Short-time Dynamics of Percolation Observables

TL;DR

This study investigates the early-time behavior of magnetic and percolation order parameters in the 2D and 3D Ising model using Monte Carlo simulations, revealing distinct dynamic behaviors and implications for interpreting cluster observables.

Contribution

It demonstrates that percolation observables do not follow power-law dynamics like magnetization in short-time evolution, highlighting the need for careful interpretation in phase transition studies.

Findings

Percolation order parameter develops a scale related to relaxation time.

Magnetization exhibits power-law behavior in early dynamics.

Differences impact the analysis of phase transitions using cluster 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 (local) 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 equilibrium) 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 interprete 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.