# Number of distinct sites visited by N random walkers.

@article{Larralde1992NumberOD, title={Number of distinct sites visited by N random walkers.}, author={Larralde and Trunfio and Havlin and Stanley and Weiss}, journal={Physical review. A, Atomic, molecular, and optical physics}, year={1992}, volume={45 10}, pages={ 7128-7138 } }

We study the number of distinct sites visited by N random walkers after t steps Siv(t) under the condition that all the walkers are initially at the origin. We derive asymptotic expressions for the mean number of distinct sites (Siv(t)) in one, two, and three dimensions. We find that (Siv(t)) passes through several growth regimes; at short times (Siv(t)) ~ t" (regime I), for t» & t & t'„we find that (Siv(t)) ~ (t ln[N Si(t)/t ])" (regime II), and for t & t'„,(Siv(t)) ~ NSi(t) (regime III). The… Expand

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#### References

SHOWING 1-3 OF 3 REFERENCES