##
**Rigidity percolation and boundary conditions.**
*(English)*
Zbl 1062.60098

The author discusses rigidity properties of components in the bond percolation on the two-dimensional triangular lattice T. While finite rigid components here are straightforward to define, the notion of infinite rigid components is not so obvious. In this paper, the latter are defined as limits of rigid components in a finite region of the lattice with specified boundary conditions (i.e., with fixed states of edges outside the region) as the region approaches T.

Among various boundary conditions there are two “extremal” choices – the “free” and “wired” boundary conditions – with all edges outside the region being closed and open, respectively. While theoretically such choices might give rise to different definitions of rigidity, the main result of the paper claims that, in two dimensions, the corresponding critical probabilities coincide. Moreover, it is shown that for all values of the tuning parameter \(p\) (with possible exception of the common critical value) the sets of “free” and “wired” rigid components are identical, almost surely w.r.t. the percolation measure \({\mathbf P}_p\).

Among various boundary conditions there are two “extremal” choices – the “free” and “wired” boundary conditions – with all edges outside the region being closed and open, respectively. While theoretically such choices might give rise to different definitions of rigidity, the main result of the paper claims that, in two dimensions, the corresponding critical probabilities coincide. Moreover, it is shown that for all values of the tuning parameter \(p\) (with possible exception of the common critical value) the sets of “free” and “wired” rigid components are identical, almost surely w.r.t. the percolation measure \({\mathbf P}_p\).

Reviewer: Ostap Hryniv (Durham)

### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

52C25 | Rigidity and flexibility of structures (aspects of discrete geometry) |

82B43 | Percolation |

PDF
BibTeX
XML
Cite

\textit{A. E. Holroyd}, Ann. Appl. Probab. 11, No. 4, 1063--1078 (2001; Zbl 1062.60098)

Full Text:
DOI

### References:

[1] | Graver, J., Servatius, B. and Servatius, H. (1993). Combinatorial Rigidity. Amer. Math. Soc., Providence, RI. · Zbl 0788.05001 |

[2] | Grimmett, G. R. (1999). Percolation, 2nd. ed. Springer, Berlin. · Zbl 0926.60004 |

[3] | Grimmett, G. R. and Holroyd, A. E. (2000). Entanglement in percolation. Proc. London Math. Soc. (3) 81, 484-512. · Zbl 1026.60109 |

[4] | Häggstr öm, O. (2001). Uniqueness in two-dimensional rigidity percolation. Math. Proc. Cambridge Philos. Soc. · Zbl 0974.60100 |

[5] | Häggstr öm, O. (2001). Uniqueness of the infinite entangled component in threedimensional bond percolation. Ann. Probab. · Zbl 1013.60083 |

[6] | Holroyd, A. E. (2000). Existence of a phase transition in entanglement percolation. Math. Proc. Cambridge Philos. Soc. 129 231-251. · Zbl 0963.60099 |

[7] | Holroyd, A. E. (1998). Existence and uniqueness of infinite components in generic rigidity percolation. Ann. Appl. Probab. 8 944-973. · Zbl 0932.60093 |

[8] | Jacobs, D. J. and Thorpe, M. F. (1996). Generic rigidity percolation in two dimensions. Phys. Rev. E 53 3682-3693. |

[9] | Welsh, D. J. A. (1976). Matroid Theory. Academic Press, New York. · Zbl 0343.05002 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.