No-enclave percolation corresponds to holes in the cluster backbone

Abstract

The no-enclave percolation (NEP) model introduced recently by Sheinman et al. can be mapped to a problem of holes within a standard percolation backbone, and numerical measurements of these holes gives the size-distribution exponent $\tau = 1.82(1)$ of the NEP model. An argument is given that $\tau=1 + d_B/2 \approx 1.822$ where $d_B$ is the backbone dimension. On the other hand, a model of simple holes within a percolation cluster implies $\tau = 1 + d_f/2 = 187/96 \approx 1.948$, where $d_f$ is the fractal dimension of the cluster, and this value is consistent with Sheinman et al.'s experimental results of gel collapse which gives $\tau = 1.91(6)$. Both models yield a discontinuous maximum hole size at $p_c$, signifying explosive percolation behavior. At $p_c$, the largest hole fills exactly half the system, due to symmetry. Extensive numerical simulations confirm our results.