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  • 1
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    Springer
    Letters in mathematical physics 46 (1998), S. 207-218 
    ISSN: 1573-0530
    Keywords: Ising model ; boundary conditions ; Boltzmann weights.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract In this Letter, we analyse the boundary conditions of the planar Ising model and determine the boundary Boltzmann weights in terms of bulk Boltzmann weights. The commutativity of the transfer matrices and their functional relations are shown.
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  • 2
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    Letters in mathematical physics 37 (1996), S. 137-143 
    ISSN: 1573-0530
    Keywords: 82B20 ; 82B26 ; 82B43 ; Bethe lattice ; FK representation ; Ising model
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We give a simple proof that the limit Ising Gibbs measure with free boundary conditions on the Bethe lattice with the forward branching ratio k≥2 is extremal if and only if β is less or equal to the spin glass transition value, given by tanh(β c SG = 1/√k.
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  • 3
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    Journal of statistical physics 78 (1995), S. 285-297 
    ISSN: 1572-9613
    Keywords: Ising model ; Lee-Yang zeros ; edge singularities ; nonperiodic systems ; phase transitions ; gap labeling
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The study of zeros of partition functions, initiated by Yang and Lee, provides an important qualitative and quantitative tool in the study of critical phenomena. This has frequently been used for periodic as well as hierarchical lattices. Here, we consider magnetic field and temperature zeros of Ising model partition functions on several aperiodic structures. In 1D, we analyze aperiodic chains obtained from substitution rules, the most prominent example being the Fibonacci chain. In 2D, we focus on the tenfold symmetric triangular tiling which allows efficient numerical treatment by means of corner transfer matrices.
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  • 4
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    Journal of statistical physics 78 (1995), S. 731-757 
    ISSN: 1572-9613
    Keywords: Ising model ; renormalization group ; finite-size conditions ; critical point
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    Topics: Physics
    Notes: Abstract We study the block spin transformation for the 2D Ising model at the critical temperatureT c . We consider the model with the constraint that the total spin in each block is zero. An old argument by Cassandro and Gallavotti strongly supports the Gibbsianness of the transformed measure, provided that such model has a critical temperatureT′ c lower thanT c . After describing a possible rigorous approach to the problem, we present numerical evidence that indeedT′ c 〈T c and study the Dobrushin-Shlosman uniqueness condition.
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  • 5
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    Journal of statistical physics 78 (1995), S. 1311-1324 
    ISSN: 1572-9613
    Keywords: Random-cluster model ; Ising model ; Potts model ; comparison inequality ; BK inequality ; FKG inequality
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    Topics: Physics
    Notes: Abstract A principal technique for studying percolation, (ferromagnetic) Ising, Potts, and random-cluster models is the FKG inequality, which implies certain stochastic comparison inequalities for the associated probability measures. The first result of this paper is a new comparison inequality, proved using an argument developed elsewhere in order to obtain strict inequalities for critical values. As an application of this inequality, we prove that the critical pointp c (q) of the random-cluster model with cluster-weighting factorq (≥1) is strictly monotone inq. Our second result is a “BK inequality” for the disjoint occurrence of increasing events, in a weaker form than that available in percolation theory.
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  • 6
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    Journal of statistical physics 78 (1995), S. 1131-1138 
    ISSN: 1572-9613
    Keywords: Kac potential ; Ising model ; critical fluctuations ; Euclidean field theory
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    Topics: Physics
    Notes: Abstract We consider ad=2 Ising system with a Kac potential whose mean-field critical temperature is 1. Calling γ〉0 the Kac parameter, we prove that there existsc *〉0 so that the true inverse critical temperature βcr(γ) 〉 1 +by 2 log γ-1, for anyb〈c * and γ correspondingly small. We also show that if γ→0 andb→c *, suitably, then the correlation functions (normalized and rescaled) converge to those of a non-Gaussian Euclidean field theory.
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  • 7
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    Journal of statistical physics 82 (1996), S. 1299-1326 
    ISSN: 1572-9613
    Keywords: Cluster algorithms ; computational complexity ; diffusion-limited aggregation ; Ising model ; Metropolis algorithm ; P-completeness
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We examine a number of models that generate random fractals. The models are studied using the tools of computational complexity theory from the perspective of parallel computation. Diffusion-limited aggregation and several widely used algorithms for equilibrating the Ising model are shown to be highly sequential; it is unlikely they can be simulated efficiently in parallel. This is in contrast to Mandelbrot percolation, which can be simulated in constant parallel time. Our research helps shed light on the intrinsic complexity of these models relative to each other and to different growth processes that have been recently studied using complexity theory. In addition, the results may serve as a guide to simulation physics.
