Correlation Decay up to Uniqueness in Spin Systems

Abstract

We give a complete characterization of the two-state anti-ferromagnetic spin systems which are of strong spatial mixing on general graphs. We show that a two-state anti-ferromagnetic spin system is of strong spatial mixing on all graphs of maximum degree at most ∆ if and only if the system has a unique Gibbs measure on infinite regular trees of degree up to ∆, where ∆ can be either bounded or unbounded. As a consequence, there exists an FPTAS for the partition function of a two-state anti-ferromagnetic spin system on graphs of maximum degree at most ∆ when the uniqueness condition is satisfied on infinite regular trees of degree up to ∆. In particular, an FPTAS exists for arbitrary graphs if the uniqueness is satisfied on all infinite regular trees. This covers as special cases all previous algorithmic results for two-state anti-ferromagnetic systems on general-structure graphs.Combining with the FPRAS for two-state ferromagnetic spin systems of Jerrum-Sinclair and Goldberg-Jerrum-Paterson, and the very recent hardness results of Sly-Sun and independently of Galanis-Stefankovic-Vigoda, this gives a complete classification, except at the phase transition boundary, of the approximability of all two-state spin systems, on either degree-bounded families of graphs or family of all graphs.

Spatial mixing and the connective constant: optimal bounds

“…The approach of [3,31] (and similar results in [15,16,24]) for establishing this last condition takes the following general form: one shows that at each step of the recurrence, the correlation decay condition implies that the error at the parent node is less than a constant factor (less than 1) times the maximum ( ∞ norm) of the errors at the children of the node. Intuitively, this strategy loses information about the structure of the tree by explaining the error at the parent in terms of only one of its children, and hence it is not surprising that the results obtained from it are only in terms of a local parameter such as the maximum degree.…”

“…(2.1) for both the hard core and monomer-dimer models: such a result basically affirms that truncating the recurrence at a small depth is sufficient in order to approximate F ρ with good accuracy. Our proof will use the message approach [16,22,24], which proceeds by defining an appropriate function φ of the marginals being computed and then showing a decay of correlation result for this function. Message [16, 22, 24]).…”

“…The first step of our approach is similar to that taken in the papers [15,16,22,24] in that we will use an appropriate message-along with the estimate in Lemma 3.2-to argue that the "distance" between two input message vectors x and y at the children of a vertex shrinks by a constant factor at each step of the recurrence. Previous works [15,16,22,24] showed such a decay on some version of the ∞ norm of the "error" vector x − y: this was achieved by bounding the appropriate dual 1 norm of the gradient of the recurrence.…”

“…Previous works [15,16,22,24] showed such a decay on some version of the ∞ norm of the "error" vector x − y: this was achieved by bounding the appropriate dual 1 norm of the gradient of the recurrence. Our intuition is that in order to achieve a bound in terms of a global quantity such as the connective constant, it should be advantageous to use a more global measure of the error such as an q norm for some q < ∞.…”

A complexity classification of spin systems with an external field

“…For example, the algorithm of Jerrum and Sinclair (14) gives an FPRAS for Z λ A Ising in the ferromagnetic case λ > 1 as long as the field acts in a consistent manner in the sense that the sign of h w ð2Þ − h w ð1Þ is the same for all vertices (15). In the antiferromagnetic regime it is known that tractability corresponds to the uniqueness threshold when there are just two spins (16). The ferromagnetic regime is not fully understood (17), even with just two spins, and the multispin case is open.…”