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We consider spectral properties of sparse sample covariance matrices, whichincludes biadjacency matrices of the bipartite Erd\H{o}s-R\'enyi graph model.We prove a local law for the eigenvalue density up to the upper spectral edge.Under a suitable condition on the sparsity, we also prove that the limitingdistribution of the rescaled, shifted extremal eigenvalues is given by the GOETracy-Widom law with an explicit formula on the deterministic shift of thespectral edge. For the biadjacency matrix of an Erd\H{o}s-R\'enyi graph withtwo vertex sets of comparable sizes $M$ and $N$, this establishes Tracy-Widomfluctuations of the second largest eigenvalue when the connection probability$p$ is much larger than $N^{-2/3}$ with a deterministic shift of order$(Np)^{-1}$.

Local law and Tracy-Widom limit for sparse sample covariance matrices