The short-time critical dynamics of propagation of damage in the Ising ferromagnet in two dimensions is studied by means of Monte Carlo simulations. Starting with equilibrium configurations at $T=\ensuremath{\infty}$ and magnetization
M=0, an initial damage is created by flipping a small amount of spins in one of the two replicas studied. In this way, the initial damage is proportional to the initial magnetization
M0 in one of the configurations upon quenching the system at
TC, the Onsager critical temperature of the ferromagnetic-paramagnetic transition. It is found that, at short times, the damage increases with an exponent ${\ensuremath{\theta}}_{D}=1.915(3)$, which is much larger than the exponent $\ensuremath{\theta}=0.197$ characteristic of the initial increase of the magnetization
M(t). Also, an epidemic study was performed. It is found that the average distance from the origin of the epidemic
(⟨R2(t)⟩) grows with an exponent ${z}^{\ensuremath{\ast}}\ensuremath{\approx}\ensuremath{\eta}\ensuremath{\approx}1.9$, which is the same, within error bars, as the exponent ${\ensuremath{\theta}}_{D}$. However, the survival probability of the epidemics reaches a plateau so that $\ensuremath{\delta}=0$. On the other hand, by quenching the system to lower temperatures one observes the critical spreading of the damage at ${T}_{D}\ensuremath{\simeq}0.51{T}_{C}$, where all the measured observables exhibit power laws with exponents ${\ensuremath{\theta}}_{D}=1.026(3)$, $\ensuremath{\delta}=0.133(1)$, and ${z}^{\ensuremath{\ast}}=1.74(3)$.
Spin glasses in dimensions 2, 3, 4 and mean field are studied numerically as a function of the degree of asymmetry of the interactions between spins. Damage spreading data are used to assess the effects on the spin glass ordering of introducing asymmetry in the couplings. In all dimensions the critical temperature for damage spreading is rather insensitive to the asymmetry, but the spin glass ordering is strongly suppressed by any finite degree of asymmetry. We discuss the physical significance of the damage spreading parameter.
The authors show that in the three-dimensional Ising model at the critical temperature the fractal dimension of the touching damage, determined by the box counting method, asymptotically converges to the expected value d f =d- beta /v. In contrast, the finite-size scaling analysis indicates for the touching damage an effective fractal dimension whose value is 20% smaller. The expected d f is, however, recovered again when instead of the system size L the quantity L/ln L is used as a scaling variable. This behaviour could be explained by the recently discovered dynamical multiscaling.