# 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…

## 66 Citations

### Number of common sites visited by N random walkers.

- MathematicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2012

The mean number of common sites, W(N)(t), visited by N independent random walkers each of length t and all starting at the origin at t = 0 in d dimensions is computed analytically.

### Exact distributions of the number of distinct and common sites visited by N independent random walkers.

- MathematicsPhysical review letters
- 2013

It is shown that these two random variables can be mapped onto extreme value quantities associated with N independent random walkers and compute exactly their probability distributions for any value of N in the limit of large time t, where the randomWalkers can be described by Brownian motions.

### Record statistics for multiple random walks.

- MathematicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2012

The statistics of the number of records R(n,N) for N identical and independent symmetric discrete-time random walks of n steps in one dimension find that the mean record number grows universally as ~α(N) sqrt[n] for large n, but with a very different behavior of the amplitude α(N] for N>1 in the two cases.

### Order statistics of Rosenstock's trapping problem in disordered media.

- MathematicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2003

Simulation results for the two-dimensional incipient percolation aggregate confirm the predictions of the approach and show that the ratio between the nth cumulant and the n fourth moment of S(N)(t) is, for large N, very large in comparison with the same ratio in Euclidean media, and almost constant.

### Expected number of distinct sites visited by N random walks in the presence of an absorbing boundary

- Mathematics
- 2003

In earlier work we have studied the expected number of distinct sites (ENDS) visited by N random walkers in time t on a translationally invariant lattice. Optical applications suggest the interest in…

### Diffusion of a set of random walkers in Euclidean media. First passage times

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- 2000

When a large numberN of independent random walkers diffuse on ad-dimensional Euclidean substrate, what is the expectation valueht1;Ni of the time spent by the first random walker to cross a given…

### Probability distribution of the number of distinct sites visited by a random walk on the finite-size fully-connected lattice

- Mathematics
- 2014

The probability distribution of the number s of distinct sites visited up to time t by a random walk on the fully-connected lattice with N sites is first obtained by solving the eigenvalue problem…

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