Proof of Straley's argument for bootstrap percolation

Enter, Aernout van

doi:10.1007/bf01019705

Cited by 122 publications

(82 citation statements)

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“…Nontrivially, Schonmann [261] was able to prove rigorously that all models with the intermediate values 2 ≤ f ≤ d are also effectively irreducible. Enter [262] had earlier proved the result for the special case of the 2, 2-SFM, formalizing an earlier unpublished argument due to Straley; Schonmann [263,261] gave a generalization to BP-like models with more complicated rules. Fredrickson and Andersen [74] had earlier given a non-rigorous argument for irreducibility of the 3, 3-SFM; Reiter [82] also constructed irreducibility proofs for the 2, 2-SFM and 3, 3-SFM.…”

Section: Irreducibilitymentioning

confidence: 69%

Glassy dynamics of kinetically constrained models

Ritort

Sollich

2003

Advances in Physics

741

1,055

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We review the use of kinetically constrained models (KCMs) for the study of dynamics in glassy systems. The characteristic feature of KCMs is that they have trivial, often non-interacting, equilibrium behaviour but interesting slow dynamics due to restrictions on the allowed transitions between configurations. The basic question which KCMs ask is therefore how much glassy physics can be understood without an underlying "equilibrium glass transition". After a brief review of glassy phenomenology, we describe the main model classes, which include spin-facilitated (Ising) models, constrained lattice gases, models inspired by cellular structures such as soap froths, models obtained via mappings from interacting systems without constraints, and finally related models such as urn, oscillator, tiling and needle models. We then describe the broad range of techniques that have been applied to KCMs, including exact solutions, adiabatic approximations, projection and mode-coupling techniques, diagrammatic approaches and mappings to quantum systems or effective models. Finally, we give a survey of the known results for the dynamics of KCMs both in and out of equilibrium, including topics such as relaxation time divergences and dynamical transitions, nonlinear relaxation, aging and effective temperatures, cooperativity and dynamical heterogeneities, and finally non-equilibrium stationary states generated by external driving. We conclude with a discussion of open questions and possibilities for future work.

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Section: Irreducibilitymentioning

confidence: 69%

Glassy dynamics of kinetically constrained models

Ritort

Sollich

2003

Advances in Physics

741

1,055

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“…Models with thresholds of p c = 1 where there are no finite clusters and first order transitions, approach the thermodynamic limit in complex logarithmic ways. The first results for the models with p c = 1 were obtained by Straley, [18] rigorised by van Enter [19] and then extended by Aizenman and Lebowitz, [20]. There is also an earlier proof of Frobose and Jackle.…”

Section: Exact Results and Scaling For Large Systemsmentioning

confidence: 93%

Bootstrap Percolation: visualizations and applications

Adler¹,

Lev²

2003

Braz. J. Phys.

209

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Bootstrap percolation models describe systems as diverse as magnetic materials, fluid flow in rocks and computer storage systems. The models have a common feature of requiring not just a simple connectivity of neighbouring sites, but rather an environment of other suitably occupied sites. Different applications as well as the connection with the mathematical literature on these models is presented. Visualizations that show the compact nature of the clusters are provided. I IntroductionIn Bootstrap Percolation (BP) [1] the lattice is occupied randomly with probability p, and then all sites that do not have at least m neighbours are iteratively removed. For m = 1 isolated sites are removed and for m = 2 both isolated sites and dangling ends are removed. For m = 3 and above the situation becomes interesting and lattice dependent and is currently quite hot amongst probabilists. For sufficiently high m, no occupied sites remain for an infinite lattice, and the interest is in the scaling as a function of system size.The conference talk on which this manuscript is based came about as follows: Uri Lev asked for a project topic just as Joan Adler was thinking about BP because Scott Kirpatrick's [2] new application to computer memory storage had reminded her that BP was an interesting topic. (Uri Lev's visualizations and website [3] are described in the final section of this paper.) Other recent activity, also motivated from ref [2] The models have connections to rigidity percolation, [11]. Time developments according to the local rules of Bootstrap Percolation can be viewed as cellular automata [12] , and there are also metastable bootstrap models, not considered further here. Models closely related to canonical BP have been proposed as descriptions of fluid flow in porous media. In these cases crack development [13] and/or advance of the fluid front [14] are assumed to be strongly dependent on the local microstructure. II Applications of Bootstrap PercolationWe show here the importance of the bootstrap principle for several applications. Adler, Palmer and Meyer, [10] showed that environmental effects of a bootstrap type play a crucial role in the orientational ordering process of quadrupoles in solid molecularOrtho hydrogen and para deuterium are quadrupolar molecules, whereas para hydrogen and or- The recent application by Kirkpatrick [2] is of a completely different nature. It relates to the failure of units in a cubic structured collection of computer memories, called a Dense Storage Array. Because it is too expensive to fix individual memory units that fail in such a system, the units are allowed to "fail in place", and it is necessary to ensure that even with a relatively high density of failed units there will still be percolation of data communications. III VisualizationsSince the Computational Physics Group at the Technion developed animation software (AViz) for atomistic simulations it was interesting to apply this also to Bootstrap Percolation. AViz [16] enables a user to draw and animate atoms (or spins, or...

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“…• 0 • • In that case p c = 0 for the infinite lattice [31] (so for each initial density the lattice will fill up). The finite-size effects at small p are such that if the size of a square is of order exp(C 1 × 1 p ), with C 1 Cst for an explicit, computable constant Cst, the square tends to be occupied in the end, while if it is of order exp(C 2 × 1 p ), with C 2 Cst, it tends to remain mostly empty [5,17] with overwhelming probability.…”

Section: Definition and Some Known Properties Of Bootstrap Percolatiomentioning

confidence: 99%