Spectral atoms of unimodular random trees

We use the Mass Transport Principle to analyze the local recursion governing the resolvent $(A-z)^{-1}$ of the adjacency operator of unimodular random trees. In the limit where the complex parameter $z$ approaches a given location $λ$ on the real axis, we show that this recursion induces a decomposition of the tree into finite blocks whose geometry directly determines the spectral mass at $λ$. We then exploit this correspondence to obtain precise information on the pure-point support of the spectrum, in terms of expansion properties of the tree. In particular, we deduce that the pure-point support of the spectrum of any unimodular random tree with minimum degree $δ\ge 3$ and maximum degree $Δ$ is restricted to finitely many points, namely the eigenvalues of trees of size less than $\frac{Δ-2}{δ-2}$. More generally, we show that the restriction $δ\ge 3$ can be weakened to $δ\ge 2$, as long as the anchored isoperimetric constant of the tree remains bounded away from $0$. This applies in particular to any unimodular Galton-Watson tree without leaves, allowing us to settle a conjecture of Bordenave, Sen and Virág (2013).

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