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

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 y_t ≡ 1 ∕ ν , the fractal dimension d_f, the backbone exponent d_B and the shortest-path exponent d_min. 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 y_t = 1 . 461(5) , d_f = 3 . 044 6(7) , d_B = 1 . 984 4(11) , d_min = 1 . 604 2(5) , b = 0 . 62(1) for 4D; and y_t = 1 . 743(10) , d_f = 3 . 526 0(14) , d_B = 2 . 022 6(27) , d_min = 1 . 813 7(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 d_B is observed near d ≈ 5 . It is suggested that, as expected, critical percolation clusters become more and more dendritic as d increases.