Persistence in the two dimensional ferromagnetic Ising model

TL;DR

This paper provides precise numerical estimates of the zero-temperature local persistence in the 2D ferromagnetic Ising model, revealing universal decay exponents dependent on initial conditions and lattice types.

Contribution

It introduces highly accurate numerical estimates of persistence exponents in the 2D Ising model, demonstrating their dependence on initial correlations and lattice boundary conditions.

Findings

Persistence exponent is approximately 0.199 for short-range initial states.

Persistence exponent is approximately 0.033 for long-range (critical) initial states.

Finite size effects are more pronounced with free boundary conditions.

Abstract

We present very accurate numerical estimates of the time and size dependence of the zero-temperature local persistence in the $2d$ ferromagnetic Ising model. We show that the effective exponent decays algebraically to an asymptotic value $\theta$ that depends upon the initial condition. More precisely, we find that $\theta$ takes one universal value $0.199(2)$ for initial conditions with short-range spatial correlations as in a paramagnetic state, and the value $0.033(1)$ for initial conditions with the long-range spatial correlations of the critical Ising state. We checked universality by working with a square and a triangular lattice, and by imposing free and periodic boundary conditions. We found that the effective exponent suffers from stronger finite size effects in the former case.