Spectral properties of partial automorphisms of a binary rooted tree

Abstract

We study asymptotics of the spectral measure of a randomly chosen partial automorphism of a rooted tree. To every partial automorphism x we assign its action matrix Ax. It is shown that the uniform distribution on eigenvalues of Ax converges weakly in probability to δ₀ as n → ∞, where δ₀ is the delta measure concentrated at 0.