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On High Moments and the Spectral Norm of Large Dilute Wigner Random Matrices
O. Khorunzhiy Laboratoire de Mathématiques Université de Versailles-Saint-Quentin, 45, Avenue des Etats-Unis 78035 Versailles, France
Abstract:
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
$\rho_n^{-1}$.
Key words and phrases:
random matrices, Wigner ensemble, dilute random matrices, spectral norm.
MSC: 15B52 Received: 08.08.2011
Revised: 17.07.2013
Language: English
DOI:
10.15407/mag10.01.064
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