Spectral properties of partial automorphisms of 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 $0$.