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  • 8
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    Journal of statistical physics 85 (1996), S. 297-361 
    ISSN: 1572-9613
    Keywords: Ashkin-Teller model ; Ising model ; Potts model ; Monte Carlo ; dynamical critical behavior ; cluster algorithm ; Swedsen-Wang algorithm ; Li-Sokal bound ; critical slowing down ; autocorrelation time, fitting correlated data
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    Topics: Physics
    Notes: Abstract We study the dynamic critical behavior of a Swendsen-Wang-type algorithm for the Ashkin-Teller model. We find that the Li-Sokal bound on the autocorrelation time (τint.δ≥ const xC H ) holds along the self-dual curve of the symmetric Ashkin-Teller model, and is almost, but not quite sharp. The ratio τint.δ/C H appears to tend to infinity either as a logarithm or as a small power (0.05≲p≲0.12). In an appendix we discuss the problem of extracting estimates of the exponential autocorrelation time.
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  • 9
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    Journal of statistical physics 85 (1996), S. 607-637 
    ISSN: 1572-9613
    Keywords: Ising model ; renormalization group pathologies ; Dobrushin uniqueness theorem ; completely analytic potentials
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We consider real-space renormalization group transformations for Ising-type systems which are formally defined by $$\exp \left[ { - H'(\sigma ')} \right] = \sum\limits_\sigma {T(\sigma ,\sigma ')} \exp \left[ { - H(\sigma )} \right]$$ whereT(σ, σ′) is a probability kernel, i.e., ∑σ′ T(σ,σ′) = 1 for every configuration σ. For each choice of the block spin configuration σ′, let σ′, let μσ′ be the measure on spin configurations σ which is formally given by taking the probability of σ to be proportional toT(σ, σ′) exp[−H(σ)]. We give a condition which is sufficient to imply that the renormalized HamiltonianH′ is defined. Roughly speaking, the condition is that the collection of measures μσ′ is in the high-temperature phase uniformly in the block spin configuration σ′. The proof of this result uses methods of Olivieri and Picco. We use our theorem to prove that the first iteration of the renormalization group transformation is defined in the following two examples: decimation with spacingb = 2 on the square lattice with β 〈 1.36β c and the Kadanoff transformation with parameterp on the trian gular lattice in a subset of the β,p plane that includes values of β greater than β c .
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  • 10
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    Journal of statistical physics 84 (1996), S. 1351-1361 
    ISSN: 1572-9613
    Keywords: Random-cluster model ; quasilocality ; almost sure quasilocality ; tree ; Gibbs measure ; Ising model
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We study the random-cluster model on a homogeneous tree, and show that the following three conditions are equivalent for a random-cluster measure: quasilocality, almost sure quasilocality, and the almost sure nonexistence of infinite clusters. As a consequence of this, we find that the plus measure for the Ising model on a tree at sufficiently low temperatures can be mapped, via a local stochastic transformation, into a measure which fails to be almost surely quasilocal.
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  • 11
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    Journal of statistical physics 88 (1997), S. 991-995 
    ISSN: 1572-9613
    Keywords: History of statistical physics ; Ising model
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The Ising model is one of the standard models in statistical physics. Since 1969 more than 13800 publications using this model have appeared. In 1997 Ernst Ising celebrated his 97th birthday. Some biographical notes and milestones of the development of the Ising model are given.
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  • 12
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    Journal of statistical physics 88 (1997), S. 795-805 
    ISSN: 1572-9613
    Keywords: Cellular automata ; Ising model ; voting models ; single sin-flip dynamics ; computational complexity ; parallel computation ; P-completeness
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We study cellular automata where the state at each site is decided by a majority vote of the sites in its neighborhood. These are equivalent, for a restricted set of initial conditions, to nonzero probability transitions in single spin-flip dynamics of the Ising model at zero temperature. We show that in three or more dimensions these systems can simulate Boolean circuits of AND and OR gates, and are therefore P-complete. That is, predicting their state t time-steps in the future is at least as hard as any other problem that takes polynomial time on a serial computer. Therefore, unless a widely believed conjecture in computer science is false, it is impossible even with parallel computation to predict majority-vote cellular automata, or zero-temperature single spin-flip Ising dynamics, qualitatively faster than by explicit simulation.
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  • 13
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    Journal of statistical physics 90 (1998), S. 1015-1035 
    ISSN: 1572-9613
    Keywords: Ising model ; mixing conditions ; Basuev region ; boundary conditions ; Glauber dynamics ; exponential relaxation ; spectral gap
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We consider Glauber dynamics on a finite cube in d-dimensional lattice (d≥2), which is associated with basic Ising model at temperature T=1/β≪1 under a magnetic field h 〉 0. We prove that if the “effective magnetic field” is positive, then the relaxation of the Glauber dynamics in the uniform norm is exponentially fast, uniformly over the size of underlying cube. The result covers the case of the free-boundary condition with arbitrarily small positive magnetic field. This paper is a continuation of an attempt initiated earlier by Schonmann and Yoshida to shed more light on the relaxation of the finite-volume Glauber dynamics when the thermodynamic parameter (β, h) is so near the phase transition line, (β, h); β c 〈 β&h = 0, that the Dobrushin–Shlosman mixing condition is no longer available.
