Unlimited number of spanning clusters in three-dimensional percolation above the critical point

Inspired by the formation of bigels, we developed a bond percolation model that yields multiple percolating clusters in three dimensions not only at the critical point, but also above it. Our simulations suggest that, in the thermodynamic limit, there is no upper limit to the number of percolating clusters. We show that in finite systems the maximum number of percolating clusters that can be obtained grows logarithmically with the lattice size. For equal initial densities in the thermodynamic limit, all clusters percolate at the same threshold and exhibit critical exponents consistent with the critical exponents of standard percolation. The threshold depends linearly on the initial density of species and the maximum and minimum initial densities decay exponentially with the maximal number of spanning clusters.

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Publication Details

Journal
Physical Review Research
Published
2025-06-16
DOI
https://doi.org/10.1103/physrevresearch.7.l022067
Primary Topic
Stochastic processes and statistical mechanics
Type
article
Field-Weighted Citation Impact
0.00

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article

Unlimited number of spanning clusters in three-dimensional percolation above the critical point

Carlos Germano Lima de Sousa, Erika Eiser, José S. Andrade, Alex Hansen et al.
Physical Review Research
Stochastic processes and statistical mechanics
article

Unlimited number of spanning clusters in three-dimensional percolation above the critical point

Carlos Germano Lima de Sousa, Erika Eiser, José S. Andrade, Alex Hansen, Cesar I. N. Sampaio Filho, Hans J. Herrmann, André A. Moreira
article en

Abstract

Inspired by the formation of bigels, we developed a bond percolation model that yields multiple percolating clusters in three dimensions not only at the critical point, but also above it. Our simulations suggest that, in the thermodynamic limit, there is no upper limit to the number of percolating clusters. We show that in finite systems the maximum number of percolating clusters that can be obtained grows logarithmically with the lattice size. For equal initial densities in the thermodynamic limit, all clusters percolate at the same threshold and exhibit critical exponents consistent with the critical exponents of standard percolation. The threshold depends linearly on the initial density of species and the maximum and minimum initial densities decay exponentially with the maximal number of spanning clusters.

Physical Review ResearchVol. 7(2)
Norwegian University of Science and Technology (NO), Universidade Federal do Ceará (BR), Physique et Mécanique des Milieux Hétérogènes (FR), ESPCI Paris (FR)
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, Conselho Nacional de Desenvolvimento Científico e Tecnológico, Fundação Cearense de Apoio ao Desenvolvimento Científico e Tecnológico, Norges Forskningsråd, Instituto Nacional de Ciência e Tecnologia de Sistemas Complexos, European Research Council
Openalex Percentile: Top 80%
Stochastic processes and statistical mechanics
DOIOpen Access PDFOpenAlex
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Unlimited number of spanning clusters in three-dimensional percolation above the critical point — Carlos Germano Lima de Sousa, Erika Eiser, et al. · Physical Review Research (2025) | TGRS Research Map | TGRS