Statistics of overtakes by a tagged agent
Santanu Das, Deepak Dhar, and Sanjib Sabhapandit

TL;DR
This paper analyzes the displacement statistics of a tagged agent in a one-dimensional lattice with velocity-dependent exchange rules, revealing different limiting distributions depending on the exchange rate mechanism.
Contribution
It introduces a detailed study of a Markov process with velocity-dependent exchanges, deriving explicit displacement distributions for different exchange rate models.
Findings
For rate-1 exchange, displacement scaled by time is uniformly distributed on [-1,1].
For velocity-dependent exchange rate, displacement distribution matches the velocity distribution shifted by mean velocity.
When exchange occurs at a constant rate regardless of velocities, displacement follows a Gaussian distribution with variance decreasing as t^{-1/2}.
Abstract
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 on the lattice, in a given duration , for two different kinds of rates: one in which a pair of agents at sites and exchange their sites with rate , 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 for large . In the first case, for a…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
