Editorial

Weitz, Robert D.

doi:10.1300/j294v02n02_01

Cited by 1 publication

(2 citation statements)

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“…Remark: After obtaining our message-decay proof, we received a sketch of Weitz's original unpublished proof [11]. It is interesting to note that that proof is quite different from ours, and employs a delicate two-step analysis of the tree recursion described in Section 2.…”

Section: Contributionsmentioning

confidence: 98%

“…To see that such a choice of parameters is possible, we consider the exact form of λ c (β, d). Translating the results in [3, p. 250] to our notation using Appendix A, we have We note that P (d) is an increasing function of d. Thus, the required conditions (11) and (12) Again, since Q(β, d) is o β (1), and P (d) is an increasing function, it is possible to find λ satisfying both the inequalities (14) and (15) for β large enough. Thus, for any d ≥ 3, we can find (β, λ) with β > 1 such that weak spatial mixing for the d-ary tree does not imply strong spatial mixing.…”

Section: Remarkmentioning

confidence: 99%

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Approximation Algorithms for Two-State Anti-Ferromagnetic Spin Systems on Bounded Degree Graphs

Sinclair

Srivastava

Thurley³

2014

J Stat Phys

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In a seminal paper [12], Weitz gave a deterministic fully polynomial approximation scheme for counting exponentially weighted independent sets (equivalently, approximating the partition function of the hard-core model from statistical physics) on graphs of degree at most d, up to the critical activity for the uniqueness of the Gibbs measure on the infinite d-regular tree. More recently Sly [10] showed that this is optimal in the sense that if there is an FPRAS for the hard-core partition function on graphs of maximum degree d for activities larger than the critical activity on the infinite d-regular tree then NP = RP. In this paper, we extend Weitz's approach to derive a deterministic fully polynomial approximation scheme for the partition function of the anti-ferromagnetic Ising model with arbitrary field on graphs of maximum degree d, up to the corresponding critical point on the d-regular tree. The main ingredient of our result is a proof that for two-state anti-ferromagnetic spin systems on the d-regular tree, weak spatial mixing implies strong spatial mixing. This in turn uses a message-decay argument which extends a similar approach proposed recently for the hard-core model by Restrepo et al [9] to the case of the anti-ferromagnetic Ising model with arbitrary field. By a standard correspondence, these results translate to arbitrary two-state anti-ferromagnetic spin systems with soft constraints.

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Section: Contributionsmentioning

confidence: 98%

Section: Remarkmentioning

confidence: 99%