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Universal hypotrochoidic law for random matrices with cyclic correlations

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Article number010302
<mark>Journal publication date</mark>16/07/2019
<mark>Journal</mark>Physical Review E
Issue number1
Volume100
Number of pages5
Publication StatusPublished
<mark>Original language</mark>English

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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© 2019 American Physical Society