3D random walk between two end points

Question

Say I have 2 endpoints A and B. I would like to use W. Mathematica to generate a random walk between these two endpoints in a three-dimensional space. Es.: A=Point[{0,0,0}] and B=Point[{1,1,1}]. I would like to have as an output the random trajectory of the particle. How can I do this? If I use

Line[Accumulate[RandomChoice[{-1, 1}, {1000, 3}]]]

As suggested in the documentation, I would have to complete each output with a trajectory segment that would bring the particle to point B. Yet in this way the last segment may be "long" and not "enough" random. How can I avoid that or simply generate a random trajectory that links two points?

Answer 1

Here's one possible way using the function BrownianBridgeProcess and RandomFunction (you will have to check to make sure it's statistically what you need - in particular the motion in each dimensions is independent of the others).

walk[a_List, b_List, n_] := 
 Thread[First@RandomFunction[#, {0, 1, 1/n}]["ValueList"] & /@ 
   MapThread[BrownianBridgeProcess[{0, #1}, {1, #2}] &, {a, b}]]

Here you specify a starting point, an ending point, and the number of steps to take between them.

a = {0, 0, 0};
b = {1, 1, 1};
Graphics3D[{Black, Arrow[Tube@walk[a, b, 100]], PointSize[Large], 
  Red, Point /@ {a, b}}, Axes -> True]

a = {4, 10, 2};
b = {-3, 12, 0};
Graphics3D[{Black, Arrow[Tube@walk[a, b, 100]], PointSize[Large], 
  Red, Point /@ {a, b}}, Axes -> True]

Answer 2

Another very slow approach.

Create a random walk trajectory and then take all the pairs whose distance is the one you need

l = Accumulate[RandomChoice[{-1, 1}, {1000, 3}]];
Graphics3D@Line[l]

Show[Graphics3D@Line[l], 
 If[EuclideanDistance[#[[1]], #[[2]]] == 25 Sqrt[3], 
    Graphics3D@{Red, Line[#]}, Nothing] & /@ Subsets[l, {2}]]

All those pairs have the distance you want and can use it for statistical purposes. Subsets goes as n^2 so it's very slow.

Answer 3

I'd leave a comment, but not enough reputation, so here is a partial answer.

Assuming step size is 1 or -1 in each direction, then in each direction you will end up with zero or more {1,-1} pairs plus a "1" move. This means you will always have an odd number of moves, with one more in the "1" direction than in the "-1" direction.

So for a target of $2n+1$ moves, for each dimension, generate a list of $n$ moves on the "-1" direction and $n+1$ "1" moves and then scramble the list using RandomSample[ ]. Then combine the moves in each dimension.

This allows for earlier visits to the {1,1,1} point, but you didn't say that wasn't OK.