Native ultrametricity of sparse random ensembles

TL;DR

This paper studies the spectral properties of large sparse Bernoulli matrices, revealing ultrametric structures, Lifshitz singularities, and connections to number theory, with implications for complex systems and rare-event statistics.

Contribution

It uncovers the ultrametric nature of spectral densities in sparse random matrices and links spectral features to number theory and localization phenomena.

Findings

95% of finite subgraphs are linear near the percolation threshold

Spectral density tail exhibits Lifshitz singularity

Ultrametricity is linked to complex systems with sparse statistics

Abstract

We investigate the eigenvalue density in ensembles of large sparse Bernoulli random matrices. We demonstrate that the fraction of linear subgraphs just below the percolation threshold is about 95\% of all finite subgraphs, and the distribution of linear chains is purely exponential. We analyze in detail the spectral density of ensembles of linear subgraphs, discuss its ultrametric nature and show that near the spectrum boundary, the tail of the spectral density exhibits a Lifshitz singularity typical for Anderson localization. We also discuss an intriguing connection of the spectral density to the Dedekind $\eta$-function. We conjecture that ultrametricity is inherit to complex systems with extremal sparse statistics and argue that a number-theoretic ultrametricity emerges in any rare-event statistics.