Short-time dynamics of percolation observables

Abstract

We consider the critical short-time evolution of magnetic and droplet-percolation order parameters for the Ising model in two and three dimensions, through Monte Carlo simulations with the ͑local͒ heat-bath method. We find qualitatively different dynamic behaviors for the two types of order parameters. More precisely, we find that the percolation order parameter does not have a power-law behavior as encountered for the magnetization, but develops a scale ͑related to the relaxation time to equilibrium͒ in the Monte Carlo time. We argue that this difference is due to the difficulty in forming large clusters at the early stages of the evolution. Our results show that, although the descriptions in terms of magnetic and percolation order parameters may be equivalent in the equilibrium regime, greater care must be taken to interpret percolation observables at short times. In particular, this concerns the attempts to describe the dynamics of the deconfinement phase transition in QCD using cluster observables.

Short-time behavior of percolation observables at O(3) spin models: a preliminary analysis

“…This function is similar to the solution of the diffusion equation [3,11]. On the other hand the magnetic order parameter follows a power law M ∼ t θ as expected.…”

“…Indeed, the magnetization follows a power law while the percolation order parameter follows a function solution similar to the diffusion equation. This fact has already been observed for the Ising model [3]. We must conclude our simulations soon using largest lattices and heat-bath algorithm [7,12].…”

“…Recently was shown that percolation order parameter and magnetization do not have the same behavior at the heat-bath short-time dynamics in the bidimensional Ising models [3]. This raised some questions about the evolution of the percolation order parameters to more complex spin models.…”

“…On the other hand, until now, the short-time evolution of percolation order parameter has received a few attention. This is due to, perhaps, these quantities present the same behavior at thermodynamic equilibrium [2] or because the theory of percolation does not have a proper dynamics.Recently was shown that percolation order parameter and magnetization do not have the same behavior at the heat-bath short-time dynamics in the bidimensional Ising models [3]. This raised some questions about the evolution of the percolation order parameters to more complex spin models.…”

Dynamic behavior of cluster observables for the 2d Ising model

“…(6), which seems to be related to the relaxation time of the algorithm. We are currently investigating the possibility that the observable Ω may suffer from finite-size effects related to the difficulty in forming a percolating cluster at the early stages of the simulation [10].…”