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We argue that the four-state Potts antiferromagnet has a finite-temperature phase transition on any Eulerian plane triangulation in which one sublattice consists of vertices of degree 4. We furthermore predict the universality class of this transition. We then present transfer-matrix and Monte?Carlo data confirming these predictions for the cases of the Union Jack and bisected hexagonal lattices. 相似文献
13.
We study the dynamic critical behavior of the local bond-update (Sweeny) dynamics for the Fortuin-Kasteleyn random-cluster model in dimensions d=2, 3 by Monte Carlo simulation. We show that, for a suitable range of q values, the global observable S2 exhibits "critical speeding-up": it decorrelates well on time scales much less than one sweep. In some cases the dynamic critical exponent for the integrated autocorrelation time is negative. We also show that the dynamic critical exponent zexp is very close (possibly equal) to the rigorous lower bound alpha/nu and quite possibly smaller than the corresponding exponent for the Chayes-Machta-Swendsen-Wang cluster dynamics. 相似文献
14.
The chromatic polynomial PG(q) of a loopless graph G is known to be non-zero (with explicitly known sign) on the intervals (−∞,0), (0,1) and (1,32/27]. Analogous theorems hold for the flow polynomial of bridgeless graphs and for the characteristic polynomial of loopless matroids. Here we exhibit all these results as special cases of more general theorems on real zero-free regions of the multivariate Tutte polynomial ZG(q,v). The proofs are quite simple, and employ deletion–contraction together with parallel and series reduction. In particular, they shed light on the origin of the curious number 32/27. 相似文献
15.
Alan D. Sokal 《Expositiones Mathematicae》2010,28(2):179-185
We obtain a characterization of generalized Stieltjes functions of any order λ>0 in terms of inequalities for their derivatives on (0,∞). When λ=1, this provides a new and simple proof of a characterization of Stieltjes functions first obtained by Widder in 1938. 相似文献
16.
We prove that theq-state Potts antiferromagnet on a lattice of maximum coordination numberr exhibits exponential decay of correlations uniformly at all temperatures (including zero temperature) wheneverq>2r. We also prove slightly better bounds for several two-dimensional lattices: square lattice (exponential decay forq7), triangular lattice (q11), hexagonal lattice (q4), and Kagomé lattice (q6). The proofs are based on the Dobrushin uniqueness theorem. 相似文献
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Roman?KoteckyEmail author Alan?D.?Sokal Jan?M.?Swart 《Communications in Mathematical Physics》2014,330(3):1339-1394
We prove the existence of long-range order at sufficiently low temperatures, including zero temperature, for the three-state Potts antiferromagnet on a class of quasi-transitive plane quadrangulations, including the diced lattice. More precisely, we show the existence of (at least) three infinite-volume Gibbs measures, which exhibit spontaneous magnetization in the sense that vertices in one sublattice have a higher probability to be in one state than in either of the other two states. For the special case of the diced lattice, we give a good rigorous lower bound on this probability, based on computer-assisted calculations that are not available for the other lattices. 相似文献
19.
Alan D. Sokal 《Advances in Mathematics》2012,229(5):2603-2621
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