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 $\lambda$ 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 $\lambda$. 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 $\delta\ge 3$ and maximum degree $\Delta$ is restricted to finitely many points, namely the eigenvalues of trees of size less than $\frac{\Delta-2}{\delta-2}$. More generally, we show that the restriction $\delta\ge 3$ can be weakened to $\delta\ge 2$, 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 Vir\'ag (2013).