Journal ArticleDOI
Discontinuity of the magnetization in one-dimensional 1/¦x−y¦2 Ising and Potts models
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TLDR
In this paper, results from percolation theory are used to study phase transitions in one-dimensional Ising and q-state Potts models with couplings of the asymptotic form.Abstract:
Results from percolation theory are used to study phase transitions in one-dimensional Ising andq-state Potts models with couplings of the asymptotic formJ
x,y≈ const/¦x−y¦2. For translation-invariant systems with well-defined lim
x→∞
x
2
J
x
=J
+ (possibly 0 or ∞) we establish: (1) There is no long-range order at inverse temperaturesβ withβJ
+⩽1. (2) IfβJ
+>q, then by sufficiently increasingJ
1 the spontaneous magnetizationM is made positive. (3) In models with 0<J
+<∞ the magnetization is discontinuous at the transition point (as originally predicted by Thouless), and obeysM(β
c
)⩾1/(β
c
J
+)1/2. (4) For Ising (q=2) models withJ
+<∞, it is noted that the correlation function decays as 〈σxσy〉(β)≈c(β)/|x−y|2 wheneverβ<β
c
. Points 1–3 are deduced from previous percolation results by utilizing the Fortuin-Kasteleyn representation, which also yields other results of independent interest relating Potts models with different values ofq.read more
Citations
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References
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Journal ArticleDOI
Nonuniversal critical dynamics in Monte Carlo simulations
TL;DR: A new approach to Monte Carlo simulations is presented, giving a highly efficient method of simulation for large systems near criticality, despite the fact that the algorithm violates dynamic universality at second-order phase transitions.
Journal ArticleDOI
On the random-cluster model: I. Introduction and relation to other models
C.M. Fortuin,P.W. Kasteleyn +1 more
TL;DR: It is shown that the function which for the random-cluster model plays the role of a partition function, is a generalization of the dichromatic polynomial earlier introduced by Tutte, and related polynomials.
Journal ArticleDOI
Association of Random Variables, with Applications
TL;DR: In this paper, it was shown that a random variable can be associated with another random variable if the test functions are either (a) binary or (b) bounded and continuous.
Journal ArticleDOI
Correlation inequalities on some partially ordered sets
TL;DR: In this article, it was shown that increasing functions on a finite distributive lattice are positively correlated by positive measures satisfying a suitable convexity property, and applications to Ising ferromagnets in an arbitrary magnetic field and to the random cluster model were given.
Journal ArticleDOI
Existence of a phase-transition in a one-dimensional Ising ferromagnet
TL;DR: In this paper, the existence of a phase transition for an infinite linear chain of spins with an interaction energy was proved for the case where ρ is positive and monotone decreasing, and the sums ρJ(n) and ρ (log logn) [n 3 ρ(n)]−1 both converged.
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