2009/04/30 by S. Ayadi, Slim Ayadi, O. Khorunzhiy
Mathematics · Physics and Astronomy · #Mathematical analysis #Mathematical physics #Mathematics #Pure mathematics #Random Matrices and Applications #Spectral Theory in Mathematical Physics #Stochastic processes and statistical mechanics #math-ph #math.MP #math.PR #msc:15A52
paper · pdf · doi:10.1016/s0034-4877(10)00016-9
published as Reports on Mathematical Physics, Vol. 65 (2010) 297-335 · 37 pages; corrected and imroved version
arxiv created 2009/05/22 · openalex publication_date 2010/06/01 · arxiv updated 2015/05/13 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We study the spectral properties of the dilute Wigner random real symmetric n-dimensional matrices H such that the entries H(i,j) take zero value with probability 1-p/n. We prove that under rather general conditions on the probability distribution of H(i,j) the semicircle law is valid for the dilute Wigner ensemble in the limit of infinite n and p. In the second part of the paper we study the leading term of the correlation function of the resolvent G(z) of H with large enough Im z in the limit of infinite n and p such that 3/5 log n <log p < log n. We show that this leading term, when considered on the local spectral scale, converges to the same limit as that of the resolvent correlation function of the Wigner ensemble of random matrices. This shows that the moderate dilution of the Wigner ensemble does not alter its universality class.