Local Marchenko–Pastur law for sparse rectangular random matrices

We consider sparse sample covariance matrices with sparsity probability $p_n\ge c_0\log^{\frac2\kappa}n/n$ with $\kappa>0$. Assuming that the distribution of matrix elements has a finite absolute moment of order $4+\delta$, $\delta>0$, it is shown that the distance between the Stieltjes transforms of the empirical spectral distribution function and the Marchenko–Pastur law is of order $\log n(1/(nv)+1/(np_n))$, где where $v$ is the distance to the real axis in the complex plane.