We prove that the q-state Potts antiferromagnet on a lattice of maximum coordination number r exhibits exponential decay of correlations uniformly at all temperatures (including zero temperature) whenever q > 2r. We also prove slightly better bounds for several two-dimensional lattices: square lattice (exponential decay for q ≥ 7), triangular lattice (q ≥ 11), hexagonal lattice (q ≥ 4), and Kagomé lattice (q ≥ 6). The proofs are based on the Dobrushin uniqueness theorem.
SummaryStreptomycetes form hydrophobic aerial hyphae that eventually septate into hydrophobic spores. Both aerial hyphae and spores possess a typical surface layer called the rodlet layer. We present here evidence that rodlet formation is conserved in the streptomycetes. The formation of the rodlet layer is the result of the interplay between rodlins and chaplins. A strain of Streptomyces coelicolor in which the rodlin genes rdlA and/or rdlB were deleted no longer formed the rodlet layer. Instead, these surfaces were decorated with fine fibrils. Deletion of all eight chaplin genes (strain D D D D chpABCDEFGH ) resulted in the absence of the rodlet layer as well as the fibrils at surfaces of aerial hyphae and spores. Apart from coating these surfaces, chaplins are involved in the escape of hyphae into the air, as was shown by the strong reduction in the number of aerial hyphae in the D D D D chpABC-DEFGH strain. The decrease in the number of aerial hyphae correlated with a lower expression of the rdl genes in the colony. Yet, expression per aerial hypha was similar to that in the wild-type strain, indicating that expression of the rdl genes is initiated after the hypha has sensed that it has grown into the air.
We study the chromatic polynomial P G (q) for m × n square-and triangular-lattice strips of widths 2 ≤ m ≤ 8 with cyclic boundary conditions. This polynomial gives the zero-temperature limit of the partition function for the antiferromagnetic q-state Potts model defined on the lattice G. We show how to construct the transfer matrix in the Fortuin-Kasteleyn representation for such lattices and obtain the accumulation sets of chromatic zeros in the complex q-plane in the limit n → ∞. We find that the different phases that appear in this model can be characterized by a topological parameter. We also compute the bulk and surface free energies and the central charge.
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