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Abstract: We examine the adjacency matrices of three-regular graphs representingone-face maps. Numerical studies reveal that the limiting eigenvalue statisticsof these matrices are the same as those of much larger, and more widely studiedclasses from Random Matrix Theory. We present an algorithm for generatingmatrices corresponding to maps of genus zero, and find the eigenvaluestatistics in the genus zero case differ strikingly from those of higher genus.These results lead us to conjecture that the eigenvalue statistics depend onthe rigidity of the underlying map, and the distribution of scaled eigenvaluespacings shifts from that of the Gaussian Orthogonal Ensemble to theexponential distribution as the map size increases relative to the genus.

Autor: E. M. McNicholas

Fuente: https://arxiv.org/

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