Outliers in spectrum of sparse Wigner matrices

In this paper, we study the effect of sparsity on the appearance of outliers in the semi‐circular law. Let be a sequence of random symmetric matrices such that each W n is n × n with i.i.d. entries above and on the main diagonal equidistributed with the product , where is a real centered uniformly bounded random variable of unit variance and b n is an independent Bernoulli random variable with a probability of success p n . Assuming that , we show that for the random sequence given by , the ratio converges to one in probability. A noncentered counterpart of the theorem allows to obtain asymptotic expressions for eigenvalues of the Erdős–Renyi graphs, which were unknown in the regime . In particular, denoting by A n the adjacency matrix of the Erdős–Renyi graph and by its k th largest (by the absolute value) eigenvalue, under the assumptions and we have (1) (No non‐trivial outliers): if then for any fixed k ≥ 2 , converges to 1 in probability; and (2) (Outliers): if then there is ε > 0 such that for any , we have . On a conceptual level, our result reveals similarities in appearance of outliers in spectrum of sparse matrices and the so‐called BBP phase transition phenomenon in deformed Wigner matrices.