Percolation threshold, Fisher exponent, and shortest path exponent for four and five dimensions

We develop a method of constructing percolation clusters that allows us to build very large clusters using very little computer memory by limiting the maximum number of sites for which we maintain state information to a number of the order of the number of sites in the largest chemical shell of the cluster being created. The memory required to grow a cluster of mass s is of the order of s^θ bytes where θ ranges from 0.4 for two-dimensional (2D) lattices to 0.5 for six (or higher)-dimensional lattices. We use this method to estimate dₘᵢₙ, the exponent relating the minimum path l to the Euclidean distance r, for 4D and 5D hypercubic lattices. Analyzing both site and bond percolation, we find dₘᵢₙ=1.607±0.005 (4D) and dₘᵢₙ=1.812±0.006 (5D). In order to determine dₘᵢₙ to high precision, and without bias, it was necessary to first find precise values for the percolation threshold, pc: pc=0.196889±0.000003 (4D) and pc=0.14081±0.00001 (5D) for site and pc=0.160130±0.000003 (4D) and pc=0.118174±0.000004 (5D) for bond percolation. We also calculate the Fisher exponent τ determined in the course of calculating the values of pc: τ=2.313±0.003 (4D) and τ=2.412±0.004 (5D).