Uniqueness thresholds on trees versus graphs

In this paper we provide an example of a modified antiferromagnetic Potts model which has uniqueness on the d-regular tree but does not have uniqueness on some infinite d-regular graphs. This example runs counter to the notion that the regular tree has the slowest decay of correlation amongst d-regular graphs. Sokal [7] conjectured that uniqueness in the hard-core model on the d-regular tree implies uniqueness on any d-regular graph. He also speculated that this might also be true for random colourings. Mossel [4] suggested that this may in fact hold for every spin system. Determining the threshold for uniqueness on regular trees can often be done through recursions and so can be much easier than on more complicated graphs. These conjectures then would allow us to take the uniqueness threshold for regular trees as a bound for general d-regular graphs. The intuition behind such conjectures is that the regular tree has the most vertices at distance n from the root and so this boundary has the greatest influence on the root. In this sense loops constitute waste. However, loops in the graph create extra dependence between the states of the neighbors. This is crucial in the construction of our counterexample. Weitz in [8] showed that marginals of the hardcore model on a d-regular graph could be exactly evaluated by calculating the marginals on a tree of self-avoiding random walks. This approach establishes efficient deterministic polynomial time algorithms for approximately counting independent sets on d-regular graphs. His approach was generalized in [3] to general 2-spin systems. Tree based constructions for spin systems have also been used in [1], [2] and [6]. As an immediate consequence of this construction [8] shows that any 2-spin system which has strong spacial mixing on the d-regular tree also has strong spacial mixing on all graphs of maximum degree d. That is, the worst case for strong spacial mixing is the d-regular tree. In the hardcore and antiferromagnetic Ising models [8] also showed that uniqueness on the d-regular tree in fact implies strong spacial mixing on the d-regular tree and so implies uniqueness on all graphs of maximum degree d proving the conjecture of Sokal. This approach does not apply more generally to all 2-spin systems, in particular in the ferromagnetic Ising model uniqueness on the d-regular tree does not in general imply strong spacial mixing, see e.g. [8]. It is unknown whether in 2-spin systems uniqueness on the d-regular tree implies uniqueness on all d-regular graphs.