The fractal dimension and other percolation exponents in four and five dimensions

Abstract

For percolation the fractal dimension d_f, which is identical to the magnetic field scaling power y_h, has never been calculated for hypercubic lattices of dimension d=5. Percolation is studied for systems of high dimensionality using the method of large-cell Monte Carlo position-space renormalisation group. The estimate d_f=y_h=3.69+or-0.02 (d=5) and 3.12+or-0.02 (d=4) is obtained. The thermal scaling power y_T=1/ nu where nu is the correlation length exponent is calculated. y_T^-1= nu =0.51+or-0.05 (d=5) and nu =0.64+or-0.02 (d=4). The results are compared with the epsilon expansions of d_f and nu .