Comparison of rigidity and connectivity percolation in two dimensions

Abstract

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. 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 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 ~ (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, \nu_{gr} = 1.16 +/- 0.03 for generic rigidity compared to the exact value for connectivity \nu_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.