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.)