Comparative study of spanning cluster distributions in different dimensions

The probability distributions of the masses of the clusters spanning from top to bottom of a percolating lattice at the percolation threshold are obtained in all dimensions from two to five. The first two cumulants and the exponents for the universal scaling functions are shown to have simple power law variations with the dimensionality. The cases where multiple spanning clusters occur are discussed separately and compared. Percolation is a subject which has been studied extensively for the last few decades. The relevance of percolation in various areas of physics is also well established. Although many of the properties of percolating systems are well understood and studied, there still remains a lot of details to be explored and intricate questions to be addressed[1, 2]. At the critical point (percolation threshold) there appears for the first time a cluster spanning the whole lattice. The spanning cluster is a fractal in the sense its mass M scales with the length as L where D < d, d being the spatial dimension and D the fractal dimension. Distribution of cluster masses at and away from criticality has been studied in detail [1]. Conditional probability distributions for spanning cluster (SC) masses, their moments, and other variables like the shortest path etc. also appear in the literature [3, 4, 5, 6, 7, 8]. While the distribution for the