Comparison of rigidity and connectivity percolation in two dimensions

Using a recently developed algorithm for generic rigidity of two-dimensional graphs, we analyze rigidity and connectivity percolation transitions in two dimensions on lattices of linear size up to $L=4096.\mathrm{}$ We compare three different universality classes: the generic rigidity class, the connectivity class, and the generic “braced square net”(GBSN). We analyze the spanning cluster density $P_{\infty },$ the backbone density $P_B$, and the density of dangling ends $P_D.$ In the generic rigidity (GR) and connectivity cases, the load-carrying component of the spanning cluster, the backbone, is fractal at $p_c,$ so that the backbone density behaves as $B\sim (p-p_c{)}^{{\beta }^{\prime }}$ for $p>\mathrm{}{p}_c.$ We estimate ${\beta }_{\mathrm{gr}}^{\prime }=0.25\mathrm{}\pm 0.02$ for generic rigidity and ${\beta }_c^{\prime }=0.467\mathrm{}\pm 0.007$ for the connectivity case. We find the correlation length exponents ${\nu }_{\mathrm{gr}}=1.16\mathrm{}\pm 0.03$ for generic rigidity compared to the exact value for connectivity, ${\nu }_c=\frac{4}{3}.$ In contrast the GBSN undergoes a first-order rigidity transition, with the backbone density being extensive at $p_c,$ and undergoing a jump discontinuity on reducing p across the transition. We define a model which tunes continuously between the GBSN and GR classes, and show that the GR class is typical.