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

TL;DR

This paper unifies the analysis of the expected maximum of a one-dimensional random walk and the flux to a spherical trap in three dimensions, deriving explicit constants and revealing deep connections between these problems.

Contribution

It introduces a unified approach to two related random walk problems, explicitly derives a key constant, and proves a new universal result analogous to Sparre Andersen's theorem.

Findings

Explicit derivation of the constant c = 0.29795219 in flux problem

Unified solution approach using Wiener-Hopf method

New universal result analogous to Sparre Andersen theorem

Abstract

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.