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Infinite system of random walkers: winners and losers

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We study an infinite system of particles initially occupying a half-line y ⩽ 0 and undergoing random walks on the entire line. The right-most particle is called a leader. Surprisingly,… Click to show full abstract

We study an infinite system of particles initially occupying a half-line y ⩽ 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 kth particle attains the leadership with probability e−2 k −1(ln  k)−1/2 when k ≫ 1. This decay law provides a quantitative measure of the correlation between earlier misfortune proportional to the label k and eternal failure. We also show that the winner defined as the first walker overtaking the initial leader has label k ≫ 1 with probability decaying as exp−12(lnk)2 .

Keywords: system; winners losers; system random; infinite system; random walkers; walkers winners

Journal Title: Journal of Physics A: Mathematical and Theoretical
Year Published: 2020

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