George H. Weiss’s research while affiliated with National Eye Institute, National Institutes of Health and other places

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Publications (1)


Random Walks on Lattices. II
  • Article

February 1965

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287 Reads

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3,050 Citations

Elliott W. Montroll

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George H. Weiss

Formulas are obtained for the mean first passage times (as well as their dispersion) in random walks from the origin to an arbitrary lattice point on a periodic space lattice with periodic boundary conditions. Generally this time is proportional to the number of lattice points. The number of distinct points visited after n steps on a k-dimensional lattice (with k ≥ 3) when n is large is a1n + a2n½ + a3 + a4n−½ + …. The constants a1 − a4 have been obtained for walks on a simple cubic lattice when k = 3 and a1 and a2 are given for simple and face-centered cubic lattices. Formulas have also been obtained for the number of points visited r times in n steps as well as the average number of times a given point has been visited. The probability F(c) that a walker on a one-dimensional lattice returns to his starting point before being trapped on a lattice of trap concentration c is F(c) = 1 + [c/(1 − c)] log c. Most of the results in this paper have been derived by the method of Green's functions.

Citations (1)


... To the best of our knowledge, the diffusion limit of a random walk system, modeled as a discrete stochastic process with nonconstant sojourn times, has not yet been obtained. For continuous-time scenarios, the Montroll-Weiss theory of continuous-time random walks (CTRW) [19,23] provides a framework, which we will briefly outline below. When t varies spatially, the resulting random walk system loses its Markovian property, rendering most Markov chain tools inapplicable. ...

Reference:

Random Walk with Heterogeneous Sojourn Time
Random Walks on Lattices. II
  • Citing Article
  • February 1965