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

Abstract

For percolation the fractal dimension df, which is identical to the magnetic field scaling power yh, 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 df=yh=3.69+or-0.02 (d=5) and 3.12+or-0.02 (d=4) is obtained. The thermal scaling power yT=1/ nu where nu is the correlation length exponent is calculated. yT-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 df and nu .