Dear Woorim Choi . I'm sorry I answer this in other question but I copy here for better discussion of this topic with coleagues...

I am not sure of the approach you want to give to the problem of the self-avoiding walk. As I understand it, the Monte Carlo method does not offer the best solutions, but any solution (let's say brute force). If you consider getting the SAW with many steps, possibly the time to obtain an acceptable result can be long. Perhaps with better algorithms you can achieve better paths, and less time processing.

CASE 1. Free randomization. Assuming the most complicated case, which is that the step can be done in any direction, (without restricting to a specific x, y, or z axis) would say the following:

As you intend to work in the Euclidean space, I would recommend to choose randomly the next point in geographic coordinates and then, having the direction vector, obtaining the point at a distance = step from the previous point, checking in each case new point that the distance to the center does not is greater than the radius. After this you could obtain spheric coordinates of each point.

Problem: How to verify the condition of self-avoider. This is complicated for the random Euclidean space, an algorithm should be designed to find intersections between each generated segment instead of point coincidence as you work with grids, nothing that I could solve "bird flight".

CASE 2 Three-dimensional grid SAW. Restricting the path only to a movement in spherical coordinates (x, y and z), is easier. In this case, you randomly choose a random axis and advance a distance of 1 step on that axis. Then you compare if any of the previous points match the 3 coordinates and, if so, discard the point and sample another.

PROBLEM: how to get a SAW of length = n

1) Get each point as described above.

2) If the sequence reaches a point < n where there is no option to achieve the self-preventing condition for following, move back one position in the sequence and re-sample between the two remaining coordinates.

3) If the same happens with the three axes of this penultimate point, discard the sequences to the previous point and so on.

This brute force method should work, however it can be quite inefficient especially for very long chains.

Regards

Dear Woorim Choi . I'm sorry I answer this in other question but I copy here for better discussion of this topic with coleagues...

I am not sure of the approach you want to give to the problem of the self-avoiding walk. As I understand it, the Monte Carlo method does not offer the best solutions, but any solution (let's say brute force). If you consider getting the SAW with many steps, possibly the time to obtain an acceptable result can be long. Perhaps with better algorithms you can achieve better paths, and less time processing.

CASE 1. Free randomization. Assuming the most complicated case, which is that the step can be done in any direction, (without restricting to a specific x, y, or z axis) would say the following:

As you intend to work in the Euclidean space, I would recommend to choose randomly the next point in geographic coordinates and then, having the direction vector, obtaining the point at a distance = step from the previous point, checking in each case new point that the distance to the center does not is greater than the radius. After this you could obtain spheric coordinates of each point.

Problem: How to verify the condition of self-avoider. This is complicated for the random Euclidean space, an algorithm should be designed to find intersections between each generated segment instead of point coincidence as you work with grids, nothing that I could solve "bird flight".

CASE 2 Three-dimensional grid SAW. Restricting the path only to a movement in spherical coordinates (x, y and z), is easier. In this case, you randomly choose a random axis and advance a distance of 1 step on that axis. Then you compare if any of the previous points match the 3 coordinates and, if so, discard the point and sample another.

PROBLEM: how to get a SAW of length = n

1) Get each point as described above.

2) If the sequence reaches a point < n where there is no option to achieve the self-preventing condition for following, move back one position in the sequence and re-sample between the two remaining coordinates.

3) If the same happens with the three axes of this penultimate point, discard the sequences to the previous point and so on.

This brute force method should work, however it can be quite inefficient especially for very long chains.

Regards

More interesting info about this topic:

https://researchers.ms.unimelb.edu.au/~guttmann@unimelb/lectures/MAV2003.pdf

yes

You can obtain a random value separately within the desired range for each coordinate. After this value you can choose any algorithm for a particular path, even SAW.