Gradient Gibbs measures for some models on trees.

In this talk, we study the construction of gradient Gibbs measures on locally finite trees corresponding to gradient specifications. More precisely, gradient Gibbs measures are obtained by mapping Gibbs measures associated with the fuzzy specification from the fuzzy height configuration space to the corresponding gradient configuration space. Our results identify critical thresholds where these systems undergo phase transitions, moving from the existence of a unique gradient state to a regime of non-uniqueness.