Unlimited number of spanning clusters in three-dimensional percolation above the critical point - Carlos G. L. Sousa; José S. Andrade, Jr.; André A. Moreira; Cesar I. N. Sampaio Filho; Erika Eiser; Alex Hansen; Hans J. Herrmann (Physical Review Research)
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.