Counter 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.
"Uniqueness thresholds on trees versus graphs." Ann. Appl. Probab. 18 (5) 1897 - 1909, October 2008. https://doi.org/10.1214/07-AAP508