Aging in 1D Discrete Spin Models and Equivalent Systems

We derive exact expressions for a number of aging functions that are scaling limits of nonequilibrium correlations, $R(t_w{,t}_w+t)$ as $t_w\to \infty$, ${t/t}_w\to \theta$, in the 1D homogenous $q$-state Potts model for all $q$ with $T=0\mathrm{}$ dynamics following a quench from $T=\infty$. One such quantity is $〈{\overrightarrow{\sigma }}_0\left(t_w\right)̇{\overrightarrow{\sigma }}_n(t_w+t)〉$ when $n/\sqrt{t_w}\to z$. Exact, closed-form expressions are also obtained when an interlude of $T=\infty$ dynamics occurs. Our derivations express the scaling limit via coalescing Brownian paths and a “Brownian space-time spanning tree,” which also yields other aging functions, such as the persistence probability of no spin flip at $0$ between $t_w$ and $t_w+t$.