Spectral atoms of unimodular random trees

Source abstract

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 Virág (2013). Finally, we produce explicit examples of non-regular trees whose spectrum is completely atom-free.