Semicircle Law for Random Matrices of Long-Range Percolation Model
| dc.creator | Slim, Ayadi | |
| dc.date | 2008-06-27 | |
| dc.date.accessioned | 2026-07-07T09:47:10Z | |
| dc.date.available | 2026-07-07T09:47:10Z | |
| dc.description | We study the normalized eigenvalue counting measure dσof matrices of long-range percolation model. These are (2n+1)\times (2n+1) random real symmetric matrices H=\{H(i,j)\}_{i,j} whose elements are independent random variables taking zero value with probability 1-ψ[(i-j)/b], b\in \mathbb{R}^{+}, where ψis an even positive function ψ(t)\le{1} vanishing at infinity. It is shown that if the third moment of \sqrt{b}H(i,j), i\leq{j} is uniformly bounded then the measure dσ:=dσ_{n,b} weakly converges in probability in the limit n,b\to\infty, b=o(n) to the semicircle (or Wigner) distribution. The proof uses the resolvent technique combined with the cumulant expansions method. We show that the normalized trace of resolvent g_{n,b}(z) converges in average and that the variance of g_{n,b}(z) vanishes. In the second part of the paper, we estimate the rate of decreasing of the variance of g_{n,b}(z), under further conditions on the moments of \sqrt{b}H(i,j), \ i\le{j}. | |
| dc.description | The author is grateful to Prof. Dr. O. Khorunzhy at University of Versailles (France), where present paper was completed, who proposed use to study the problems described in this paper | |
| dc.identifier | https://arxiv.org/abs/0806.4497 | |
| dc.identifier | http://arxiv.org/abs/0806.4497 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/163782 | |
| dc.subject | Probability | |
| dc.subject | Mathematical Physics | |
| dc.title | Semicircle Law for Random Matrices of Long-Range Percolation Model | |
| dc.type | text |