### Abstract

A random two-dimensional checkerboard of squares of conductivities 1 and δ in proportions p and 1-p is considered. Classical duality implies that the effective conductivity obeys σ*= δ at p=1/2. It is rigorously found here that to leading order as δ→0, this exact result holds for all p in the interval (1-pc,pc), where pc0.59 is the site percolation probability, not just at p=1/2. In particular, σ*(p,δ)= δ +O(δ), as δ→0, which is argued to hold for complex δ as well. The analysis is based on the identification of a ''symmetric'' backbone, which is statistically invariant under interchange of the components for any p(1-pc,pc), like the entire checkerboard at p=1/2. This backbone is defined in terms of ''choke points'' for the current, which have been observed in an experiment.

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

Pages (from-to) | 2114-2117 |

Number of pages | 4 |

Journal | Physical Review B |

Volume | 50 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 1994 |

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### All Science Journal Classification (ASJC) codes

- Condensed Matter Physics

### Cite this

*Physical Review B*,

*50*(4), 2114-2117. https://doi.org/10.1103/PhysRevB.50.2114

}

*Physical Review B*, vol. 50, no. 4, pp. 2114-2117. https://doi.org/10.1103/PhysRevB.50.2114

**Exact result for the effective conductivity of a continuum percolation model.** / Berlyand, Leonid V.; Golden, K.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Exact result for the effective conductivity of a continuum percolation model

AU - Berlyand, Leonid V.

AU - Golden, K.

PY - 1994/1/1

Y1 - 1994/1/1

N2 - A random two-dimensional checkerboard of squares of conductivities 1 and δ in proportions p and 1-p is considered. Classical duality implies that the effective conductivity obeys σ*= δ at p=1/2. It is rigorously found here that to leading order as δ→0, this exact result holds for all p in the interval (1-pc,pc), where pc0.59 is the site percolation probability, not just at p=1/2. In particular, σ*(p,δ)= δ +O(δ), as δ→0, which is argued to hold for complex δ as well. The analysis is based on the identification of a ''symmetric'' backbone, which is statistically invariant under interchange of the components for any p(1-pc,pc), like the entire checkerboard at p=1/2. This backbone is defined in terms of ''choke points'' for the current, which have been observed in an experiment.

AB - A random two-dimensional checkerboard of squares of conductivities 1 and δ in proportions p and 1-p is considered. Classical duality implies that the effective conductivity obeys σ*= δ at p=1/2. It is rigorously found here that to leading order as δ→0, this exact result holds for all p in the interval (1-pc,pc), where pc0.59 is the site percolation probability, not just at p=1/2. In particular, σ*(p,δ)= δ +O(δ), as δ→0, which is argued to hold for complex δ as well. The analysis is based on the identification of a ''symmetric'' backbone, which is statistically invariant under interchange of the components for any p(1-pc,pc), like the entire checkerboard at p=1/2. This backbone is defined in terms of ''choke points'' for the current, which have been observed in an experiment.

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U2 - 10.1103/PhysRevB.50.2114

DO - 10.1103/PhysRevB.50.2114

M3 - Article

AN - SCOPUS:0001244566

VL - 50

SP - 2114

EP - 2117

JO - Physical Review B

JF - Physical Review B

SN - 0163-1829

IS - 4

ER -