Peculiar spectral statistics of ensembles of trees and star-like graphs

TL;DR

This paper explores the spectral properties of ensembles of trees and star graphs, revealing unique ultrametric structures and localization phenomena, with implications for understanding dendrimers and disordered systems.

Contribution

It uncovers the peculiar spectral statistics and ultrametric structures of binary trees and star graphs, linking them to localization and number-theoretic effects in disordered systems.

Findings

Spectral densities exhibit ultrametric structure.

Lifshitz singularity appears in binary tree spectra.

Spectral properties depend on graph topology and size.

Abstract

In this paper we investigate the eigenvalue statistics of exponentially weighted ensembles of full binary trees and $p$-branching star graphs. We show that spectral densities of corresponding adjacency matrices demonstrate peculiar ultrametric structure inherent to sparse systems. In particular, the tails of the distribution for binary trees share the "Lifshitz singularity" emerging in the one-dimensional localization, while the spectral statistics of $p$-branching star-like graphs is less universal, being strongly dependent on $p$. The hierarchical structure of spectra of adjacency matrices is interpreted as sets of resonance frequencies, that emerge in ensembles of fully branched tree-like systems, known as dendrimers. However, the relaxational spectrum is not determined by the cluster topology, but has rather the number-theoretic origin, reflecting the peculiarities of the rare-event statistics typical for one-dimensional systems with a quenched structural disorder. The similarity of spectral densities of an individual dendrimer and of ensemble of linear chains with exponential distribution in lengths, demonstrates that dendrimers could be served as simple disorder-less toy models of one-dimensional systems with quenched disorder.