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Entropy
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Percolation in the 21st Century
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Percolation in the 21st Century

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Statistical Physics".

Deadline for manuscript submissions: 15 June 2025 | Viewed by 3322

Special Issue Editor

Prof. Dr. Robert M. Ziff

E-Mail Website
Guest Editor
Department of Chemical Engineering, University of Michigan, Ann Arbor, MI 48109, USA
Interests: statistical physics; percolation; Bose–Einstein condensation; aggregation

Special Issue Information

Dear Colleagues,

The percolation model was introduced by Broadbent and Hammersley in the 1950s, but its antecedents can be traced back to the 19th century—namely, to the Galton–Watson process of branching theory, which originally concerned the extinction of family surnames, and the gelation theories of Flory and Stockmayer in the 1940s. The model captured the attention of physicists in the 1970s and 1980s; however, the work of Cardy and others using conformal field theory in the 1990s led to an explosion of interest among mathematicians starting around 2000, which continues today. Other notable areas of interest include percolation, explosive percolation, and percolation on complex networks.

In this volume, we focus on the theory and applications of percolation and its many generalizations. This includes the fields of physics, mathematics, and engineering. We solicit papers both reviews of recent work and new developments and directions for the future.

Prof. Dr. Robert M. Ziff
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • percolation
  • explosive percolation
  • percolation on complex networks
  • physics
  • mathematics
  • engineering

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Published Papers (5 papers)

