Critical exponents and universal excess cluster number of percolation in four and five dimensions

Abstract

We study critical bond percolation on periodic four-dimensional (4D) and five-dimensional (5D) hypercubes by Monte Carlo simulations. By classifying the occupied bonds into branches, junctions and non-bridges, we construct the whole, the leaf-free and the bridge-free clusters using the breadth-first search algorithm. From the geometric properties of these clusters, we determine a set of four critical exponents, including the thermal exponent yt≡1∕ν, the fractal dimension df, the backbone exponent dB and the shortest-path exponent dmin. We also obtain an estimate of the excess cluster number b which is a universal quantity related to the finite-size scaling of the total number of clusters. The results are yt=1.461(5), df=3.0446(7), dB=1.9844(11), dmin=1.6042(5), b=0.62(1) for 4D; and yt=1.743(10), df=3.5260(14), dB=2.0226(27), dmin=1.8137(16), b=0.62(2) for 5D. The values of the critical exponents are compatible with or improving over the existing estimates, and those of the excess cluster number b have not been reported before. Together with the existing values in other spatial dimensions d, the d-dependent behavior of the critical exponents is obtained, and a local maximum of dB is observed near d≈5. It is suggested that, as expected, critical percolation clusters become more and more dendritic as d increases.