We use the Mass Transport Principle to analyze the local recursion governing the resolvent of the adjacency operator of unimodular random trees. In the limit where the complex parameter approaches a given location 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 . 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 and maximum degree is restricted to finitely many points, namely the eigenvalues of trees of size less than . More generally, we show that the restriction can be weakened to , 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.
Cite this article
Justin Salez, Spectral atoms of unimodular random trees. J. Eur. Math. Soc. 22 (2020), no. 2, pp. 345–363DOI 10.4171/JEMS/923