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      The Limiting Spectral Measure for Ensembles of Symmetric Block Circulant Matrices

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          Abstract

          Given an ensemble of NxN random matrices, a natural question to ask is whether or not the empirical spectral measures of typical matrices converge to a limiting spectral measure as N --> oo. While this has been proved for many thin patterned ensembles sitting inside all real symmetric matrices, frequently there is no nice closed form expression for the limiting measure. Further, current theorems provide few pictures of transitions between ensembles. We consider the ensemble of symmetric m-block circulant matrices with entries i.i.d.r.v. These matrices have toroidal diagonals periodic of period m. We view m as a "dial" we can "turn" from the thin ensemble of symmetric circulant matrices, whose limiting eigenvalue density is a Gaussian, to all real symmetric matrices, whose limiting eigenvalue density is a semi-circle. The limiting eigenvalue densities f_m show a visually stunning convergence to the semi-circle as m tends to infinity, which we prove. In contrast to most studies of patterned matrix ensembles, our paper gives explicit closed form expressions for the densities. We prove that f_m is the product of a Gaussian and a degree 2m-2 polynomial; the formula equals that of the m x m Gaussian Unitary Ensemble (GUE). The proof is by the moments. The new feature, which allows us to obtain closed form expressions, is converting the central combinatorial problem in the moment calculation into an equivalent counting problem in algebraic topology. We end with a generalization of the m-block circulant pattern, dropping the assumption that the m random variables be distinct. We prove that the limiting spectral distribution exists and is determined by the pattern of the independent elements within an m-period, depending on not only the frequency at which each element appears, but also the way the elements are arranged.

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          Statistical Theory of the Energy Levels of Complex Systems. I

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            The Euler characteristic of the moduli space of curves

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              • Record: found
              • Abstract: not found
              • Article: not found

              The expected eigenvalue distribution of a large regular graph

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                Author and article information

                Journal
                27 August 2010
                2011-06-28
                Article
                1008.4812
                fe5fa30d-6079-4d21-aae0-74e1c9eddc97

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                Custom metadata
                39 pages, 10 figures; version 3.1 (includes a new appendix on generalized m-block circulant ensembles)
                math.PR math-ph math.MP

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