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Research

25 pages, 657 KiB  
Article
Finite-Size Corrections from the Subleading Magnetic Scaling Field for the Ising and Potts Models in Two Dimensions
by Yihao Xu, Jesús Salas and Youjin Deng
Entropy 2025, 27(4), 418; https://doi.org/10.3390/e27040418 - 11 Apr 2025
Viewed by 244
Abstract
In finite-size scaling analyses of critical phenomena, proper consideration of correction terms, which can come from different sources, plays an important role. For the Fortuin–Kasteleyn representation of the Q-state Potts model in two dimensions, although the subleading magnetic scaling field, with exactly [...] Read more.
In finite-size scaling analyses of critical phenomena, proper consideration of correction terms, which can come from different sources, plays an important role. For the Fortuin–Kasteleyn representation of the Q-state Potts model in two dimensions, although the subleading magnetic scaling field, with exactly known exponent, is theoretically expected to give rise to finite-size-scaling analyses, numerical observation remains elusive, probably due to the mixing of various corrections. We simulate the O(n) loop model on the hexagonal lattice, which is in the same universality class as the Q=n2 Potts model but has suppressed corrections from other sources and provides strong numerical evidence for the attribution of the subleading magnetic field in finite-size corrections. Interestingly, it is also observed that the corrections in small- and large-cluster-size regions have opposite magnitudes, and, for the special n=2 case, they compensate with each other in observables like the second moment of the cluster-size distribution. Our finding reveals that the effect of the subleading magnetic field should be taken into account in finite-size-scaling analyses, which was unfortunately ignored in many previous studies. Full article
(This article belongs to the Special Issue Percolation in the 21st Century)
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14 pages, 1352 KiB  
Article
Applications of Percolation Theory to Prevent the Propagation of Phytopathogens and Pests on Plantations
by J. Alonso Tlali, J. R. Alvarado García, B. Cardenas Castro, A. Fernández Téllez, E. G. García Prieto, J. F. López-Olguín, Y. Martínez Laguna, J. E. Ramírez, D. Rosales Herrera and J. D. Silva Montiel
Entropy 2025, 27(4), 386; https://doi.org/10.3390/e27040386 - 5 Apr 2025
Viewed by 680
Abstract
One of the most important problems in agroecology is designing eco-friendly strategies to minimize the propagation of phytopathogens and pests. In this paper, we explore some strategies based on the modification of the plantation configuration together with percolation theory to prevent the propagation [...] Read more.
One of the most important problems in agroecology is designing eco-friendly strategies to minimize the propagation of phytopathogens and pests. In this paper, we explore some strategies based on the modification of the plantation configuration together with percolation theory to prevent the propagation of phytopathogens and pests that move over nearest neighbor plants, such as the case of Phytophthora zoospores or pest mites. The percolation threshold is determined for well-mixed and intercropping plantations modeled in nearest neighbor square lattices. Our main result is that the best agroecology strategy consists of designing polyculture plantations to raise the net production yield. Full article
(This article belongs to the Special Issue Percolation in the 21st Century)
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17 pages, 958 KiB  
Article
Lower Limit of Percolation Threshold on Square Lattice with Complex Neighborhoods
by Antoni Piotr Ciepłucha, Marcin Utnicki, Maciej Wołoszyn and Krzysztof Malarz
Entropy 2025, 27(4), 361; https://doi.org/10.3390/e27040361 - 29 Mar 2025
Viewed by 314
Abstract
In this paper, the 60-year-old concept of long-range interaction in percolation problems introduced by Dalton, Domb and Sykes is reconsidered. With Monte Carlo simulation—based on the Newman–Ziff algorithm and the finite-size scaling hypothesis—we estimate 64 percolation thresholds for a random site percolation problem [...] Read more.
In this paper, the 60-year-old concept of long-range interaction in percolation problems introduced by Dalton, Domb and Sykes is reconsidered. With Monte Carlo simulation—based on the Newman–Ziff algorithm and the finite-size scaling hypothesis—we estimate 64 percolation thresholds for a random site percolation problem on a square lattice with neighborhoods that contain sites from the seventh coordination zone. The percolation thresholds obtained range from 0.27013 (for the neighborhood that contains only sites from the seventh coordination zone) to 0.11535 (for the neighborhood that contains all sites from the first to the seventh coordination zone). Similarly to neighborhoods with smaller ranges, the power-law dependence of the percolation threshold on the effective coordination number with an exponent close to −1/2 is observed. Finally, we empirically determine the limit of the percolation threshold on square lattices with complex neighborhoods. This limit scales with the inverse square of the mean radius of the neighborhood. The boundary of this limit is touched for threshold values associated with extended (compact) neighborhoods. Full article
(This article belongs to the Special Issue Percolation in the 21st Century)
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12 pages, 12984 KiB  
Article
Scaling and Clustering in Southern California Earthquake Sequences: Insights from Percolation Theory
by Zaibo Zhao, Yaoxi Li and Yongwen Zhang
Entropy 2025, 27(4), 347; https://doi.org/10.3390/e27040347 - 27 Mar 2025
Viewed by 247
Abstract
Earthquake activity poses significant risks to both human survival and economic development. However, earthquake forecasting remains a challenge due to the complex, poorly understood interactions that drive seismic events. In this study, we construct an earthquake percolation model to examine the relationships between [...] Read more.
Earthquake activity poses significant risks to both human survival and economic development. However, earthquake forecasting remains a challenge due to the complex, poorly understood interactions that drive seismic events. In this study, we construct an earthquake percolation model to examine the relationships between earthquakes and the underlying patterns and processes in Southern California. Our results demonstrate that the model can capture the spatiotemporal and magnitude characteristics of seismic activity. Through clustering analysis, we identify two distinct regimes: a continuous increase driven by earthquake clustering, and a discontinuous increase resulting from the merging of clusters dominated by large, distinct mega-earthquakes. Notably, in the continuous increase regime, we observe that clusters exhibit a broader spatiotemporal distribution, suggesting long-range and long-term correlations. Additionally, by varying the magnitude threshold, we explore the scaling behavior of earthquake percolation. The robustness of our findings is confirmed through comparison with multiple shuffling tests. Full article
(This article belongs to the Special Issue Percolation in the 21st Century)
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9 pages, 1869 KiB  
Article
Physical Realizations of Interdependent Networks: Analogy to Percolation
by Bnaya Gross and Shlomo Havlin
Entropy 2025, 27(2), 109; https://doi.org/10.3390/e27020109 - 23 Jan 2025
Viewed by 855
Abstract
Percolation on interdependent networks generalizes the well-studied percolation model in a single network to multiple interacting systems, unveiling spontaneous cascading failures, abrupt collapses, and high vulnerability. The main novelty of interdependent networks has been the introduction of two types of links, connectivity within [...] Read more.
Percolation on interdependent networks generalizes the well-studied percolation model in a single network to multiple interacting systems, unveiling spontaneous cascading failures, abrupt collapses, and high vulnerability. The main novelty of interdependent networks has been the introduction of two types of links, connectivity within networks and the dependency between them. The interplay between these two types of interactions results in novel critical phenomena and phase transitions. This abstract percolation paradigm was recently applied to magnetic networks, as an experimentally testable method for interdependent superconducting networks as well as to other systems like k-core percolation and overloaded networks. Here, we will review these physical applications and provide insights into several potential directions for the field of physically interdependent networks. Full article
(This article belongs to the Special Issue Percolation in the 21st Century)
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