Can a Lamb Reach a Haven Before Being Eaten by Diffusing Lions?

Details Can a Lamb Reach a Haven Before Being Eaten by Diffusing Lions? Can a Lamb Reach a Haven Before Being Eaten by Diffusing Lions?

2012-03-14

arxiv.org/abs/1203.2985v1

Physics

Data Analysis, Statistics and Probability

13 pages, 6 figures, IOP format

Scientific paper

We study the survival of a single diffusing lamb on the positive half line in the presence of N diffusing lions that all start at the same position L to the right of the lamb and a haven at x=0. If the lamb reaches this haven before meeting any lion, the lamb survives. We investigate the survival probability of the lamb, S_N(x,L), as a function of N and the respective initial positions of the lamb and the lions, x and L. We determine S_N(x,L) analytically for the special cases of N=1 and N--->oo. For large but finite N, we determine the unusual asymptotic form whose leading behavior is S_N(z)\simN^{-z^2}, with z=x/L. Simulations of the capture process very slowly converge to this asymptotic prediction as N reaches 10^{500}.

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