Random homomorphisms and Lipschitz functions on trees

A graph homomorphism is an integer-valued function on the vertex set of a graph that assigns values differing by exactly one to adjacent vertices. We consider uniformly random homomorphisms on general finite trees, conditioned to take the value zero at all leaves, and study the distribution of the value at the root. Our main result is a stochastic comparison, both from above and below, between the absolute values of the homomorphism value at the root and certain discrete Gaussian-like random variables. In particular, we obtain a subgaussian tail bound valid for all deviations, a matching lower bound that holds up to a certain threshold, and upper and lower variance bounds that differ by a constant factor. These bounds depend solely on the effective resistance between the root and the leaves in the associated electrical network. As a consequence, in the setting of infinite locally finite trees, we obtain that the homomorphism model is localized on transient trees and delocalized on recurrent trees. Analogous results are obtained for random integer-valued Lipschitz functions. Our results extend previous results of Benjamini--H\"aggstr\"om--Mossel on homomorphisms on regular trees, of Peled--Samotij--Yehudayoff on Lipschitz functions on regular trees, and of Lammers--Toninelli on homomorphisms on trees of minimum degree at least three.