Ordering of Random Walks: The Leader and the Laggard

Abstract

We investigate two complementary problems related to maintaining the relative positions of N random walks on the line: (i) the leader problem, that is, the probability {\cal L}_N(t) that the leftmost particle remains the leftmost as a function of time and (ii) the laggard problem, the probability {\cal R}_N(t) that the rightmost particle never becomes the leftmost. We map these ordering problems onto an equivalent (N-1)-dimensional electrostatic problem. From this construction we obtain a very accurate estimate for {\cal L}_N(t) for N=4, the first case that is not exactly soluble: {\cal L}_4(t) ~ t^{-β_4}, with β_4=0.91342(8). The probability of being the laggard also decays algebraically, {\cal R}_N(t) ~ t^{-γ_N}; we derive γ_2=1/2, γ_3=3/8, and argue that γ_N--> ln N/N$ as N-->oo.

arXiv comment7 pages, 4 figures, 2-column revtex 4 format

References in corpus (7)

• Capture of the Lamb: Diffusing Predators Seeking a Diffusing Prey
• Vicious walkers, friendly walkers and Young tableaux II: With a wall
• Go with the Winners: a General Monte Carlo Strategy
• Kinetics of a Diffusive Capture Process: Lamb Besieged by a Pride of Lions
• Scaling limit of vicious walks and two-matrix model
• Families of Vicious Walkers
• A Hexagon Model for 3D Lorentzian Quantum Cosmology

Cited by in corpus (17)

• Persistence and First-Passage Properties in Non-equilibrium Systems
• Vicious walk with a wall, noncolliding meanders, and chiral and Bogoliubov-deGennes random matrices
• Can a Lamb Reach a Haven Before Being Eaten by Diffusing Lions?
• First-Passage Exponents of Multiple Random Walks
• Kinetics of First Passage in a Cone
• Anomalous Dimension in a Two-Species Reaction-Diffusion System
• Slow Kinetics of Brownian Maxima
• On the Mixing of Diffusing Particles
• Maximum Distance Between the Leader and the Laggard for Three Brownian Walkers
• Gradual Diffusive Capture: Slow Death by Many Mosquito Bites
• Scaling Exponents for Ordered Maxima
• First Passage in Infinite Paraboloidal Domains
• Ordering statistics of 4 random walkers on the line
• Statistical properties of sites visited by independent random walks
• Infinite System of Random Walkers: Winners and Losers
• Diffusion in a fluid flow generated by a source at the apex of a wedge
• Vicious accelerating walkers