The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-Brownian excursion
TLDR
For independent nearest-neighbor bond percolation on Zd with d ≥ 6, it was shown in this paper that the incipient infinite cluster's two-point function and threepoint function converge to those of integrated super-Brownian excursion (ISE) in the scaling limit.Abstract:
For independent nearest-neighbor bond percolation on Zd with d≫6, we prove that the incipient infinite cluster’s two-point function and three-point function converge to those of integrated super-Brownian excursion (ISE) in the scaling limit. The proof is based on an extension of the new expansion for percolation derived in a previous paper, and involves treating the magnetic field as a complex variable. A special case of our result for the two-point function implies that the probability that the cluster of the origin consists of n sites, at the critical point, is given by a multiple of n−3/2, plus an error term of order n−3/2−e with e>0. This is a strong version of the statement that the critical exponent δ is given by δ=2.read more
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References
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Dietrich Stauffer,Amnon Aharony +1 more
TL;DR: In this paper, a scaling solution for the Bethe lattice is proposed for cluster numbers and a scaling assumption for cluster number scaling assumptions for cluster radius and fractal dimension is proposed.
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Dietrich Stauffer,Amnon Aharony +1 more
TL;DR: In this article, a scaling solution for the Bethe lattice is proposed for cluster numbers and a scaling assumption for cluster number scaling assumptions for cluster radius and fractal dimension is proposed.
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S.R. Broadbent,J. M. Hammersley +1 more
TL;DR: In this paper, the authors study how the random properties of a medium influence the percolation of a fluid through it, in a general way, in which the treatment diifers from conventional diffusion theory.
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The Continuum Random Tree III
TL;DR: The notion of convergence in distribution was introduced in this paper, which is based on the assumption that, for fixed k, the subtrees of a random tree determined by k randomly chosen vertices converge to a limit continuum random tree.
Journal ArticleDOI
Singularity analysis of generating functions
Philippe Flajolet,Andrew Odlyzko +1 more
TL;DR: This work presents a class of methods by which one can translate, on a term-by-term basis, an asymptotic expansion of a function around a dominant singularity into a corresponding asymPTotic expansion for the Taylor coefficients of the function.