My question is somehow related to the one here [First Collision Time for k Random Walkers on a Torus](https://mathoverflow.net/questions/169297/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.



First collision time of $n$ random walkers on a cycle
Suppose the initial distribution of walkers has each independently choosing a site with equal probabilities (note that multiply-occupied sites are allowed).
This distribution is invariant under the process. The probability of a collision at any step is the probability that the site the moving walker enters is occupied, which is 
$1 - (1-1/k)^{n-1}$. By the elementary renewal theorem, the expected time between collisions is $1/(1 - (1-1/k)^{n-1})$.


