Examples of self-trapping random walks

Hugo Pfoertner, Dec 2018

Relative to a continuation of the walk in the same direction as the previous segment,
the direction of the next step in an "n - Walk" is drawn with equal probability from the angles in the following table.
For n = 3, 4, and 6 this is equivalent to the usual lattice walks.
Instead of the rounded numbers, the exact directions are taken as multiples of Pi/n.

7
Turning angles (Degrees)
n
3-60  60
4-90 0     90
5-108 -36 36 108
6-120 -60 0     60 120
7-128.57 -77.14 -25.71 25.71 77.14 128.5
8-135 -90 -45 0     45 90 135
9-140 -100 -60 -20 20 60 100 140
10-144 -108 -72 -36 0     36 72 108 144
11 -147.27 -114.55 -81.82 -49.09 -16.3616.36 49.09 81.82 114.55 147.27
12 -150 -120 -90 -60 -30 0     30 60 90 120 150

Honeycomb Net (n=3)
Shortest, P ~= 0.0078, L=9
Most probable P ~= 0.012, L=33
P ~= 10^-3, L=183

Square Lattice (n=4)
Shortest, P ~= 0.0028, L=7
Most probable P ~= 0.012, L=29
P ~= 10^-3, L=173

Penta - Walk (n=5)
Shortest, P ~= 0.0077, L=5
Most probable P ~= 0.02, L=18
P ~= 10^-3, L=113

Hexagonal Lattice (n=6)
Shortest, P ~= 0.0008, L=6
Most probable P ~= 0.0105, L=35
P ~= 10^-3, L=185

Hepta - Walk (n=7)
Shortest, P ~= 0.00407, L=5
Most probable P ~= 0.0181, L=22
P ~= 10^-3, L=123

Octa - Walk (n=8)
Shortest, P ~= 0.0033, L=5
Most probable P ~= 0.0158, L=23
P ~= 10^-3, L=136

9 - Walk (n=9)
Shortest, P ~= 0.00446, L=4
Most probable P ~= 0.0191, L=20
P ~= 10^-3, L=117

10 - Walk (n=10)
Shortest, P ~= 0.0015, L=5
Most probable P ~= 0.0145, L=27
P ~= 10^-3, L=145

11 - Walk (n=11)
Shortest, P ~= 0.0027, L=5
Most probable P ~= 0.0169, L=23
P ~= 10^-3, L=130

12 - Walk (n=12)
Shortest, P ~= 0.002, L=4
Most probable P ~= 0.0156, L=24
P ~= 10^-3, L=136

Back to OEIS A322831.