Abstract
A random walk starts from the origin of ad-dimensional lattice. The occupation numbern(x,t) equals unity if aftert steps site x has been visited by the walk, and zero otherwise. We study translationally invariant sumsM(t) of observables defined locally on the field of occupation numbers. Examples are the numberS(t) of visited sites, the areaE(t) of the (appropriately defined) surface of the set of visited sites, and, in dimension d=3, the Euler index of this surface. Ind≤ 3, theaverages - M(t) all increase linearly witht ast ® ∞. We show that in d=3, to leading order in an asymptotic expansion int, thedeviations from average ΔM(t) = M(t) -M(t) are, up to a normalization, allidentical to a single “universal” random variable. This result resembles an earlier one in dimensiond=2; we show that this universality breaks down ford>3.
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This paper is dedicated to Bernard Jancovici on the occasion of his 65th birthday.
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van Wijland, F., Hilhorst, H.J. Universal fluctuations in the support of the random walk. J Stat Phys 89, 119–134 (1997). https://doi.org/10.1007/BF02770757
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DOI: https://doi.org/10.1007/BF02770757