Results for the 4D Self-Trapping Random Walk

Average number of steps before termination: 3.961*106 +-11800
Average number of failed attempts: 777150 +-2300
Number of walks in simulation 1.9*106(complete),


Average lenght of walk as a function of domain size.

The simulations were performed on a 4-d domain with cyclic boundary conditions. For small domain sizes the increased trapping probability leads to shorter walks. The figures indicate that for larger domain sizes the walk lengths reach some asymptotic value, which is an estimate for the expected average walk length on an infinite domain. The domain size corresponding to curves labeled as is (2*xxx+1)4. The largest domain size was (2*285+1)4=1.063*1011.

Probability density for the number of steps before trapping occurs

The figures show the combined results of different simulations. The asymptotic behavior for very long walks was determined from 16717 walks on a 74G domain, whereas the behavior for very short walks was determined from 1.2*1010 walks that were truncated after 25 moves. The shortest possible self-trapping walk in 4 dimensions has 15 steps. The probability for this walk found in the simulation is 33/1.2*1010~=1/363636363. The exact probability for the occurrence of the shortest self-trapping walk is 1728/7^14~=1/392490204. A table of the 36 shortest walks remaining after elimination of symmetries and permutations of coordinates is given here.

How far away from the start point is a walker after 25 steps?

The "spiky" behavior of the landing counts is a consequence of the number of available lattice points for a given radius (see OEIS A000118).

Counting self-avoiding 4D walks

A "brute force" Fortran program to count the 4D walks is available here.
The results agree with OEIS A010575.
A table of A010575/8 is given here.
Illustration of the asymptotic behavior of A010575.

Back to Random Walk Home