Percolation processes in d-dimensions

Gaunt, D S; Sykes, M F; Ruskin, Heather J.

doi:10.1088/0305-4470/9/11/015

Cited by 127 publications

(66 citation statements)

References 18 publications

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“…in reasonable agreement with the exponent x = 0.75 f 0.05 from Gaunt and Sykes (1976) and consistent with the renormalisation group result that the correction exponent is of order unity (Houghton et a1 1978; G Grest, private communication). (If we were to neglect the correction term, treat T as a free exponent and fit it on n, between s = 10' and s = lo4, we would get T = 2.02, which is somewhat too low .)…”

Section: ( 5 6 )supporting

confidence: 88%

Monte Carlo experiments on cluster size distribution in percolation

Hoshen

Stauffer

Bishop

et al. 1979

J. Phys. A: Math. Gen.

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Section: ( 5 6 )supporting

confidence: 88%

Monte Carlo experiments on cluster size distribution in percolation

Hoshen

Stauffer

Bishop

et al. 1979

J. Phys. A: Math. Gen.

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“…Gaunt, Sykes, and Ruskin [GSR76] provided the numbers of 3-to 7-dimensional polycubes up to size 13, 11, 10, 9, and 9, respectively (with a slight error in the count of 3-dimensional polycubes of size 13). The counts of 3-dimensional polycubes of up to size 17 can be derived from data obtained by Martin [Ma90] and published by Madras et al [MSW+90].…”

Section: Introductionmentioning

confidence: 99%

Formulae and Growth Rates of High-Dimensional Polycubes

Barequet

Rote

2009

Electronic Notes in Discrete Mathematics

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“…Arguments of this kind have already been applied to percolation on hypercubic lattices [25] making use of the fact that loops in random graphs on sparsely occupied high dimensional lattices are infrequent. Indeed, the first terms of a high dimension expansion on a hypercubic lattice of dimension d are given by the exact values of the percolation threshold for the Bethe-lattice with coordination number d.…”

Section: Scaling Laws and Bethe Approximationmentioning

confidence: 99%

“…This gives an additional argument that the connectivity thresholds of both sequences of graphs are equal which can also be verified by evaluating eq. (25). Mean value t of the resulting great fragments after application of the edge-elimination procedure for G 1 9 depending on the occupation probability p. …”

Section: Appendix A: Renormalization Group Approachmentioning

confidence: 99%

Architecture of idiotypic networks: Percolation and scaling Behavior

Brede

Behn

2001

Phys. Rev. E

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We investigate a model where idiotypes (characterizing B-lymphocytes and antibodies of an immune system) and anti-idiotypes are represented by complementary bitstrings of a given length d allowing for a number of mismatches (matching rules). In this model, the vertices of the hypercube in dimension d represent the potential repertoire of idiotypes. A random set of (with probability p) occupied vertices corresponds to the expressed repertoire of idiotypes at a given moment. Vertices of this set linked by the above matching rules build random clusters. We give a structural and statistical characterization of these clusters -or in other words -of the architecture of the idiotypic network. Increasing the probability p one finds at a critical p a percolation transition where for the first time a large connected graph occures with probability one. Increasing p further, there is a second transition above which the repertoire is complete in the sense that any newly introduced idiotype finds a complementary anti-idiotype. We introduce structural characteristics such as the mass distributions and the fragmentation rate for random clusters, and determine the scaling behaviour of the cluster size distribution near the percolation transition, including finite size corrections. We find that slightly above the percolation transition the large connected cluster (the central part of the idiotypic network) consists typically of one highly connected part and a number of weakly connected constituents and coexists with a number of small, isolated clusters. This is in accordance with the picture of a central and a peripheral part of the idiotypic network and gives some support to idealized architectures of the central part used in recent dynamical mean field models.

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Percolation processes in d-dimensions

Cited by 127 publications

References 18 publications

Monte Carlo experiments on cluster size distribution in percolation

Monte Carlo experiments on cluster size distribution in percolation

Formulae and Growth Rates of High-Dimensional Polycubes

Architecture of idiotypic networks: Percolation and scaling Behavior

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