# Random Walk on Circle

Suppose there is a Markov chain with K states {1,2,...,K} arranged clockwise on a circle. $$Z_0$$ = 1 The probability of moving clockwise is $$p$$ and the probability of moving counterclockwise is $$1-p$$. Then what's the probability that the Markov chain complete a clockwise circle before a counterclockwise one? In addition, I'm also interested in the long run average number of loops completed, i.e., $$\lim_{n\to\infty} L(n)/n = ?$$, where $$L(n)$$ is the number of loops completed by time n.

I don't know how I should approach this question. Any help is appreciated!

• When you talk about "number of loops" do you mean to keep track of the loop direction, so $L(n)$ is a winding number? – kimchi lover Jun 8 at 22:00
• No just the number of loops including clockwise and counterclockwise – Chloe Jun 9 at 3:09

The first question is equivalent to a random walk on $$\{-K,-(K-1),\dots,0,\dots,K-1,K\}$$ starting at $$0$$ and taking a step to the right with probability $$p$$ or to the left with probability $$1-p$$, and asking the probability that the walk reaches $$K$$ before $$-K$$; presumably this is a standard problem that has been answered before.
For the second question, the expected number of (signed) clockwise steps per turn is $$p-(1-p) = 1-2p$$, and so the expected number of (signed) clockwise loops in $$n$$ steps is $$n(1-2p)/K$$. It becomes more difficult if you care about unsigned loops, "starting over" every time a loop is completed.
• $p-(1-p) = 2p-1$. – Fabio Somenzi Jun 8 at 23:46