Random matrices: Localization of the eigenvalues and the necessity of four moments

Details Random matrices: Localization of the eigenvalues and the necessity of four moments Random matrices: Localization of the eigenvalues and the necessity of four moments

2010-05-17

arxiv.org/abs/1005.2901v4

Mathematics

Probability

19 pages, one figure, to appear, Acta Math. Vietnamica. A conjectured asymptotic for the dependence on the fourth moment has b

Scientific paper

Consider the eigenvalues $\lambda_i(M_n)$ (in increasing order) of a random Hermitian matrix $M_n$ whose upper-triangular entries are independent with mean zero and variance one, and are exponentially decaying. By Wigner's semicircular law, one expects that $\lambda_i(M_n)$ concentrates around $\gamma_i \sqrt n$, where $\int_{-\infty}^{\gamma_i} \rho_{sc} (x) dx = \frac{i}{n}$ and $\rho_{sc}$ is the semicircular function. In this paper, we show that if the entries have vanishing third moment, then for all $1\le i \le n$ $$\E |\lambda_i(M_n)-\sqrt{n} \gamma_i|^2 = O(\min(n^{-c} \min(i,n+1-i)^{-2/3} n^{2/3}, n^{1/3+\eps})) ,$$ for some absolute constant $c>0$ and any absolute constant $\eps>0$. In particular, for the eigenvalues in the bulk ($\min \{i, n-i\}=\Theta (n)$), $$\E |\lambda_i(M_n)-\sqrt{n} \gamma_i|^2 = O(n^{-c}). $$ \noindent A similar result is achieved for the rate of convergence. As a corollary, we show that the four moment condition in the Four Moment Theorem is necessary, in the sense that if one allows the fourth moment to change (while keeping the first three moments fixed), then the \emph{mean} of $\lambda_i(M_n)$ changes by an amount comparable to $n^{-1/2}$ on the average. We make a precise conjecture about how the expectation of the eigenvalues vary with the fourth moment.

Affiliated with

Also associated with

No associations

Rating

Random matrices: Localization of the eigenvalues and the necessity of four moments does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Random matrices: Localization of the eigenvalues and the necessity of four moments, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Random matrices: Localization of the eigenvalues and the necessity of four moments will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-298525