Results for the 2D Self-Trapping Random Walk

Average number of steps before termination: 70.7598 +-0.001
Average number of failed attempts: 45.0295 +-0.001
Number of walks in simulation 1.2*1010


Probability density for the number of steps before trapping occurs

Examples for walks with the maximum number of constrained steps
Conjecture: A n-step 2D STW has never more than n-floor(sqrt(4*n+1))-2 steps
with only two choices for the possible next direction.
n-floor(sqrt(4*n+1) is Sequence A076874 in Sloane's OEIS.

5 Unconstrained and 7 maximally 2-constrained walks of length 10
Manhattan Distance Start-End for n-step Self-Avoiding Walks
Asymptotic Behavior of Mean Square Displacement
Asymptotic Behavior of Mean Manhattan Displacement
End-to-End Euclidean Distance Distribution of all 25-Step Walks


Results of simulation, comparison with exact probabilities
Count self-trapping walks up to length 23
Enumeration of all short self-trapping walks
7 8 9 10 11 12 13 14 15 16 17

Distribution of end point distance
Average Euclidean and squared end point distance
Average Manhattan end point distance
Comparison of average Euclidean and Manhattan displacements

Unsolved Problem: Is there an asymptotic value for the difference between the average displacement of all self-avoiding n-step walks and the subset of self-trapping n-step walks for large n.
See figure: Manhattan displacement difference SAW-STW.


Fortran program to determine exact probabilities
Fortran program for distance counting
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