Scaling of self-avoiding walks in high dimensions

We examine self-avoiding walks in dimensions 4 to 8 using high-precision Monte Carlo simulations up to length N = 16 384, providing the first such results in dimensions d > 4 on which we concentrate our analysis. We analyse the scaling behaviour of the partition function and the statistics of nearest-neighbour contacts, as well as the average geometric size of the walks, and compare our results to 1/ d -expansions and to excellent rigorous bounds that exist. In particular, we obtain precise values for the connective constants, µ 5 = 8.838 544(3), µ 6 = 10.878 094(4), µ 7 = 12.902 817(3), µ 8 = 14.919 257(2) and give a revised estimate of µ 4 = 6.774 043(5). All of these are by at least one order of magnitude more accurate than those previously given (from other approaches in d > 4 and all approaches in d = 4). Our results are consistent with most theoretical predictions, though in d = 5 we find clear evidence of anomalous N -1/2 -corrections for the scaling of the geometric size of the walks, which we understand as a non-analytic correction to scaling of the general form N (4- d )/2 (not present in pure Gaussian random walks).