# Reference: probability distribution of first meeting time of two random walks on a cycle graph

I am looking for a reference or derivation for the following question:

Consider a cycle graph $$G$$ with $$N$$ vertices (see example here). Let two independent continuous-time random walkers$$^\star$$ start on node $$i$$ and node $$j$$. Let $$T$$ be the time when the two walkers meet for the first time.

What will be the probability density function of $$T$$?

I am looking for the exact expression of that pdf for finite $$N$$. I am sure this is a well known question with a precise formula but I have not been able to find a clear reference solving it. This is a new field for me so perhaps I am using the wrong vocabulary? With the hope that this post will help. Thanks!

($$^\star$$: each walker jumps to a neighboring vertex (chosen at random) with rate one.)

Assume that $$i You can reduce this to the following simpler-looking question: Consider a continuous-time RW on the integers, moving at rate two, started at $$k:=j-i$$. Let $$T$$ be the hitting time of $$\{0,N\}$$ by this walk, also known as as the exit time from $$[1,N-1]$$. Let $$\tau$$ denote the hitting time of $$\{0,N\}$$ by discrete-time random walk. The distribution of $$\tau$$ can be determined from the arguments in Chapter 21 of . (Spitzer gives the general method, and then provides all the details only for the case $$k=N/2$$.) Then you can deduce the distribution of $$T$$ via $$P_k[T>q] = \sum_n P[{\rm Poisson}(2q)=m] \cdot P_k[\tau>m] \,.$$