We analyze the critical properties of the three-dimensional Ising model with linear parallel extended defects. Such a form of disorder produces two distinct correlation lengths, a parallel correlation length $xi_parallel$ in the direction along defec
ts, and a perpendicular correlation length $xi_perp$ in the direction perpendicular to the lines. Both $xi_parallel$ and $xi_perp$ diverge algebraically in the vicinity of the critical point, but the corresponding critical exponents $ u_parallel$ and $ u_perp$ take different values. This property is specific for anisotropic scaling and the ratio $ u_parallel/ u_perp$ defines the anisotropy exponent $theta$. Estimates of quantitative characteristics of the critical behaviour for such systems were only obtained up to now within the renormalization group approach. We report a study of the anisotropic scaling in this system via Monte Carlo simulation of the three-dimensional system with Ising spins and non-magnetic impurities arranged into randomly distributed parallel lines. Several independent estimates for the anisotropy exponent $theta$ of the system are obtained, as well as an estimate of the susceptibility exponent $gamma$. Our results corroborate the renormalization group predictions obtained earlier.
We study the survival of a prey that is hunted by N predators. The predators perform independent random walks on a square lattice with V sites and start a direct chase whenever the prey appears within their sighting range. The prey is caught when a p
redator jumps to the site occupied by the prey. We analyze the efficacy of a lazy, minimal-effort evasion strategy according to which the prey tries to avoid encounters with the predators by making a hop only when any of the predators appears within its sighting range; otherwise the prey stays still. We show that if the sighting range of such a lazy prey is equal to 1 lattice spacing, at least 3 predators are needed in order to catch the prey on a square lattice. In this situation, we establish a simple asymptotic relation ln(Pev)(t) sim (N/V)2ln(Pimm(t)) between the survival probabilities of an evasive and an immobile prey. Hence, when the density of the predators is low N/V<<1, the lazy evasion strategy leads to the spectacular increase of the survival probability. We also argue that a short-sighting prey (its sighting range is smaller than the sighting range of the predators) undergoes an effective superdiffusive motion, as a result of its encounters with the predators, whereas a far-sighting prey performs a diffusive-type motion.
