$ be a non-random $n\times n$ matrix with $\|\mathbf M_n\|\le Kn^Q$, for some positive constants $K>0$ 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.