Unified Solution of the Expected Maximum of a Discrete Time Random Walk and the Discrete Flux to a Spherical Trap |
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Authors: | Satya N Majumdar Alain Comtet Robert M Ziff |
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Institution: | (1) Laboratoire de Physique Théorique et Modèles Statistiques, Université Paris-Sud. Bat, Orsay Cedex, 100. 91405, France;(2) Institut Henri Poincaré, 11 rue Pierre et Marie Curie, Paris, 75005, France;(3) Michigan Center for Theoretical Physics and Department of Chemical Engineering, University of Michigan, Ann Arbor, MI 48109-2136, USA |
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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 analytically. 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. |
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Keywords: | Random walk adsorption to a trap Wiener-Hopf diffusion Sparre Anderson theorem |
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