Abstract
A numerical study is presented for the eigensolution statistics of largeN×N real and symmetric sparse random matrices as a function of the mean numberp of nonzero elements per row. The model shows classical percolation and quantum localization transitions atp c =1 andp q >1, respectively. In the rigid limitp=N we demonstrate that the averaged density of states follows the Wigner semicircle law and the corresponding nearest energy-level-spacing distribution functionP(S) obeys the Wigner surmise. In the very sparse matrix limitp≪N, withp>p q a singularity 〈ρ(E))∝1/¦E¦ is found as¦E¦→ 0 and exponential tails develop in the high-¦E¦ regions, but theP(S) distribution remains consistent with level repulsion. The localization properties of the model are examined by studying both the eigenvector amplitude and the density fluctuations. The valuep q 1.4 is roughly estimated, in agreement with previous studies of the Anderson transition in dilute Bethe lattices.
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Evangelou, S.N. A numerical study of sparse random matrices. J Stat Phys 69, 361–383 (1992). https://doi.org/10.1007/BF01053797
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DOI: https://doi.org/10.1007/BF01053797