Percolation threshold, Fisher exponent, and shortest path exponent for four and five dimensions - art. no. 026115

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 numbe r 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 o f the order of s(theta) bytes where theta ranges from 0.4 for two-dimension al (2D) lattices to 0.5 for six (or higher) -dimensional lattices. We use t his method to estimated d(min), the exponent relating the minimum path, l t o the Euclidean distance r, for 4D and 5D hypercubic lattices. Analyzing bo th site and bond percolation, we find d(min) = 1.607 +/- 0.005 (4D) and d(m in) = 1.812 +/- 0.006 (5D). In order to determine d(min) to high precision, and without bias, it was necessary to first find precise values for the pe rcolation threshold, p(c) : p(c)=0.196889 +/- 0.000003 (4D) and p(c)=0.1408 1 +/- 0.00001 (5D) for site and p(c) = 0.160130 +/- 0.000003 (4D) and p(c) = 0.118174 +/- 0.000004 (5D) for bond percolation. We also calculate the Fi sher exponent tau determined in the course of calculating the values of p(c ): tau =2.313 +/- 0.003 (4D) and tau =2.412 +/- 0.004 (5D).