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Absence of Phase Transition for Antiferromagnetic Potts Models via the Dobrushin Uniqueness
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Absence of Phase Transition
for Antiferromagnetic Potts Models
via the Dobrushin Uniqueness Theorem
Jesu?s Salas
Alan D. Sokal
Department of Physics
New York University
4 Washington Place
New York, NY 10003 USA
SALAS@MAFALDA.PHYSICS.NYU.EDU, SOKAL@NYU.EDU
February 1, 2008
Dedicated to the memory of R.L. Dobrushin
Abstract
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 Kagome? lattice (q ≥ 6). The proofs are based on the Dobrushin
uniqueness theorem.
Key Words: Dobrushin uniqueness theorem, antiferromagnetic Potts models, phase
transition.
1 Introduction
Dobrushin’s uniqueness theorem [1, 2, 3, 4] provides a simple but powerful method
for proving the uniqueness of the infinite-volume Gibbs measure, as well as the ex-
ponential decay of correlations in this unique Gibbs measure, for classical-statistical-
mechanical systems deep in a single-phase region. The basic idea underlying this
theorem is that if the probability distribution of a single spin σi depends “sufficiently
weakly” on the remaining spins {σj}j 6=i, then one can deduce (by a clever iterative
argument) uniqueness of the Gibbs measure and exponential decay of correlations.
The principal applications of this method have been in two regimes:
1) High temperature. Here σi depends weakly on the {σj}j 6=i because of the strong
thermal fluctuations.
2) Large magnetic field. Here σi tends to follow the magnetic field, no matter what
the other spins are doing; so the probability distribution of σi again depends
weakly on the {σj}j 6=i.
However, Kotecky? (cited in [3, pp. 148–149, 457]) has pointed out that
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