Unified Solution of the Expected Maximum of a Random Walk and the Discrete Flux to a Spherical Trap

Satya N. Majumdar ¹, Alain Comtet ^1, 2, Robert M. Ziff ³

Journal of Statistical Physics 122 (2006) 833-856

Two random-walk related problems which have been studied independently in the past, the expected maximum of a random walker in one dimension and the flux to a spherical trap of particles undergoing discrete jumps in three dimensions, are shown to be closely related to each other and are studied using a unified approach as a solution to a Wiener-Hopf problem. For the flux problem, this work shows that a constant c = 0.29795219 which appeared in the context of the boundary extrapolation length, and was previously found only numerically, can be derived explicitly. The same constant enters in higher-order corrections to the expected-maximum asymptotics. As a byproduct, we also prove a new universal result in the context of the flux problem which is an analogue of the Sparre Andersen theorem proved in the context of the random walker’s maximum.

• 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
• 3. Michigan Center for Theoretical Physics and Department of chemical Engineering,
  University of Michigan-Ann Arbor