We examine self-avoiding walks in dimensions 4 to 8 using high-precision
Monte-Carlo simulations up to length N=16384, 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, μ5=8.838544(3), μ6=10.878094(4), μ7=12.902817(3),
μ8=14.919257(2) and give a revised estimate of μ4=6.774043(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).Comment: 14 pages, 2 figure