Infinite System of Random Walkers: Winners and Losers

Abstract

We study an infinite system of particles initially occupying a half-line $y\leq 0$ and undergoing random walks on the entire line. The right-most particle is called a leader. Surprisingly, every particle except the original leader may never achieve the leadership throughout the evolution. For the equidistant initial configuration, the $k^{\text{th}}$ particle attains the leadership with probability $e^{-2} k^{-1} (\ln k)^{-1/2}$ when $k\gg 1$. This provides a quantitative measure of the correlation between earlier misfortune (represented by $k$) and eternal failure. We also show that the winner defined as the first walker overtaking the initial leader has label $k\gg 1$ with probability decaying as $\exp\!\left[-\tfrac{1}{2}(\ln k)^2\right]$.