Statistics of overtakes by a tagged agent

We consider a one-dimensional infinite lattice where at each site there sits an agent carrying a velocity, which is drawn initially for each agent independently from a common distribution. This system evolves as a Markov process where a pair of agents at adjacent sites exchange their positions with a specified rate, while retaining their respective velocities, only if the velocity of the agent on the left site is higher. We study the statistics of the net displacement of a tagged agent $m(t)$ on the lattice, in a given duration $t$, for two different kinds of rates: one in which a pair of agents at sites $i$ and $i+1$ exchange their sites with rate $1$, independent of the velocity difference between the neighbors, and another in which a pair exchange their sites with a rate equal to their relative speed. In both cases, we find $m(t)sim t$ for large $t$. In the first case, for a randomly picked agent, $m/t$, in the limit $tto infty$, is distributed uniformly on $[-1,1]$ for all continuous distributions of velocities. In the second case, the distribution is given by the distribution of the velocities itself, with a Galilean shift by the mean velocity. We also find the large time approach to the limiting forms and compare the results with numerical simulations. In contrast, if the exchange of velocities occurs at unit rate, independent of their values, and irrespective of which is faster, $m(t)/t$ for large $t$ is has a gaussian distribution, whose width varies as $t^{-1/2}$.