On High Moments and the Spectral Norm of Large Dilute Wigner Random Matrices ∗†‡

We consider a dilute version of the Wigner ensemble of n × n random real symmetric matrices H(n,ρ), where ρ denotes the average number of non-zero elements per row. We study the asymptotic properties of the spectral norm ‖H(n,ρn) ‖ in the limit of infinite n with sn = O(n2/3) and ρn = n2/3(1+ε), ε> 0. Our main result is that the probability P ‖H(n,ρn) ‖> 1 + xn−2/3, x> 0 is bounded for any ε ∈ (ε0, 1/2], ε0> 0 by an expression that does not de-pend on the particular values of the first several moments V2l, 2 ≤ l ≤ 6 + φ0, φ0 = φ(ε0) of the matrix elements of H (n,ρ) provided they exist and the prob-ability distribution of the matrix elements is symmetric. The proof is based on the detailed study of the upper bound of the moments of random matrices with truncated random variables Ln = E {Tr (Ĥ(n,ρn))2sn}. We also consider the lower bound of Ln and show that in the complementary asymptotic regime, when ρn = n with ∈ (0, 2/3], the fourth moment V4 is involved into the estimates and the scale n−2/3 at the border of the limiting spectrum should be replaced by ρ−1n