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Finite time and asymptotic behaviour of the maximal excursion of a random walk

Roger Bidaux·Jerome Chave·Radim Vocka
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Abstract

We evaluate the limit distribution of the maximal excursion of a random walk in any dimension for homogeneous environments and for self-similar supports under the assumption of spherical symmetry. This distribution is obtained in closed form and is an approximation of the exact distribution comparable to that obtained by real space renormalization methods. Then we focus on the early time behaviour of this quantity. The instantaneous diffusion exponent $ν_n$ exhibits a systematic overshooting of the long time exponent. Exact results are obtained in one dimension up to third order in $n^{-1/2}$. In two dimensions, on a regular lattice and on the Sierpiński gasket we find numerically that the analytic scaling $ν_n \simeq ν+A n^{-ν}$ holds.

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Roger Bidaux, Jerome Chave, Radim Vocka. 1999-05-20. Finite time and asymptotic behaviour of the maximal excursion of a random walk. https://doi.org/10.1088/0305-4470%2F32%2F27%2F302

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