Dynamic behavior of cluster observables for the 2d Ising model

Abstract

We present results of our numerical study of the critical dynamics of percolation observables for the twodimensional Ising model. We consider the (Monte Carlo) short-time evolution of the system with small initial magnetization and heat-bath dynamics. We find qualitatively different dynamic behaviors for the magnetization M and for Ω, the so-called strength of the percolating cluster, which is the order parameter of the percolation transition. More precisely, we obtain a (leading) exponential form for Ω as a function of the Monte Carlo time t, to be compared with the power-law increase encountered for M at short times. Our results suggest that, although the descriptions in terms of magnetic or percolation order parameters may be equivalent in the equilibrium regime, greater care must be taken to interpret percolation observables at short times.

Short-time dynamics of percolation observables

“…( 6) and (7) are well fitted by the data in two and three dimensions, for all volumes considered. (We note however that the two-dimensional data for Ω were fitted in [16] considering also a second exponential term, adding a third parameter to this fit.) The above data for τ and τ ′ can be fitted to the form L z , in the two-and three-dimensional cases separately.…”

“…We thus consider here the short-time dynamics of the two-and three-dimensional Ising model and focus on the dynamic critical behavior of the percolation order parameter. (Preliminary results of our study were presented in [15,16].) Note that the dynamic behavior of cluster observables has been considered for the droplet clusters in Ising and Potts models in various studies (see e.g.…”