What question did this study set out to answer?

To establish a general mathematical formalism for calculating the average number of distinct lattice sites visited during an n-step random walk across different dimensions.

September 1, 1963Journal of Mathematical PhysicsOpen Access

The Number of Distinct Sites Visited in a Random Walk on a Lattice

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GVGeorge H. VineyardSouthern Illinois University Carbondale

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Implication

Mathematical analysis reveals asymptotic scaling of distinct sites visited during lattice random walks, indicating direct correspondence with cubic resistor network resistance.

Key Points

To establish a general mathematical formalism for calculating the average number of distinct lattice sites visited during an n-step random walk across different dimensions.
Developed a theoretical framework to derive asymptotic visitation limits for random walks in one, three, and higher dimensions.
Evaluated exact proportionality constants for simple cubic, body-centered cubic, and face-centered cubic lattices using Watson's integrals.
Mapped the random walk problem to an electrical network where nearest-neighbor bonds are represented by unit resistors.
Demonstrated that the asymptotic average number of distinct sites visited scales as (8n/π)^(1/2) in one dimension and linearly as cn in three or more dimensions.
Evaluated the exact constant c for simple cubic, body-centered cubic, and face-centered cubic lattice structures using Watson's integrals.
Calculated the exact effective electrical resistance across infinite networks for all three cubic lattice configurations.

Cite This Study

George H. Vineyard (1963) studied this question.

synapsesocial.com/papers/6a154bc079ff98d0de4e661b https://doi.org/10.1063/1.1704049

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Implication

Mathematical analysis reveals asymptotic scaling of distinct sites visited during lattice random walks, indicating direct correspondence with cubic resistor network resistance.

The Number of Distinct Sites Visited in a Random Walk on a Lattice

Discussion

Member takes

Implication

Key Points

Cite This Study

The Number of Distinct Sites Visited in a Random Walk on a Lattice

Discussion

Member takes

Implication

Key Points

Cite This Study

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