Universal hypotrochoidic law for random matrices with cyclic correlations.

Area of Science:

• Mathematical Physics
• Random Matrix Theory
• Complex Systems

Background:

• The elliptic law describes eigenvalue distributions in random matrices with pairwise correlations.
• This law has broad applications in physics and biology.
• Generalizing this law is crucial for understanding more complex systems.

Purpose of the Study:

• To investigate the generalization of the elliptic law to random matrices with higher-order cyclic correlations.
• To determine the spectral boundaries for these generalized random matrices.
• To explore the universality of the findings in both full and sparse matrices.

Main Methods:

• Analysis of random matrix ensembles with k-tuples of cyclic correlations.
• Derivation of spectral boundary properties using mathematical techniques.
• Extension of the analysis to matrices and graphs with competing cycle motifs.

Main Results:

• The eigenvalue spectrum of these matrices is bounded by a hypotrochoid curve with k-fold rotational symmetry.
• This hypotrochoid law exhibits universality, applying to both full and sparse matrices.
• Polytrochoid spectral boundaries were identified for matrices and graphs with competing cycle motifs.

Conclusions:

• The study successfully generalizes the elliptic law to higher-order cyclic correlations.
• Hypotrochoid and polytrochoid curves provide a new framework for understanding eigenvalue distributions in complex random matrix systems.
• The findings highlight the universality and broad applicability of these spectral laws.