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  • 14
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    Journal of statistical physics 98 (2000), S. 1115-1134 
    ISSN: 1572-9613
    Keywords: Ising model ; correlation inequalities ; surface tension ; disorder
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract For the semiinfinite Ising model with quenched boundary disorder, we prove concavity inequalities for the difference of wall tensions associated with the minus and plus phases. These inequalities generalize phenomenological equalitiesknown as Cassie's law.
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  • 15
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    Journal of statistical physics 78 (1995), S. 575-584 
    ISSN: 1572-9613
    Keywords: Ising model ; correlation functions ; spontaneous magnetization ; Toeplitz determinants ; Szegö limit theorems ; Toeplitz operators
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract This is not primarily a paper about applications of mathematics to statistical physics, but rather a report on how a particular problem of statistical physics has resulted in an extensive mathematical theory. The problem alluded to is the computation of the spontaneous magnetizationM o (T) of the two-dimensional Ising model with nearest-neighbor interactions, whose solution for temperaturesT below the Curie pointT c was given by the famous formula of Lars Onsager in 1948. The theory grown out of this formula is the edifice of Toeplitz determinants, matrices, and operators.
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  • 16
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    Journal of statistical physics 78 (1995), S. 893-916 
    ISSN: 1572-9613
    Keywords: Dynamic critical phenomena ; disordered spin systems ; Ising model ; finite size ; scaling
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    Topics: Physics
    Notes: Abstract Finite-size scaling effects of the Ising model with quenched random impurities are studied, focusing on critical dynamics. In contrast to the pure Ising model, disordered systems are characterized by continuous relaxation time spectra. Dynamic field theory is applied to compute the spectral densities of the magnetizationM(t) and ofM 2(t). In addition, universal cumulant ratios are calculated to second order in ε1/4, where ε=4−d andd〈4 denotes the spatial dimension.
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  • 17
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    Journal of statistical physics 79 (1995), S. 25-42 
    ISSN: 1572-9613
    Keywords: Renormalization group ; decimation ; non-Gibbsianness ; Ising model
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We investigate the stability and instability of pathologies of renormalization group transformations for lattice spin systems under decimation. In particular we show that, even if the original renormalization group transformation gives rise to a non-Gibbsian measure, Gibbsianness may be restored by applying an extra decimation transformation. This fact is illustrated in detail for the block spin transformation applied to the Ising model. We also discuss the case of another non-Gibbsian measure with nicely decaying correlations functions which remains non-Gibbsian after arbitrary decimation.
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  • 18
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    Journal of statistical physics 82 (1996), S. 87-113 
    ISSN: 1572-9613
    Keywords: Onsager's algbera ; loop algebras ; Ising model ; integrability
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    Topics: Physics
    Notes: Abstract We define ansl(N) analog of Onsager's algebra through a finite set of relations that generalize the Dolan-Grady defining relations for the original Onsager's algebra. This infinite-dimensional Lie algebra is shown to be isomorphic to a fixed-point subalgebra ofsl(N) loop algebra with respect to a certain involution. As the consequence of the generalized Dolan-Grady relations a Hamiltonian linear in the generators ofsl(N) Onsager's algebra is shown to posses an infinite number of mutually commuting integrals of motion.
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  • 19
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    Journal of statistical physics 84 (1996), S. 655-696 
    ISSN: 1572-9613
    Keywords: Ising model ; Glauber dynamics ; relaxation time
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We consider a Glauber dynamics reversible with respect to the two-dimensional Ising model in a finite square of sideL with open boundary conditions, in the absence of an external field and at large inverse temperature β. We prove that the gap in the spectrum of the generator restricted to the invariant subspace of functions which are even under global spin flip is much larger than the true gap. As a consequence we are able to show that there exists a new time scalet even, much smaller than the global relaxation timet rel, such that, with large probability, any initial configuration first relaxes to one of the two “phases” in a time scale of ordert even and only after a time scale of the order oft rel does it reach the final equilibrium by jumping, via a large deviation, to the opposite phase. It also follows that, with large probability, the time spent by the system during the first jump from one phase to the opposite one is much shorter than the relaxation time.
