Absence of phase transition for antiferromagnetic Potts models via the Dobrushin uniqueness theorem

Salas, José Valero; Sokal, Alan D.

doi:10.1007/bf02199113

Cited by 228 publications

(311 citation statements)

References 41 publications

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“…As we know that the true value for A should be bounded from above by q c = 3, we can try to fit the data to the Ansatz: 3 − q 0 = Bm −∆ . Again, we do not find any stable fit: as m min increases from 5 to 11, the estimate B monotonicly goes from 2.47978(4) to 5.3807 (12), and the exponent ∆ grows from 1.28668(1) to 1.62265 (9). Again the χ 2 is poor: it goes from 1.72 × 10 8 for m min = 5 to 5.22 × 10 5 for m min = 9.…”

Section: Discussioncontrasting

confidence: 56%

“…One interesting question is the value of the number q c (G) for the most common regular lattices G. There are general analytic bounds in terms of the coordination number ∆ of the lattice (e.g., q c ≤ 2∆ [9]), but they are not very sharp. An alternative approach, inspired by the Lee-Yang picture of phase transitions [10], is to study the zeros of the chromatic polynomial when the parameter q is allowed to take complex values (see Ref.…”

Section: Introductionmentioning

confidence: 99%

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Jacobsen

Salas

2001

Journal of Statistical Physics

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We study the chromatic polynomials for m × n square-lattice strips, of width 9 ≤ m ≤ 13 (with periodic boundary conditions) and arbitrary length n (with free boundary conditions). We have used a transfer matrix approach that allowed us also to extract the limiting curves when n → ∞. In this limit we have also obtained the isolated limiting points for these square-lattice strips and checked some conjectures related to the Beraha numbers.

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Section: Discussioncontrasting

confidence: 56%

Section: Introductionmentioning

confidence: 99%

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Jacobsen

Salas

2001

Journal of Statistical Physics

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“…From (3) and (5), the first of these is the scaling law (2). From (17), the second can be conveniently expressed as another relation between the correction exponents, namelŷ…”

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confidence: 99%

“…To summarize thus far, the three standard scaling laws (1)-(3) have been recovered and three analogous relations for the logarithmic corrections (19), (21), and (27) presented. Furthermore, the standard formula (5) for the edge has been recovered and its logarithmic-correction counterpart is given in (17). While the standard scaling laws for the leading critical exponents are well established, it is now necessary to confront the scaling relations for corrections with results from the literature, and a variety of models with logarithmic corrections are examined on a case-by-case basis.…”

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Scaling Relations for Logarithmic Corrections

2006

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Multiplicative logarithmic corrections to scaling are frequently encountered in the critical behavior of certain statistical-mechanical systems. Here, a Lee-Yang zero approach is used to systematically analyse the exponents of such logarithms and to propose scaling relations between them. These proposed relations are then confronted with a variety of results from the literature. 05.50.+q, 05.70.Jk, 75.10.Hk Conventional leading scaling behavior at a secondorder phase transition is described by power laws in the reduced temperature t and field h. With h = 0, the correlation length, specific heat, and susceptibility behave as ξ ∞ (t) ∼ |t| −ν , C ∞ (t) ∼ |t| −α , and χ ∞ (t) ∼ |t| −γ , while the magnetization in the broken phase has m ∞ (t) ∼ |t| β . Here the subscript indicates the extent of the system. At t = 0 the magnetization scales as m ∞ (h) ∼ h 1/δ while the anomalous dimension η characterizes the correlation function at criticality. In the 1960's, it was shown that these six critical exponents are related via four scaling relations (see e.g. Ref.[1] and references therein), which are now firmly established and fundamentally important in the theory of critical phenomena. With d representing the dimensionality of the system, the scaling relations areIn the conventional scaling scenario, (2) and (3) can, in fact, be deduced from the Widom scaling hypothesis that the Helmholtz free energy is a homogeneous function [2]. Widom scaling and the remaining two laws can, in turn, be derived from the Kadanoff block-spin construction [3] and ultimately from Wilson's renormalization group (RG) [4]. The relation (1) can also be derived from the hyperscaling hypothesis, namely, that the free energy behaves near criticality as the inverse correlation volume:. Twice differentiating this relation recovers (1).The scaling relations, (2) and (3), were both rederived using an alternative route by Abe [5] and Suzuki [6] exploiting the fact that the even and odd scaling fields can be linked by Lee-Yang zeros [7]. The locus of these zeros in the magnetic-field plane is controlled by the temperature. In the t > 0 (disordered) phase this locus terminates at the Yang-Lee edge [7], the distance of which from the critical point is denoted by r YL (t). At a conventional second-order phase transition r YL (t) ∼ t ∆ for t > 0, and the gap exponent ∆ is related to the other exponents through [5,6] Logarithmic corrections are characteristic of a number of marginal scenarios (see, e.g., Ref.[8] and references therein). Hyperscaling fails at and above the upper critical dimension d c and, while (1) holds there, it too fails above d c , where mean-field behavior (which is independent of d) prevails. At d c itself, multiplicative logarithmic corrections to scaling are manifest. Such corrections are found in marginal d < d c situations too [8,9]. The q-state Potts model in d = 2 dimensions possesses a firstorder transition for q > 4 and a second-order one when q < 4. The q = 4 case is also characterized by a transition of second order, alb...

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A more rapidly mixing Markov chain for graph colorings

Dyer

Greenhill

1998

Random Struct. Alg.

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We define a new Markov chain on (proper) k-colourings of graphs, and relate its convergence properties to the maximum degree ∆ of the graph. The chain is shown to have bounds on convergence time appreciably better than those for the wellknown Jerrum/Salas-Sokal chain in most circumstances. For the case k = 2∆, we provide a dramatic decrease in running time. We also show improvements whenever the graph is regular, or fewer than 3∆ colours are used. The results are established using the method of path coupling. We indicate that our analysis is tight by showing that the couplings used are optimal in a sense which we define.1 Ω * ( • ) is the notation which hides factors of log(n)

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Absence of phase transition for antiferromagnetic Potts models via the Dobrushin uniqueness theorem

Cited by 228 publications

References 41 publications

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Scaling Relations for Logarithmic Corrections

A more rapidly mixing Markov chain for graph colorings

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