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

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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 .