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Abstract:
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We consider the largest eigenvalues of sparse sample covariance matrices, the class of random matrices that includes biadjacency matrices of the bipartite Erd?s-Rényi graph model. We prove that the rescaled, shifted extremal eigenvalues exhibit GOE Tracy-Widom fluctuations under a suitable condition on the sparsity, which is optimal. The result suggests that the sparsity should be taken into consideration in some applications of random matrix theory due to a deterministic shift of the edge from the Marchenko-Pastur law. This is a joint work with Jong Yun Hwang and Ji Oon Lee.
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