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An Isaac Newton Institute Workshop

First-Passage and Extreme Value Problems in Random Processes

A Nontrivial Constant c=0.29795219028 in One and Three Dimensional Random Walks

28th June 2006

Author: Majumdar, S (Universite Paris-Sud)

Abstract

Two different random walk problems, one in one dimension and the other in three dimensions, seem to share a nontrivial constant whose numerical value is c=0.29795219028.. In the first problem, this constant shows up in the finite size correction to the expected maximum of a discrete-time random walk on a continuous line with unform jump density. In the second problem, this constant appears as the `Milne extrapolation length' in the expression for flux of discrete-time random walkers to a spherical trap in three dimensions. We prove why the same constant appears in the two problems and derive an exact analytical formula for this constant.