D ben-Avraham et al 2003 J. Phys. A: Math. Gen. 36 1789 doi:10.1088/0305-4470/36/7/301
Ordering of random walks: the leader and the laggard
D ben-Avraham1, B M Johnson1, C A Monaco1, P L Krapivsky2 and S Redner2
Show affiliationsWe investigate two complementary problems related to maintaining the relative positions of N random walks on the line: (i) the leader problem, that is, the probability 
N(t) that the leftmost particle remains the leftmost as a function of time and (ii) the laggard problem, the probability 
N(t) that the rightmost particle never becomes the leftmost. We map these ordering problems onto an equivalent (N − 1)-dimensional electrostatic problem. From this construction we obtain a very accurate estimate for N(t) for N = 4, the first case that is not exactly solvable: 4(t) 
t−β4, with β4 = 0.91342(8). The probability of being the laggard also decays algebraically, N(t) t−γN; we derive γ2 = 1/2, γ3 = 3/8, and argue that γN → N−1 ln N as N → ∞.
- PACS
- MSC
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60Axx Foundations of probability theory
82B41 Random walks, random surfaces, lattice animals, etc. (See also 60G50, 82C41)
- Subjects
- Dates
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Issue 7 (21 February 2003)
Received 28 October 2002, in final form 6 January 2003
Published 5 February 2003
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Ordering of random walks: the leader and the laggard
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