# Random Walks in 2D

1. Jun 23, 2015

### davidmoore63@y

Consider the following random walks in 2D, starting at a point we will call the origin:
(a) random walk on a square lattice (step size 1 on the integer lattice for example)
(b) random walk on a triangular lattice (step size 1 on the lattice of equilateral triangles of side 1). Thus there are 6 equiprobable choices for each step in this walk
(c) random walk of step size 1 but with a direction selected from the uniform distribution U[0, 2pi).

My question is, we know that (a) is recurrent (returns to the origin with probability 1). What about (b) and (c) ? Are they recurrent or transient? Intuitively I would say (c) must be transient, but I am struggling with (b).

2. Jun 23, 2015

### Staff: Mentor

What is your expectation about b? Can you transform it to a different lattice, maybe with different rules?

c is transient, right.

3. Jun 23, 2015

### davidmoore63@y

I would tentatively expect (b) to be transient, because (a) is borderline recurrent - as i recall the expected number of returns for (a) is sigma (1/n) thus infinite, whereas a walk with more choices ought to render the origin less likely to be hit later. Thinking about your suggestion..

4. Jun 23, 2015

### davidmoore63@y

The 3D walk on an integer lattice has 6 choices but it doesn't seem to be isomorphic - there are no ways to get back to the origin in 3 steps, unlike the triangular lattice

5. Jun 24, 2015

### davidmoore63@y

I have convinced myself that (b) is in fact recurrent, against my intuition. There is an interesting piece of work by Mare (https://math.dartmouth.edu/~pw/math100w13/mare.pdf) which refers to earlier work by Doyle and others interpreting probability of recurrence as resistance in electrical circuits! The conclusion is that a walk is recurrent if the resistance between the origin and infinity is infinite, whereas it is transient if the resistance is finite. An amazing parallel.

Using that interpretation, all 2d lattices with finite branches will be recurrent. The resistance to infinity of (b) is infinite. Because there are six choices instead of 4 in (a), the resistance is 4/6 as much for a given diameter of lattice, but 4/6 of infinity is still infinity.

Now I'm confused about (c). The resistance of an infinite disc is infinite, but the walk is transient because the origin has zero measure.

6. Jun 24, 2015

### Staff: Mentor

Interesting correlation to the resistance, I have to check that in more detail.
The infinite disk does not represent your random walk on it.

7. Jun 24, 2015

### davidmoore63@y

Noted on (c).