Monte Carlo Simulations of Spin Systems

Janke, Wolfhard

doi:10.1007/978-3-642-85238-1_3

Cited by 31 publications

(36 citation statements)

References 109 publications

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“…However, a single-cluster variant can also be implemented with strictly O(N) run-time scaling and is hence found to be asymptotically more efficient than Luijten's approach. For the simulations close to criticality, we determine integrated autocorrelation times to ensure equilibration and sufficient independence of successive samples [35]. Our simulations indicate a dramatic reduction of autocorrelation times and also the dynamical critical exponents through the use of the cluster updates.…”

Section: Cluster-update Monte Carlo Simulationsmentioning

confidence: 99%

Finite-size scaling above the upper critical dimension in Ising models with long-range interactions

et al. 2015

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The correlation length plays a pivotal role in finite-size scaling and hyperscaling at continuous phase transitions. Below the upper critical dimension, where the correlation length is proportional to the system length, both finite-size scaling and hyperscaling take conventional forms. Above the upper critical dimension these forms break down and a new scaling scenario appears. Here we investigate this scaling behaviour by simulating one-dimensional Ising ferromagnets with long-range interactions. We show that the correlation length scales as a non-trivial power of the linear system size and investigate the scaling forms. For interactions of sufficiently long range, the disparity between the correlation length and the system length can be made arbitrarily large, while maintaining the new scaling scenarios. We also investigate the behavior of the correlation function above the upper critical dimension and the modifications imposed by the new scaling scenario onto the associated Fisher relation.

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Section: Cluster-update Monte Carlo Simulationsmentioning

confidence: 99%

Finite-size scaling above the upper critical dimension in Ising models with long-range interactions

et al. 2015

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“…Hence, the plaquette update simulates the (transcribed) dual rather than the original model itself. Figure 7 gives the exact internal energy of the original Ising model on a finite lattice with periodic boundary conditions [39,40] and that of the dual model transcribed to the original one using Eq. (38).…”

Section: Monte Carlo Studymentioning

confidence: 99%

Geometrical vs. Fortuin–Kasteleyn clusters in the two-dimensional q-state Potts model

Janke

Schakel

2004

Nuclear Physics B

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The tricritical behavior of the two-dimensional q-state Potts model with vacancies for 0 ≤ q ≤ 4 is argued to be encoded in the fractal structure of the geometrical spin clusters of the pure model. The known close connection between the critical properties of the pure model and the tricritical properties of the diluted model is shown to be reflected in an intimate relation between Fortuin-Kasteleyn and geometrical clusters: The same transformation mapping the two critical regimes onto each other also maps the two cluster types onto each other. The map conserves the central charge, so that both cluster types are in the same universality class. The geometrical picture is supported by a Monte Carlo simulation of the high-temperature representation of the Ising model (q = 2) in which closed graph configurations are generated by means of a Metropolis update algorithm involving single plaquettes.

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“…where the constant t defines the characteristic length scale at which the environmental response changes, known as the correlation length (Janke 1996). To model environmental variation in both space and time, we generate U j, x (t) with a two-parameter autoregressive process ("Modeling Details"), which introduces both spatial and temporal correlation lengths.…”

Section: Model Of Stream Communitiesmentioning

confidence: 99%

Scale-Dependent Community Theory for Streams and Other Linear Habitats

Holt

Chesson

2016

The American Naturalist

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The maintenance of species diversity occurs at the regional scale but depends on interacting processes at the full range of lower scales. Although there is a long history of study of regional diversity as an emergent property, analyses of fully multiscale dynamics are rare. Here, we use scale transition theory for a quantitative analysis of multiscale diversity maintenance with continuous scales of dispersal and environmental variation in space and time. We develop our analysis with a model of a linear habitat, applicable to streams or coastlines, to provide a theoretical foundation for the long-standing interest in environmental variation and dispersal, including downstream drift. We find that the strength of regional coexistence is strongest when local densities and local environmental conditions are strongly correlated. Increasing dispersal and shortening environmental correlations weaken the strength of coexistence regionally and shift the dominant coexistence mechanism from fitness-density covariance to the spatial storage effect, while increasing local diversity. Analysis of the physical and biological determinants of these mechanisms improves understanding of traditional concepts of environmental filters, mass effects, and species sorting. Our results highlight the limitations of the binary distinction between local communities and a species pool and emphasize species coexistence as a problem of multiple scales in space and time.

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Monte Carlo Simulations of Spin Systems

Cited by 31 publications

References 109 publications

Finite-size scaling above the upper critical dimension in Ising models with long-range interactions

Finite-size scaling above the upper critical dimension in Ising models with long-range interactions

Geometrical vs. Fortuin–Kasteleyn clusters in the two-dimensional q-state Potts model

Scale-Dependent Community Theory for Streams and Other Linear Habitats

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