Persistence in systems with algebraic interaction

Abstract

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