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  • 20
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    Journal of statistical physics 86 (1997), S. 149-164 
    ISSN: 1572-9613
    Keywords: Ising model ; Gibbs measures ; large-deviation principle
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    Topics: Physics
    Notes: Abstract In this paper we obtain the equivalence of the large deviation principle for Gibbs measures with and without an external field. For the Ising model, the equivalence allows us to study the result of competing influences of a positive external fieldh and a negative boundary condition in the cube (Λ(B/h) ash↘0 for variousB. We find a critical balance at a valueB 0 ofB.
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  • 21
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    Journal of statistical physics 90 (1998), S. 211-226 
    ISSN: 1572-9613
    Keywords: Ising model ; stochastic dynamics ; metastability ; nucleation
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    Topics: Physics
    Notes: Abstract We investigate metastability in the two dimensional Ising model in a square with free boundary conditions at low temperatures. Starting with all spins down in a small positive magnetic field, we show that the exit from this metastable phase occurs via the nucleation of a critical droplet in one of the four corners of the system. We compute the lifetime of the metastable phase analytically in the limit T → 0, h → 0 and via Monte Carlo simulations at fixed values of T and h and find good agreement. This system models the effects of boundary domains in magnetic storage systems exiting from a metastable phase when a small external field is applied.
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  • 22
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    Journal of statistical physics 90 (1998), S. 1051-1059 
    ISSN: 1572-9613
    Keywords: Ising model ; Peierls contour ; low-temperature expansion ; high dimension
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    Topics: Physics
    Notes: Abstract We consider the low-temperature expansion for the Ising model on $$\mathbb{Z}^d ,d \geqslant 2$$ , with ferromagnetic nearest neighbor interactions in terms of Peierls contours. We prove that the expansion converges for all temperatures smaller than Cd(log d)−1, which is the correct order in d.
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  • 23
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    Journal of statistical physics 92 (1998), S. 1203-1208 
    ISSN: 1572-9613
    Keywords: Loop model ; criticality ; universality ; 3–12 lattice ; self-avoiding walk ; connective constant ; Ising model
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    Topics: Physics
    Notes: Abstract The partition function of the O(n) loop model on the honeycomb lattice is mapped to that of the O(n) loop model on the 3–12 lattice. Both models share the same operator content and thus critical exponents. The critical points are related via a simple transformation of variables. When n = 0 this gives the recently found exact value μ = 1.711041... for the connective constant of self-avoiding walks on the 3–12 lattice. The exact critical points are recovered for the Ising model on the 3–12 lattice and the dual asanoha lattice at n = 1.
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  • 24
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    Journal of statistical physics 93 (1998), S. 33-78 
    ISSN: 1572-9613
    Keywords: Random external field ; Ising model ; Gibbs states ; ground states ; Bethe lattice ; residual entropy ; dipole configurations ; Griffiths singularities
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    Topics: Physics
    Notes: Abstract The ferromagnetic Ising model on the Bethe lattice of degree k is considered in the presence of a dichotomous external random field ξ x = ±α and the temperature T≥0. We give a description of a part of the phase diagram of this model in the T−α plane, where we are able to construct limiting Gibbs states and ground states. By comparison with the model with a constant external field we show that for all realizations ξ = {ξ x = ±α} of the external random field: (i) the Gibbs state is unique for T 〉 T c (k ≥ 2 and any α) or for α 〉 3 (k = 2 and any T); (ii) the ±-phases coexist in the domain {T 〈 T c, α ≤ H F(T)}, where T c is the critical temperature and H F(T) is the critical external field in the ferromagnetic Ising model on the Bethe lattice with a constant external field. Then we prove that for almost all ξ: (iii) the ±-phases coexist in a larger domain {T 〈 T c, α ≤H F(T) + ε(T)}, where ε(T)〉0; and (iv) the Gibbs state is unique for 3≥α≥2 at any T. We show that the residual entropy at T = 0 is positive for 3≥α≥2, and we give a constructive description of ground states, by so-called dipole configurations.
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  • 25
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    Journal of statistical physics 94 (1999), S. 299-320 
    ISSN: 1572-9613
    Keywords: wetting ; surface tension ; rough surfaces ; Wenzel's law ; semi-infinite systems ; Ising model ; cluster expansions
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    Notes: Abstract We consider a semi-infinite 3-dimensional Ising system with a rough wall to describe the effect of the roughness r of the substrate on wetting. We show that the difference of wall free energies Δτ(r)=τ AW(r)−τ BW(r) of the two phases behaves like Δτ(r)∼rΔτ(1), where r=1 characterizes a purely flat surface, confirming at low enough temperature and small roughness the validity of Wenzel's law, cos θ(r)≈r cos θ(1), which relates the contact angle θ of a sessile droplet to the roughness of the substrate
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  • 26
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    Journal of statistical physics 94 (1999), S. 321-345 
    ISSN: 1572-9613
    Keywords: dynamical triangulations ; quenched disorder ; Ising model
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    Notes: Abstract We study with Monte Carlo methods an ensemble of c=−5 gravity graphs, generated by coupling a conformal field theory with central charge c=−5 to two-dimensional quantum gravity. We measure the fractal properties of the ensemble, such as the string susceptibility exponent γ s and the intrinsic fractal dimension d H. We find γ s=−1.5(1) and d H=3.36(4), in reasonable agreement with theoretical predictions. In addition, we study the critical behavior of an Ising model on a quenched ensemble of the c=−5 graphs and show that it agrees, within numerical accuracy, with theoretical predictions for the critical behavior of an Ising model coupled dynamically to two-dimensional quantum gravity, with a total central charge of the matter sector c=−5.
