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

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Discussion Overview

The discussion revolves around the recurrence properties of different types of random walks in 2D, specifically focusing on a square lattice, a triangular lattice, and a random walk with uniformly chosen directions. Participants explore whether the latter two walks are recurrent or transient, building on the established knowledge of the square lattice's recurrence.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Post 1 introduces three types of 2D random walks and questions the recurrence of the triangular lattice and the uniformly directed walk.
  • Post 2 suggests transforming the triangular lattice to analyze its properties further.
  • Post 3 tentatively proposes that the triangular lattice may be transient, drawing on the expected number of returns for the square lattice.
  • Post 4 contrasts the properties of 3D walks with those of the triangular lattice, noting differences in returning to the origin.
  • Post 5 argues for the recurrence of the triangular lattice based on a resistance analogy, stating that all 2D lattices with finite branches are recurrent due to infinite resistance to infinity.
  • Post 5 also expresses confusion regarding the uniformly directed walk, noting that while the resistance is infinite, the walk is transient due to the origin having zero measure.
  • Post 6 acknowledges the correlation to resistance and suggests further investigation.
  • Post 7 simply notes the point made about the uniformly directed walk.

Areas of Agreement / Disagreement

Participants express differing views on the recurrence of the triangular lattice and the uniformly directed walk. There is no consensus on whether the triangular lattice is recurrent or transient, and the status of the uniformly directed walk remains unresolved.

Contextual Notes

Participants reference mathematical concepts such as resistance in electrical circuits and the expected number of returns, which may depend on specific definitions and assumptions not fully explored in the discussion.

davidmoore63@y
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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).
 
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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).
 

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