### Abstract

Original language | American English |
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Pages (from-to) | 2665-2672 |

Number of pages | 2397 |

Journal | Physics Letters, Section A: General, Atomic and Solid State Physics |

DOIs | |

State | Published - 5 Sep 2017 |

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**Topological Hausdorff dimension and geodesic metric of critical percolation cluster in two dimensions.** / Balankin, Alexander S.; Mena, Baltasar; Martínez Cruz, M. A.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Topological Hausdorff dimension and geodesic metric of critical percolation cluster in two dimensions

AU - Balankin, Alexander S.

AU - Mena, Baltasar

AU - Martínez Cruz, M. A.

PY - 2017/9/5

Y1 - 2017/9/5

N2 - © 2017 Elsevier B.V. In this work, we prove that the topological Hausdorff dimension of critical percolation cluster (CPC) in two dimensions is equal to DtH=Drb+1=7/4, where Drb is the Hausdorff dimension of the set of red bonds. Hence, the CPC is infinitely ramified. We also argue that the mapping from the Euclidean metric to the geodesic metric on the CPC is governed by the Hausdorff dimension of the cluster skeleton Dsc=DH/dℓ>dmin, where DH, dℓ, and dmin are the Hausdorff and the connectivity (chemical) dimensions of the CPC and the fractal dimension of the minimum path, respectively. Then we introduce the notion of the topological connectivity dimension dtℓ. This allows us to establish the exact upper and lower bounds for the connectivity dimension dℓ of the CPC in d=2. The upper and lower bounds for some other dimension numbers were established using the relations between dimension numbers. Narrow ranges defined by these bounds are much smaller than the error bars of numerical estimates reported in literature. Accordingly, the exact values of some dimension numbers are conjectured.

AB - © 2017 Elsevier B.V. In this work, we prove that the topological Hausdorff dimension of critical percolation cluster (CPC) in two dimensions is equal to DtH=Drb+1=7/4, where Drb is the Hausdorff dimension of the set of red bonds. Hence, the CPC is infinitely ramified. We also argue that the mapping from the Euclidean metric to the geodesic metric on the CPC is governed by the Hausdorff dimension of the cluster skeleton Dsc=DH/dℓ>dmin, where DH, dℓ, and dmin are the Hausdorff and the connectivity (chemical) dimensions of the CPC and the fractal dimension of the minimum path, respectively. Then we introduce the notion of the topological connectivity dimension dtℓ. This allows us to establish the exact upper and lower bounds for the connectivity dimension dℓ of the CPC in d=2. The upper and lower bounds for some other dimension numbers were established using the relations between dimension numbers. Narrow ranges defined by these bounds are much smaller than the error bars of numerical estimates reported in literature. Accordingly, the exact values of some dimension numbers are conjectured.

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U2 - 10.1016/j.physleta.2017.06.028

DO - 10.1016/j.physleta.2017.06.028

M3 - Article

SP - 2665

EP - 2672

JO - Physics Letters, Section A: General, Atomic and Solid State Physics

JF - Physics Letters, Section A: General, Atomic and Solid State Physics

SN - 0375-9601

ER -