Random walks on three-dimensional lattices: A matrix method for calculating the probability of eventual return

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Abstract
A matrix method for calculating the number of visits to the origin in random-walks on periodic lattices is developed. The method is applied to three-dimensional lattices: the f.c.c., b.c.c. and the diamond lattices. For the f.c.c. and the b.c.c. lattices the results are in good agreement with those obtained by Montroll. The probability of eventual return in the diamond lattice is evaluated for the first time: it is about 0·442. A new concept, the effective coordination number, is defined on the basis of a calculation of random walks on an imaginary lattice.

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