# Random Walks in 2D: Recurrence of (a), (b), (c)

• davidmoore63@y
In summary, we discussed three different random walks in 2D: (a) on a square lattice, (b) on a triangular lattice, and (c) with a uniform distribution of direction. While (a) is recurrent, (b) and (c) were initially thought to be transient. However, using the concept of resistance in electrical circuits, it was determined that (b) is actually recurrent due to the infinite resistance between the origin and infinity. The resistance for (c) is also infinite, but the walk is transient due to the zero measure of the origin in the infinite disk.
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).

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

c is transient, right.

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..

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

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.

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

Noted on (c).

## 1. What is a random walk in 2D?

A random walk in 2D is a mathematical model that describes the movement of a particle in two-dimensional space. It is a sequence of steps where each step is taken in a random direction from the previous step.

## 2. How is recurrence defined in the context of random walks in 2D?

In the context of random walks in 2D, recurrence refers to the property of a random walk to return to its starting point infinitely often. This means that no matter how far the particle travels, there is always a possibility for it to return to its original position.

## 3. What is the significance of recurrence in random walks in 2D?

The presence or absence of recurrence in a random walk in 2D can provide insights into the behavior of the system. For example, a recurrent random walk may have a higher probability of reaching certain points in space, while a non-recurrent random walk may not have a predictable pattern.

## 4. How is the recurrence of a random walk in 2D affected by the step size?

The step size of a random walk in 2D can affect its recurrence. For example, a smaller step size may increase the chances of recurrence, as the particle has a greater chance of returning to its starting point. Conversely, a larger step size may decrease the chances of recurrence, as the particle is more likely to move further away from its starting point.

## 5. Can the recurrence of a random walk in 2D be predicted?

The recurrence of a random walk in 2D is not always predictable, as it depends on various factors such as the step size and the number of steps taken. However, there are mathematical techniques and simulations that can be used to estimate the likelihood of recurrence in a specific random walk scenario.

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