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  • 27
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    Journal of statistical physics 96 (1999), S. 135-167 
    ISSN: 1572-9613
    Keywords: statistical entropy ; multiparticle correlations ; cumulant expansion ; lattice gases ; Ising model
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    Notes: Abstract A formula expressing the statistical entropy of a lattice-gas model as a multiparticle correlation expansion is derived in the grand-canonical and in the canonical ensembles. The differences from the analogous expansion in the continuum case are elucidated. The Ising model in one dimension is discussed as a case study.
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  • 28
    ISSN: 1572-9613
    Keywords: renormalization group ; Gibbsianness ; finite-size conditions ; complete analyticity ; strong mixing ; equivalence of ensembles ; Ising model
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    Notes: Abstract In this paper we study a renormalization-group map: the block averaging transformation applied to Gibbs measures relative to a class of finite-range lattice gases, when suitable strong mixing conditions are satisfied. Using a block decimation procedure, cluster expansion, and detailed comparison between statistical ensembles, we are able to prove Gibbsianness and convergence to a trivial (i.e., Gaussian and product) fixed point. Our results apply to the 2D standard Ising model at any temperature above the critical one and arbitrary magnetic field.
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  • 29
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    Journal of statistical physics 98 (2000), S. 131-244 
    ISSN: 1572-9613
    Keywords: Ising model ; boundary conditions ; renormalization group ; free boson ; conformal invariance ; critical phenomena
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    Notes: Abstract The partition function with boundary conditions for various two-dimensional Ising models is examined and previously unobserved properties of nonformal invariance and universality are established numerically.
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  • 30
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    Journal of statistical physics 98 (2000), S. 321-345 
    ISSN: 1572-9613
    Keywords: statistical mechanics ; variance reduction ; Monte Carlo algorithms ; Metropolis algorithm ; statistical estimators ; Ising model ; histogram methods ; transition probabilities
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    Notes: Abstract For Metropolis Monte Carlo simulations in statistical physics, efficient, easy- to-implement, and unbiased statistical estimators of thermodynamic properties are based on the transition dynamics. Using an Ising model example, we demonstrate (problem-specific) variance reductions compared to conventional histogram estimators. A proof of variance reduction in a microstate limit is presented.
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    Journal of statistical physics 99 (2000), S. 691-705 
    ISSN: 1572-9613
    Keywords: Monte Carlo methods ; Ising model ; computational physics
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    Notes: Abstract We discuss the conceptual differences between the broad histogram (BHM) and reweighting methods in general, and particularly the so-called multicanonical (MUCA) approaches. The main difference is that BHM is based on microcanonical, fixed-energy averages which depend only on the good statistics taken inside each energy level. The detailed distribution of visits among different energy levels, determined by the particular dynamic rule one adopts, is irrelevant. Contrary to MUCA, where the results are extracted from the dynamic rule itself, within BHM any microcanonical dynamics could be adopted. As a numerical test, we have used both BHM and MUCA in order to obtain the spectral energy degeneracy of the Ising model in 4×4×4 and 32×32 lattices, for which exact results are known. We discuss why BHM gives more accurate results than MUCA, even using the same Markovian sequence of states. In addition, such an advantage increases for larger systems.
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    Journal of statistical physics 78 (1995), S. 7-16 
    ISSN: 1572-9613
    Keywords: Statistical mechanics ; lattice models ; Ising model ; solvable models ; integrable systems ; Yang-Baxter relations
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    Notes: Abstract There is now a whole field in mathematical physics concerned with solvable models in statistical mechanics, field theory, and related areas. We indicate the influence that Onsager's solution of the planar Ising model has had, and continues to have, on this field.
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    Journal of statistical physics 81 (1995), S. 837-842 
    ISSN: 1572-9613
    Keywords: Ising model ; staggered field ; metastable state
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    Topics: Physics
    Notes: Abstract We report a Monte Carlo simulation of the layered Ising antiferromagnet under an external magnetic field. We show that under a staggered field, there occurs a phase transition from a metastable state which follows a Vogel-Fulcher law. For a staggered intrasublattice interaction a similar situation occurs.
