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

TL;DR

This paper analyzes the probability of a diffusing lamb reaching a safe haven before being caught by diffusing lions, providing exact solutions for specific cases and asymptotic behavior for large numbers of lions.

Contribution

It offers analytical solutions for the lamb's survival probability for N=1 and N→∞, and characterizes the asymptotic form for large N, including simulation validation.

Findings

Exact survival probability for N=1 and N→∞.

Asymptotic form S_N(z) ~ N^{-z^2} for large N.

Simulations converge slowly to the asymptotic prediction.

Abstract

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}.