Precise Asymptotics for a Random Walker’s Maximum

Alain Comtet 1, 2, Satya N. Majumdar 1

Journal of Statistical Mechanics: Theory and Experiment 06 (2005) P06013

We consider a discrete time random walk in one dimension. At each time step the walker jumps by a random distance, independent from step to step, drawn from an arbitrary symmetric density function. We show that the expected positive maximum E[M_n] of the walk up to n steps behaves asymptotically for large n as, E[M_n]/\\sigma=\\sqrt{2n/\\pi}+ \\gamma +O(n^{-1/2}), where \\sigma^2 is the variance of the step lengths. While the leading \\sqrt{n} behavior is universal and easy to derive, the leading correction term turns out to be a nontrivial constant \\gamma. For the special case of uniform distribution over [-1,1], Coffmann et. al. recently computed \\gamma=-0.516068...by exactly enumerating a lengthy double series. Here we present a closed exact formula for \\gamma valid for arbitrary symmetric distributions. We also demonstrate how \\gamma appears in the thermodynamic limit as the leading behavior of the difference variable E[M_n]-E[|x_n|] where x_n is the position of the walker after n steps. An application of these results to the equilibrium thermodynamics of a Rouse polymer chain is pointed out. We also generalize our results to L\\évy walks.

  • 1. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
    CNRS : UMR8626 – Université Paris XI - Paris Sud
  • 2. Unite mixte de service de l'institut Henri Poincaré (UMSIHP),
    CNRS : UMS839 – Université Paris VI - Pierre et Marie Curie