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    Journal of statistical physics 83 (1996), S. 867-905 
    ISSN: 1572-9613
    Keywords: Ising model ; Wulff shape ; large deviations ; boundary effects
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    Topics: Physics
    Notes: Abstract We continue our study of the behavior of the two-dimensional nearest neighbor ferromagnetic Ising model under an external magnetic fieldh, initiated in our earlier work. We strengthen further a result previously proven by Martirosyan at low enough temperature, which roughly states that for finite systems with (−)-boundary conditions under a positive external field, the boundary effect dominates in the system if the linear size of the system is of orderB/h withB small enough, while ifB is large enough, then the external field dominates in the system. In our earlier work this result was extended to every subcritical value of the temperature. Here for every subcritical value of the temperature we show the existence of a critical valueB 0 (T) which separates the two regimes specified above. We also find the asymptotic shape of the region occupied by the (+)-phase in the second regime, which turns out to be a “squeezed Wulff shape”. The main step in our study is the solution of the variational problem of finding the curve minimizing the Wulff functional, which curve is constrained to the unit square. Other tools used are the results and techniques developed to study large deviations for the block magnetization in the absence of the magnetic field, extended to all temperatures below the critical one.
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  • 35
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    Journal of statistical physics 84 (1996), S. 1077-1093 
    ISSN: 1572-9613
    Keywords: Gibbs states ; ground states ; residual entropy ; random field ; Ising model
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    Topics: Physics
    Notes: Abstract We consider the random Gibbs field formalism for the ferromagnetic ID dichotomous random-field Ising model as the simplest example of a quenched disordered system. We prove that for nonzero temperatures the Gibb state is unique for any realization of the external field. Then we prove that asT→0, the Gibbs state converges to a limit, a ground state, for almost all realizations of the external field. The ground state turns out to be a probability measure concentrated on an infinite set of configurations, and we give a constructive description of this measure.
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    Journal of statistical physics 78 (1995), S. 147-160 
    ISSN: 1572-9613
    Keywords: Foam bilayer ; phase transition ; Ising model ; mean-field ; binding energy
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    Topics: Physics
    Notes: Abstract Foam bilayers from individual and mixed phosphatidylcholines are experimentally studied at different temperatures. Occurrence of a chain-melting phase transition in the foam bilayers is detected by two independent parameters—the critical concentrationC c for formation of foam bilayer and the foam bilayer thickness. The data forC c are discussed on the basis of the hole-nucleation theory, which applies the Ising model to foam bilayers and uses the mean-field approximation for interpretation of their stability. This allows the determination of the binding energy of a phospholipid molecule in gel and liquid-crystalline foam bilayers. New possibilities to relate the microscopic and macroscopic characteristics of foam bilayers are demonstrated.
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  • 37
    ISSN: 1572-9613
    Keywords: Polytypism ; Ising model ; Order-disorder ; X-ray scattering
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    Topics: Physics
    Notes: Abstract Layered crystalline materials like K3Me(CN)6 with Me=Cr, Mn, Fe, Co may often exist in various polytypic forms, due to a variety of choices of layer stacking modes. For cases where the interlayer constellations can be limited to only two energetically almost equivalent ways, the buildup of the crystal may be described by a spin-1/2 Ising-like model. For the system presently being studied one can rationalize the layer stacking to a four-valued choice (i.e., a 1D 4-state Potts case), or use an Ising-like two-sublattice model. Previous diffraction studies of K3Me(CN)6 indicated that two long-range ordered structures prevailed, an orthohombic one named MDO1, with one double layer per repetition unit, and a monoclinic one, MDO2, with two double-layer units. Our studies reveal a more complex situation: The Fe material is for the most part of the MDO2 type. But in addition, in some crystal samples, a hitherto unobserved phase also appears, with six double-layer repetition units, in fact a hybrid of MDO1 and MDO2. The Co material is for the most part of the MDO2 type, but contains in addition a considerable contribution of stacking disorder, as evidenced by the presence of diffuse X-ray scattering lines. The lines do, however, contain distinct maxima, indicating the presence of several layer stacking modes with preference of two, three, four, five, and seven double-layer correlations. The findings can be qualitatively discussed in terms of the ANNNI model.
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  • 38
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    Journal of statistical physics 80 (1995), S. 103-123 
    ISSN: 1572-9613
    Keywords: Discrete variational problem ; Ising model ; droplets ; metastability
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    Topics: Physics
    Notes: Abstract We consider a variational problem on thed-dimensional latticeZ d which has applications in the study of the meatastable behavior of the stochastic Ising model. The problem, an isoperimetric one, is to find what is the smallest area a finite subset ofZ d can have restricted to three classes of subsets ofZ d . If ϕ is one of these subsets, we define its volume as the number of points in it and its area as the number of pairs of points inZ d which are neighbors and such that only one of them belongs to ϕ.
