Comparison of rigidity and connectivity percolation in two dimensions

Citation
C. Moukarzel et Pm. Duxbury, Comparison of rigidity and connectivity percolation in two dimensions, PHYS REV E, 59(3), 1999, pp. 2614-2622
Citations number
49
Categorie Soggetti
Physics
Journal title
PHYSICAL REVIEW E
ISSN journal
1063651X → ACNP
Volume
59
Issue
3
Year of publication
1999
Part
A
Pages
2614 - 2622
Database
ISI
SICI code
1063-651X(199903)59:3<2614:CORACP>2.0.ZU;2-N
Abstract
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].