Outliers in spectrum of sparse Wigner matrices - Tikhomirov, Konstantin; Youssef, Pierre (Random structures & algorithms, 2021)
In this paper, we study the effect of sparsity on the appearance of outliers in the semi‐circular law. Let (Wn)n=1∞ be a sequence of random symmetric matrices such that each Wn is n × n with i.i.d. entries above and on the main diagonal equidistributed with the product bnξ, where ξ is a real centered uniformly bounded random variable of unit variance and bn is an independent Bernoulli random variable with a probability of success pn. Assuming that limn→∞npn=∞, we show that for the random sequence (ρn)n=1∞ given by ρn:=θn+npnθn,θn:=max(maxi≤n‖rowi(Wn)‖22−npn,npn), the ratio ‖Wn‖ρn 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 npn=Θ(logn). In particular, denoting by An the adjacency matrix of the Erdős–Renyi graph 𝒢(n,pn) and by λ|k|(An) its kth largest (by the absolute value) eigenvalue, under the assumptions limn→∞npn=∞ and limn→∞pn=0 we have (1) (No non‐trivial outliers): if liminfnpnlogn≥1log(4/e) then for any fixed k ≥ 2, |λ|k|(An)|2npn converges to 1 in probability; and (2) (Outliers): if limsupnpnlogn<1log(4/e) then there is ε > 0 such that for any k∈ℕ, we have limn→∞ℙ|λ|k|(An)|2npn>1+ε=1. 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.