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  • 39
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    Journal of statistical physics 86 (1997), S. 1117-1151 
    ISSN: 1572-9613
    Keywords: Majority rule ; renormalization group ; non-Gibbsianness ; finite-size conditions ; complete analyticity ; Ising model
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    Topics: Physics
    Notes: Abstract We study the majority rule transformation applied to the Gibbs measure for the 2D Ising model at the critical point. The aim is to show that the renormalized Hamiltonian is well defined in the sense that the renormalized measure is Gibbsian. We analyze the validity of Dobrushin-Shlosman uniqueness (DSU) finite-size condition for the “constrained models” corresponding to different configurations of the “image” system. It is known that DSU implies, in our 2D case, complete analyticity from which, as recently shown by Haller and Kennedy. Gibbsianness follows. We introduce a Monte Carlo algorithm to compute an upper bound to Vasserstein distance (appearing in DSU) between finite-volume Gibbs measures with different boundary conditions. We get strong numerical evidence that indeed the DSU condition is verified for a large enough volumeV for all constrained models.
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  • 40
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    Journal of statistical physics 92 (1998), S. 35-45 
    ISSN: 1572-9613
    Keywords: Competing influences ; Ising model ; Gibbs measures
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    Topics: Physics
    Notes: Abstract We continue a study of Schonmann (1994), Schonmann and Shlosman (1996), and Greenwood and Sun (1997) regarding the competing influences of boundary conditions and external field for the Ising model. We find a critical point B 0 in the competing influences for low temperature in dimension d 2A7E; 2.
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  • 41
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    Journal of statistical physics 92 (1998), S. 785-808 
    ISSN: 1572-9613
    Keywords: Critical phenomena ; diluted spin systems ; Ising model ; renormalization group
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    Topics: Physics
    Notes: Abstract Within the massive field-theoretic renormalization-group approach the expressions for the β and γ functions of the anisotropic mn-vector model are obtained for general space dimension d in three-loop approximation. Resumming corresponding asymptotic series, critical exponents for the case of the weakly diluted quenched Ising model (m = 1, n = 0), as well as estimates for the marginal order parameter component number m c of the weakly diluted quenched m-vector model, are calculated as functions of d in the region 2 ≤ d 〈 4. Conclusions concerning the effectiveness of different resummation techniques are drawn.
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  • 42
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    Journal of statistical physics 93 (1998), S. 573-582 
    ISSN: 1572-9613
    Keywords: Scaling ; Ising model ; Markov property ; critical phenomena ; parallel computational procedures
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    Topics: Physics
    Notes: Abstract The idea that near the critical point each block of spins behaves just like a single big spin is investigated. The case where a diamond-shaped block of spins is embedded in a (small) sea of spins is studied. Use is made of the Markov property method to make exact computations of the various spin moments needed to test this hypothesis. The residual fluctuation about the mean value of the block spin is seen to tend to a finite fraction of the length of the mean block-spin. This result is in line with previous studies which used different types of boundary conditions.
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    Journal of statistical physics 97 (1999), S. 87-144 
    ISSN: 1572-9613
    Keywords: Ising model ; anisotropic field ; phase diagram ; cluster expansion
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    Topics: Physics
    Notes: Abstract In this paper we analyze the equilibrium phase diagram of the two-dimensional ferromagnetic n.n. Ising model when the external field takes alternating signs on different rows. We show that some of the zero-temperature coexistence lines disappear at every positive sufficiently small temperature, whereas one (and only one) of them persists for sufficiently low temperature.
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    Journal of statistical physics 98 (2000), S. 551-588 
    ISSN: 1572-9613
    Keywords: Ising model ; universal amplitude ratios ; conformal field theory ; torus ; finite-size scaling ; corrections to scaling ; Monte Carlo ; Swendsen–Wang algorithm ; cluster algorithm
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    Topics: Physics
    Notes: Abstract Using results from conformal field theory, we compute several universal amplitude ratios for the two-dimensional Ising model at criticality on a symmetric torus. These include the correlation-length ratio x ★=lim L→∞ ξ(L)/L and the first four magnetization moment ratios V 2n =〈 $$M$$ 2n 〉/〈 $$M$$ 2〉 n . As a corollary we get the first four renormalized 2n-point coupling constants for the massless theory on a symmetric torus, G*2n . We confirm these predictions by a high-precision Monte Carlo simulation.
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    Journal of statistical physics 98 (2000), S. 1063-1073 
    ISSN: 1572-9613
    Keywords: polygon statistics ; Ising model ; exact solution
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    Topics: Physics
    Notes: Abstract We calculate the number of polygons with fixed total length drawn on a square lattice with periodic boundary conditions. In addition, we study the statistics of polygons with the number of horizontal and vertical links fixed separately. The analysis is performed via a mapping to the Ising model with isotropic and anisotropic interactions. We deal with the case of finite lattice sizes as well as the thermodynamic limit.
