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Universal hypotrochoidic law for random matrices with cyclic correlations

Pau Vilimelis Aceituno, Tim Rogers, and Henning Schomerus
Phys. Rev. E 100, 010302(R) – Published 16 July 2019
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Abstract

The celebrated elliptic law describes the distribution of eigenvalues of random matrices with correlations between off-diagonal pairs of elements, having applications to a wide range of physical and biological systems. Here, we investigate the generalization of this law to random matrices exhibiting higher-order cyclic correlations between k tuples of matrix entries. We show that the eigenvalue spectrum in this ensemble is bounded by a hypotrochoid curve with k-fold rotational symmetry. This hypotrochoid law applies to full matrices as well as sparse ones, and thereby holds with remarkable universality. We further extend our analysis to matrices and graphs with competing cycle motifs, which are described more generally by polytrochoid spectral boundaries.

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  • Received 21 December 2018
  • Revised 11 February 2019

DOI:https://doi.org/10.1103/PhysRevE.100.010302

©2019 American Physical Society

Physics Subject Headings (PhySH)

NetworksStatistical Physics & Thermodynamics

Authors & Affiliations

Pau Vilimelis Aceituno*

  • Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany

Tim Rogers

  • Centre for Networks and Collective Behaviour, Department of Mathematical Sciences, University of Bath, Bath BA27AY, United Kingdom

Henning Schomerus

  • Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom

  • *pau.aceituno@mis.mpg.de
  • t.c.rogers@bath.ac.uk
  • h.schomerus@lancaster.ac.uk

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Issue

Vol. 100, Iss. 1 — July 2019

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