# Universal hypotrochoidic law for random matrices with cyclic correlations

## TL;DR

This paper generalizes the elliptic law for random matrices to include higher-order cyclic correlations, revealing that their eigenvalue spectra are bounded by hypotrochoid curves with universal applicability across different matrix types.

## Contribution

It introduces a new hypotrochoid law for eigenvalue distributions in matrices with cyclic correlations, extending the classical elliptic law to higher-order motifs.

## Key findings

- Eigenvalue spectra are bounded by hypotrochoid curves with k-fold symmetry.
- The law applies to both full and sparse matrices, demonstrating universality.
- Extension to matrices with competing cycle motifs leads to polytrochoid spectral boundaries.

## 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.