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

Abstract

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.