Comparison of rigidity and connectivity percolation in two dimensions

TL;DR

This paper investigates rigidity and connectivity percolation in two-dimensional lattices, comparing different universality classes and analyzing critical behaviors and phase transitions using advanced algorithms.

Contribution

It introduces a new algorithm for analyzing 2D graph rigidity and compares three universality classes, providing detailed critical exponents and transition types.

Findings

The backbone is fractal at the percolation threshold in rigidity and connectivity classes.

The critical exponent eta' for rigidity is approximately 0.25.

The GBSN exhibits a first-order transition with a discontinuous backbone density.

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.