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Cluster percolation in the three-dimensional ±J random-bond Ising model

L. Münster1 and M. Weigel1,2

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Phys. Rev. E 113, 024139 – Published 26 February, 2026

DOI: https://doi.org/10.1103/y464-wxbk

Abstract

Based on extensive parallel-tempering Monte Carlo simulations, we investigate the relationship between cluster percolation and equilibrium ordering phenomena in the three-dimensional ±J random-bond Ising model for different fractions of antiferromagnetic bonds. We consider a range of cluster definitions, most of which are constructed in the space of overlaps between two independent replicas of the system. In the pure ferromagnet that is contained as a limiting case in the class of problems considered, the relevant percolation point coincides with the thermodynamic ordering transition. For the disordered ferromagnet encountered first on introducing antiferromagnetic bonds and the adjacent spin-glass phase of strong disorder, this connection is altered, and one finds a percolation transition above the thermodynamic ordering point that is accompanied by the appearance of two percolating clusters of equal density. Only at the lower (disordered) ferromagnetic or spin-glass transition points the densities of these two clusters start to diverge, thus providing a percolation signature of these thermodynamic transitions. We compare the scaling behavior at this secondary percolation transition with the thermodynamic behavior at the corresponding ferromagnetic and spin-glass phase transitions.

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Phys. Rev. E 113, 024139– Published 26 February, 2026
  • Received 13 November 2025
  • Accepted 2 February 2026
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