Asymptotic Properties of Random Matrices of Long-Range Percolation Model

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dc.creatorAyadi, Slim
dc.date2009-04-18
dc.date.accessioned2026-07-07T13:05:47Z
dc.date.available2026-07-07T13:05:47Z
dc.descriptionWe study the spectral properties of matrices of long-range percolation model. These are N\times N 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 $ψ$ is an even positive function with ψ(t)\le{1} and vanishing at infinity. We study the resolvent G(z)=(H-z)^{-1}, Imz\neq{0} in the limit N,b\to\infty, b=O(N^α), 1/3<α<1 and obtain the explicit expression T(z_{1},z_{2}) for the leading term of the correlation function of the normalized trace of resolvent g_{N,b}(z)=N^{-1}Tr G(z). We show that in the scaling limit of local correlations, this term leads to the expression (Nb)^{-1}T(λ+r_{1}/N+i0,λ+r_{2}/N-i0)= b^{-1}\sqrt{N}|r_{1}-r_{2}|^{-3/2}(1+o(1)) found earlier by other authors for band random matrix ensembles. This shows that the ratio $b^{2}/N$ is the correct scale for the eigenvalue density correlation function and that the ensemble we study and that of band random matrices belong to the same class of spectral universality.
dc.descriptionNo comments
dc.identifierhttps://arxiv.org/abs/0904.2837
dc.identifierhttp://arxiv.org/abs/0904.2837
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/227603
dc.subjectMathematical Physics
dc.subjectProbability
dc.subject15A52, 45B85, 60F99
dc.titleAsymptotic Properties of Random Matrices of Long-Range Percolation Model
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