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Critical behavior of the multigel percolation model

Carlos G. L. Sousa1,*, José S. Andrade, Jr.1, Cesar I. N. Sampaio Filho1, Erika Eiser2, Alex Hansen2, Hans J. Herrmann1,3, and André A. Moreira1

  • *Contact author: germano@fisica.ufc.br
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Phys. Rev. Research 8, 033125 – Published 30 July, 2026

DOI: https://doi.org/10.1103/762d-x9gp

Abstract

In a recently introduced bond-percolation model, Sousa et al. [Phys. Rev. Res. 7, L022067 (2025)] showed that multiple spanning clusters can coexist in three dimensions, with their number remaining unbounded in the thermodynamical limit. Here, we investigate further this multigel percolation model by showing that the fractal dimension of the shortest path is the same as standard three-dimensional percolation, while the fractal dimension of the contact between percolating clusters at the critical point is consistent with that expected for the intersection of two fractal objects. When the initial density is treated as a control parameter, the behavior of both the largest cluster and the second moment follows the same scaling relations as in standard percolation. For systems with multiple species, the minimum density required for all clusters to percolate scales with system size in agreement with the fractal dimension. In the case of two species with different initial densities, we find a transition from a regime where only one species percolates to one where both percolate. These results strengthen the connection between multigel percolation and classical percolation theory and suggest possible routes for designing materials with multiple interpenetrating percolating structures, such as trigels and tetragels.

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Phys. Rev. Research 8, 033125– Published 30 July, 2026
  • Received 29 January 2026
  • Accepted 2 June 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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