My question is somehow related to the one here First Collision Time for k Random Walkers on a Torus but, unfortunately, the answer does not cover my concern.

My problem is: consider $n$ walkers on the cycle $\mathbb{Z}/k$ ($n < k$). At each step, one walker is selected with probability $1/n$ and moves by one unit counter-clockwise; the other walkers remain at their locations. The steps are independent.

I would like to have some information on the first time $T$ until two walkers collide (go into to the same site); e.g., expectation, asymptotic behaviour (e.g. $k,n \rightarrow \infty$ in some proportion), etc.

When $n = 2$, this reduces to a single random walker on the cycle moving clockwise, or counter-clockwise with probability $1/2$, and the time $T$ is simply the hitting time of the site $0$.

But for arbitrary $n$, this approach does not seem to work ...

Do you have any ideas, or references to similar problems ?

Thank you.