A322831 Average path length to self-trapping, rounded to nearest integer, of self-avoiding two-dimensional random walks using unit steps and direction changes from the set Pi*(2*k/n - 1), k = 1..n-1.

71, 71, 40, 77, 45, 51, 42, 56, 49, 51, 48, 54

Offset

3,1

Comments

The cases n = 3, 4, and 6 correspond to the usual self-avoiding random walks on the honeycomb net, the square lattice, and the hexagonal lattice, respectively. The other cases n = 5, 7, ... are a generalization using self-avoiding rooted walks similar to those defined in A306175, A306177, ..., A306182. The walk is trapped if it cannot be continued without either hitting an already visited (lattice) point or crossing or touching any straight line connecting successively visited points on the path up to the current point.

The result 71 for n=4 was established in 1984 by Hemmer & Hemmer.

The sequence data are based on the following results of at least 10^9 simulated random walks for each n <= 12, with an uncertainty of +- 0.004 for the average walk length:

n length

3 71.132

4 70.760 (+-0.001)

5 40.375

6 77.150

7 45.297

8 51.150

9 42.049

10 56.189

11 48.523

12 51.486

13 47.9 (+-0.2)

14 53.9 (+-0.2)

Links

Table of n, a(n) for n=3..14.

S. Hemmer, P. C. Hemmer, An average self‐avoiding random walk on the square lattice lasts 71 steps, J. Chem. Phys. 81, 584 (1984)

Hugo Pfoertner, Examples of self-trapping random walks.

Hugo Pfoertner, Probability density for the number of steps before trapping occurs, 2018.

Hugo Pfoertner, Results for the 2D Self-Trapping Random Walk.

Alexander Renner, Self avoiding walks and lattice polymers, Diplomarbeit, Universität Wien, December 1994.

Crossrefs

Cf. A001668, A001411, A001334, A077482, A306175, A306177, A306178, A306179, A306180, A306181, A306182.

Cf. A122223, A122224, A122226, A127399, A127400, A127401, A300665, A323141, A323560, A323562, A323699.

Sequence in context: A087075 A095936 A104843 * A127316 A322444 A339700

Adjacent sequences: A322828 A322829 A322830 * A322832 A322833 A322834

Keyword

nonn,more

Author

Hugo Pfoertner, Dec 27 2018

Status

approved