Persistence in systems with algebraic interaction

TL;DR

This study investigates persistence behavior in one-dimensional spin systems with algebraically decaying interactions, revealing a universal persistence exponent for large interaction ranges and discussing finite size effects for smaller ranges.

Contribution

The paper provides numerical evidence that the persistence exponent is universal for large interaction exponents and introduces scaling arguments to address finite size effects for smaller exponents.

Findings

Persistence decays algebraically with system size for large interaction exponents.

Persistence exponent is approximately 0.175 for large $\sigma$, independent of $\sigma$.

Finite size effects are significant for small $\sigma$, requiring system size scaling as ${[{ m O}(1/\sigma)]}^{1/\sigma}$.

Abstract

Persistence in coarsening 1D spin systems with a power law interaction $r^{-1-\sigma}$ is considered. Numerical studies indicate that for sufficiently large values of the interaction exponent $\sigma$ ($\sigma\geq 1/2$ in our simulations), persistence decays as an algebraic function of the length scale $L$, $P(L)\sim L^{-\theta}$. The Persistence exponent $\theta$ is found to be independent on the force exponent $\sigma$ and close to its value for the extremal ($\sigma \to \infty$) model, $\bar\theta=0.17507588...$. For smaller values of the force exponent ($\sigma< 1/2$), finite size effects prevent the system from reaching the asymptotic regime. Scaling arguments suggest that in order to avoid significant boundary effects for small $\sigma$, the system size should grow as ${[{\cal O}(1/\sigma)]}^{1/\sigma}$.