I am looking at (1D) random walks in discrete time and space, where a fair coin determines the direction the walker moves in. I am interested in the probability of the walk to return to the origin at every time step.
I found a question on here that asked about the average time to return to the origin (so similar to what I am interested in). One answer to that question said that a random walk on a line with reflecting boundaries at $\frac{-n}{2}$ and $\frac{n}{2}$ is equivalent to a a random walk on a circle where the positions $0$ and $n$ are glued together (see the answer to this question: 1D Random walk with reflective barriers — average time to return to origin). I have a few questions about this:
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Why is this true? The answer to that question explained it somewhat, but I was wondering if someone could explain it further/provide some references on this.
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I am in particular wondering about a walk on a circle of 8 nodes being equivalent to a walk on a line with reflecting boundaries at 1 and 5 (or 0 and 4 equivalently). Is this true?
Many thanks!
Best Answer
Direction is irrelevant to arc length, so going past halfway around a circle is the same distance as going to halfway and then bouncing back by the remainder of the distance.
Thus, the maximum distance around the circle is half the circle length, with movement past the halfway point being converted into reflection back into the same side of the circle as the movement originated from, which is identical to having a line segment with length being the circumference and reflection at the half-circumference distances from the origin.
Try making this with paper or string, a knot to indicate the origin, and a button or some other marker showing the reflector. Break the loop through the reflector, and there you have the same structure.