Comparison of rigidity and connectivity percolation in two dimensions

Using a recently developed algorithm for generic rigidity of two-dimensiona l graphs, we analyze rigidity and connectivity percolation transitions in t wo dimensions on lattices of linear size up to L = 4096. We compare three d ifferent universality classes: the generic rigidity class, the connectivity class, and the generic "braced square net''(GBSN). We analyze the spanning cluster density P-proportional to, the backbone density P-B, and the densi ty of dangling ends P-D. In the generic rigidity (GR) and connectivity case s, the lend-carrying component of the spanning cluster, the backbone, is fr actal at p(c), so that the backbone density behaves as B similar to (p - p( c))(beta') for p > p(c). We estimate beta(gr)' = 0.25 +/- 0.02 for generic rigidity and beta(c)' = 0.467 +/- 0.007 for the connectivity case. We find the correlation length exponents v(gr) = 1.16 +/- 0.03 for generic rigidity compared to the exact value for connectivity, v(c) = 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. [S1063-651X (99)12102-0].