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Variance of the range of a random walk

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Abstract

T n, the expectation of the square of the number of distinct sites occupied by a random walk in steps 1 throughn, is obtained from its relation to the dual first occupancy probabilityF ij(x, x′), and the latter quantity is obtained from a recursion with the first occupancy probabilityF k (x″). The varianceV n of the number of distinct sites occupied is calculated directly from Tn; the procedure is illustrated by the calculation ofV n (4096 /⩾n) and the derivation of asymptotic expansions forV n for a particular random walk in dimensions 1 through 3.

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Authors and Affiliations

  1. Theoretical Biology and Biophysics, Theoretical Division, Los Alamos National Laboratory, Group T-10, Mail Stop K710, 87545, Los Alamos, New Mexico, USA

    David C. Torney

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  1. David C. Torney
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Work completed under the auspices of the United States Department of Energy.

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Torney, D.C. Variance of the range of a random walk. J Stat Phys 44, 49–66 (1986). https://doi.org/10.1007/BF01010904

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  • DOI: https://doi.org/10.1007/BF01010904

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