Percolation in high dimensions is not understood

The number of spanning clusters in four to nine dimensions does not fully follow the expected size dependence for random percolation. Researchers were interested already long ago in percolation theory above the upper critical dimension of six [1, 2], and we followed [3]. At the perco-lation threshold [4], there is a theoretical consensus that the number N of spanning clusters stays finite with increasing lattice size below d = 6 dimensions , and increases with some power of the lattize size above six dimensions, for hypercubic lattices of L d sites [5]. Andronico et al.[6], however, have worrying data in five dimensions showing an increase of N with increasing L. Thus we now check this question. One Fortran program, available from [email protected], checks if a cluster spans from top to bottom and uses free boundary conditions in this and one other direction, while helical boundary conditions are used in the remaining d − 2 directions. The spanning properties are known to depend on boundary conditions and thus no quantitative agreement with [6] is expected. In three dimensions the average N is about 0.4 for L = 7 to 101, roughly independent of L as predicted; that means there is often no spanning cluster. Figure 1, however, shows for d = 5 an increase of N with increasing L = 3 to 101. Figure 2 shows for d = 7, 8 and 9 an increase of the multiplicity as L 1.65 , L 2.49 and L 3.39 , respectively. The points in Figs. 1 and 2 are averages over mostly 1000 runs. The other Fortran program uses free boundary conditions in all directions and it is available from [email protected]. Its results 1 in Figs.3 and 4, which refer mostly to a number of iterations between 10000 and 50000, are qualitatively similar to Figs.1 and 2. However, one derives instead an increase of the spanning cluster multiplicity as L 0.97 , L 1.53 and L 2.1 for d = 7, 8 and 9, respectively. We remark that this series of slopes is quite well reproduced by the simple formula (d − 5)/2, which is not predicted by any theory and which, if true, would hint the existence of infinite spanning clusters at threshold already in five dimensions. In fact, even the trend of the 6D data is quite well reproduced by a power law with exponent 0.51, which is amazingly close to …