Harnessing finite-size effects to gauge aging in the $2D$ Ising model

The relaxation behavior towards equilibrium of the $2D$ Ising model with nearest-neighbor interactions has been studied with focus on the two-time autocorrelator $C(t,s)$. Finite-size effects affecting the growing magnetic domains lead to the saturation of $C(t,s)$ with a distinct plateau of height $C_{\mathrm{\infty }}^{(2)}(s,L)$ scaling algebraically with waiting time $s$ and lattice size $L$. These scaling relations are used to produce precise estimates for the autocorrelation exponent $\lambda$ and dynamical exponent $z$ with deliberately small lattices. Treating smooth domain walls in a similar manner to the lattice boundaries, their effect on $C(t,s)$ can be understood as premature finite-size phenomenon, extending our ansatz to systems not yet in equilibrium.