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

We consider a dilute version of the Wigner ensemble of $n\times n$ random real symmetric matrices $H^{(n,\rho )}$, where $\rho$ denotes an average number of non-zero elements per row. We study the asymptotic properties of the spectral norm $\Vert H^{(n,\rho_n)}\Vert$ in the limit of infinite $n$ with $\rho_n = n^{2/3(1+\varepsilon)}$, $\varepsilon>0$. Our main result is that the probability $\mathbf{P}\left\{ \Vert H^{(n,\rho_n)} \Vert > 1+x n^{-2/3}\right\}$, $x>0$ is bounded for any $\varepsilon \in (\varepsilon_0, 1/2]$, $\varepsilon_0>0$ by an expression that does not depend on the particular values of the first several moments $V_{2l}, 2\le l\le 6$ and $V_{12+2\mathbf{P}hi_0}$, $\phi_0=\phi(\varepsilon_0)$ of the matrix elements of $H^{(n,\rho)}$ provided they exist and the probability distribution of the matrix elements is symmetric. The proof is based on the study of the upper bound of the averaged moments of random matrices with truncated random variables $ \mathbf{E}\{ \mathrm{Tr} (\hat H^{(n,\rho_n)})^{2s_n}\}$, $s_n = \lfloor \chi n^{2/3}\rfloor$ with $\chi>0$, in the limit $n\to\infty$.
We also consider the lower bound of $\mathbf{E}\{ \mathrm{Tr} ( H^{(n,\rho_n)})^{2s_n}\}$ and show that in the complementary asymptotic regime, when $\rho_n = n^\epsilon$ with $ \epsilon\in(0, 2/3]$ and $n\to\infty$, the fourth moment $V_4$ enters the estimates from below and the scaling variable $n^{-2/3}$ at the border of the limiting spectrum is to be replaced by a variable related with $\rho_n^{-1}$.