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Journal of Statistical Physics

, Volume 175, Issue 2, pp 384–401 | Cite as

Proof of a Conjecture on the Infinite Dimension Limit of a Unifying Model for Random Matrix Theory

  • Mario PerniciEmail author
  • Giovanni M. Cicuta
Article
  • 44 Downloads

Abstract

We study the large N limit of a sparse random block matrix ensemble. It depends on two parameters: the average connectivity Z and the size of the blocks d, which is the dimension of an euclidean space. In the limit of large d with \(\frac{Z}{d}\) fixed, we prove the conjecture that the spectral distribution of the sparse random block matrix converges in the case of the Adjacency block matrix to the one of the effective medium approximation, in the case of the Laplacian block matrix to the Marchenko-Pastur distribution. We extend previous analytical computations of the moments of the spectral density of the Adjacency block matrix and the Lagrangian block matrix, valid for all values of Z and d.

Keywords

Random matrix theory Random trees Block matrix Moments method 

Notes

Acknowledgements

One of us (G. M. C.) thanks Alessio Zaccone for introducing him to the sparse random block model analyzed in this paper and Giorgio Parisi for encouraging an automated evaluation of the moments.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Istituto Nazionale di Fisica Nucleare, Sezione di MilanoMilanoItaly
  2. 2.Dip. Fisica, Università di ParmaParmaItaly

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