## Results for the 4D Self-Trapping Random Walk

**
**
Average number of steps before termination: |
3.961*10^{6} |
+-11800 |

Average number of failed attempts: |
777150 |
+-2300 |

Number of walks in simulation |
1.9*10^{6}(complete),
1.2*10^{10}(short) |

### Diagrams

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 xxx.mov is (2*xxx+1)^{4}. The
largest domain size was (2*285+1)^{4}=1.063*10^{11}.

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*10^{10} 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*10^{10}~=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.