### Abstract

We study occurrence and properties of percolation of occupied bonds in systems with random interactions and, hence, frustration. We develop a general argument, somewhat like Peierls' argument, by which we show that in ℤ^{d} , d ≥ 2, percolation occurs for all possible interactions (provided they are bounded away from zero) if the parameter p ∈ (0, 1), regulating the density of occupied bonds, is high enough. If the interactions are i.i.d. random variables then we determine bounds on the values of p for which percolation occurs for all, almost all but not all almost none but some, or none of the interactions. Motivations of this work come from the rigorous analysis of phase transitions in frustrated statistical mechanics systems.

Original language | English (US) |
---|---|

Pages (from-to) | 1781-1808 |

Number of pages | 28 |

Journal | Annals of Probability |

Volume | 27 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 1999 |

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### Keywords

- Frustration
- Peierls' argument
- Percolation
- Spin glasses

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Annals of Probability*,

*27*(4), 1781-1808. https://doi.org/10.1214/aop/1022874815

**Bond percolation in frustrated systems.** / De Santis, E.; Gandolfi, Alberto.

Research output: Contribution to journal › Article

*Annals of Probability*, vol. 27, no. 4, pp. 1781-1808. https://doi.org/10.1214/aop/1022874815

}

TY - JOUR

T1 - Bond percolation in frustrated systems

AU - De Santis, E.

AU - Gandolfi, Alberto

PY - 1999/1/1

Y1 - 1999/1/1

N2 - We study occurrence and properties of percolation of occupied bonds in systems with random interactions and, hence, frustration. We develop a general argument, somewhat like Peierls' argument, by which we show that in ℤd , d ≥ 2, percolation occurs for all possible interactions (provided they are bounded away from zero) if the parameter p ∈ (0, 1), regulating the density of occupied bonds, is high enough. If the interactions are i.i.d. random variables then we determine bounds on the values of p for which percolation occurs for all, almost all but not all almost none but some, or none of the interactions. Motivations of this work come from the rigorous analysis of phase transitions in frustrated statistical mechanics systems.

AB - We study occurrence and properties of percolation of occupied bonds in systems with random interactions and, hence, frustration. We develop a general argument, somewhat like Peierls' argument, by which we show that in ℤd , d ≥ 2, percolation occurs for all possible interactions (provided they are bounded away from zero) if the parameter p ∈ (0, 1), regulating the density of occupied bonds, is high enough. If the interactions are i.i.d. random variables then we determine bounds on the values of p for which percolation occurs for all, almost all but not all almost none but some, or none of the interactions. Motivations of this work come from the rigorous analysis of phase transitions in frustrated statistical mechanics systems.

KW - Frustration

KW - Peierls' argument

KW - Percolation

KW - Spin glasses

UR - http://www.scopus.com/inward/record.url?scp=0033212330&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033212330&partnerID=8YFLogxK

U2 - 10.1214/aop/1022874815

DO - 10.1214/aop/1022874815

M3 - Article

VL - 27

SP - 1781

EP - 1808

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 4

ER -