Fractal backbone within a compact spanning cluster: A new scenario in central-force rigidity percolation

Using a recently developed algorithm for generic rigidity of two-dimensional graphs, the rigidity percolation transition is studied on randomly diluted triangular lattices of linear size up to 1024. For both site and bond dilution, and for several kinds of boundary conditions, we find that the relative number of bonds that are rigidly connected to both ends of the sample, the spanning cluster density $P_{\infty}$ jumps from zero to approximately 0.1 at the transition point $p_c$. The stress-carrying component of the spanning cluster, the elastic backbone, is fractal at $p_c$, so that the backbone density behaves as $B \sim (p-p_c)^{\beta'}$ above $p_c$. We estimate $\beta' = 0.25 \pm 0.02$. We also find a non-trivial value for the correlation-length exponent, $\nu = 1.16 \pm 0.03$, which is associated with a divergent length-scale at $p_c$ due to the fractal backbone. Therefore the rigidity percolation transition is second-order, despite the discontinuous character of the spanning cluster, which is described by $P_{\infty} \sim \Delta P + B_0 (p-p_c)^{\beta'}$ above $p_c$. We therefore propose that the backbone density, and not the spanning cluster density, is the appropriate order parameter in this problem.