2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/173890Counter to the general notion that the regular tree is the worst case for decay of correlation between sets and nodes, we produce an example of a multi-spin interacting system which has uniqueness on the $d$-regular tree but does not have uniqueness on some infinite $d$-regular graphs.Published in at http://dx.doi.org/10.1214/07-AAP508 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)Probability82B26 (Primary)text