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.
N1 - Publisher Copyright:
© 2017 Elsevier B.V.
PY - 2017/9/5
Y1 - 2017/9/5
N2 - 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 - 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.
KW - Dimension numbers
KW - Fractal geometry
KW - Geodesic metric
KW - Percolation
KW - Scaling exponents
KW - Topological Hausdorff dimension
UR - http://www.scopus.com/inward/record.url?scp=85021406199&partnerID=8YFLogxK
U2 - 10.1016/j.physleta.2017.06.028
DO - 10.1016/j.physleta.2017.06.028
M3 - Artículo
SN - 0375-9601
VL - 381
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
IS - 33
ER -