Spectrum of partial automorphisms of regular rooted tree

TL;DR

This paper investigates the eigenvalues of matrices linked to random partial automorphisms of regular rooted trees, revealing that as the tree depth increases, the proportion of non-zero eigenvalues diminishes to zero.

Contribution

It provides a probabilistic analysis of eigenvalue distributions in matrices from partial automorphisms of regular rooted trees, highlighting asymptotic behavior.

Findings

Fraction of non-zero eigenvalues approaches zero as levels increase

Eigenvalue distribution becomes sparse in large trees

Asymptotic convergence in probability of eigenvalue properties

Abstract

We study properties of eigenvalues of a matrix associated with a randomly chosen partial automorphism of a regular rooted tree. We show that asymptotically, as the numbers of levels goes to infinity, the fraction of non-zero eigenvalues converges to zero in probability.