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  • 46
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    Journal of statistical physics 80 (1995), S. 1309-1326 
    ISSN: 1572-9613
    Keywords: Renormalization group ; position-space renormalization-group transformations ; Ising model ; low-temperature expansions
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    Topics: Physics
    Notes: Abstract A method for computing low-temperature series for renormalized operators in the two-dimensional Ising model is proposed. These series are applied to the study of the properties of the truncated renormalized Hamiltonians when we start at very low temperature and zero field. The truncated Hamiltonians for majority rule, Kadanoff transformation, and decimation for 2×2 blocks depend on the how we approach the first-order phase-transition line. The renormalization group transformations are multivalued and discontinuous at this first-order transition line when restricted to some finite-dimensional interaction space.
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  • 47
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    Journal of statistical physics 84 (1996), S. 85-118 
    ISSN: 1572-9613
    Keywords: Ising model ; Markov chain ; transfer matrix ; Friedrichs model ; saddle-point method ; scattering theory ; T-matrix
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    Topics: Physics
    Notes: Abstract We find the asymptotic decrease of correlations 〈σ A +y ,σ B 〉,y∈Z v +1, |y|→∞, in the Ising model at high temperatures. For the case when monomialsσ A andσ B both are odd, using the saddle-point method, we find the asymptotics of the correlations for any dimension ν. For even monomialsσ A ,σ B we formulate a general hypothesis about the form of the asymptotics and confirm it in two cases: (1) ν=1 and the vectory has an arbitrary direction, (2)y is directed along a fixed axis and arbitrary ν. Here we use besides the saddle-point method, some arguments from scattering theory.
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    Journal of statistical physics 84 (1996), S. 295-307 
    ISSN: 1572-9613
    Keywords: Ising model ; lattice Sierpinski gasket ; Dobrushin-Shlosmann mixing condition
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    Topics: Physics
    Notes: Abstract Ferromagnetic Ising models on the lattice Sierpinski gasket are considered. We prove the Dobrushin-Shlosmann mixing condition and discuss corresponding properties of the stochastic Ising models.
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  • 49
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    Journal of statistical physics 85 (1996), S. 383-401 
    ISSN: 1572-9613
    Keywords: Ising model ; roughening transition ; Monte Carlo ; finite-size scaling ; renormalization group
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    Topics: Physics
    Notes: Abstract We study the roughening transition of an interface in an Ising system on a 3D simple cubic lattice using a finite-size scaling method. The particular method has recently been proposed and successfully tested for various solid-on-solid models. The basic idea is the matching of the renormalization-groupflow of the interface with that of the exactly solvable body-centered cubic solid-on-solid model. We unambiguously confirm the Kosterlitz-Thouless nature of the roughening transition of the Ising interface. Our result for the inverse transition temperatureK r=0.40754(5) is almost two orders of magnitude more accurate than the estimate of Mon, Landau, and Stauffer.
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    International journal of thermophysics 17 (1996), S. 641-649 
    ISSN: 1572-9567
    Keywords: coexistence curse ; critical exponents ; critical point ; Ising model ; scaling laws ; universality classes
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract Based on the available literature data on (he temperature dependence of coexisting densities near the critical point for Ne, HD, N2, and C2,H4,. the disadvantages of using the extended scaling equation of the coexistence curve with the Ising exponent are shown, Combined statistical methods are proposed to reanalyze these data. For all the above-mentioned substances, in (he range of reduced temperature 1.3 x 10-4 〈 r 〈 6 x 10-3 to 2 x 10-2, we obtain for the order parameter a single-term fit, with the common value β=0.355. The fit describes the experimental densities with .m uncertainty ol 0,06 to 0.1%.
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  • 51
    ISSN: 1572-9567
    Keywords: benzene ; coexistence curve ; critical behavior ; critical point ; Ising model ; order parameter ; scaling laws ; triple point
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    Topics: Physics
    Notes: Abstract The temperature dependence of the density along the coexistence curve of benzene in the vicinity of the critical point and in a wide temperature range down to the triple point was investigated, The original results as well as literature data were statistically treated, A regression analysis of data on the critical exponents and critical amplitudes used as fitting parameters in a model equations was curried out, An adequate description of the order parameter by the three-term scaling equation in the entire two-phase (liquid gas) region of benzene was obtained with experimental values ofβ 0 = 0.352 ± 0,003 and Δ = 1.3 ± 0.2, which are inconsistent with the Ising modelβ(0=0.325) and the Wegner exponent (Δ=0.5). respectively, It is shown that the equation with fixed classical exponents does not adequately describe the experimental data even far from the critical point.
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