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 \(A_x\). It is shown that the uniform distribution  on eigenvalues of \(A_x\) converges weakly in probability to \(\delta_0\) as \(n \to \infty\), where \(\delta_0\) is the delta measure concentrated at \(0\).