### 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 \(0\).

#### Keywords

partial automorphism, semigroup, eigenvalues, random matrix, delta measure

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