Blocking and Persistence in the Zero-Temperature Dynamics of Homogeneous and Disordered Ising Models

A “persistence” exponent $\theta$ has been extensively used to describe the nonequilibrium dynamics of spin systems following a deep quench: For zero-temperature homogeneous Ising models on the $d$-dimensional cubic lattice $Z^d$, the fraction $p\left(t\right)$ of spins not flipped by time $t$ decays to zero like $t^{-\theta \left(d\right)}$ for low $d$; for high $d$, $p\left(t\right)$ may decay to $p\left(\infty \right)>0\mathrm{}$, because of “blocking” (but perhaps still like a power). What are the effects of disorder or changes of the lattice? We show that these can quite generally lead to blocking (and convergence to a metastable configuration) even for low $d$, and then present two examples—one disordered and one homogeneous—where $p\left(t\right)$ decays exponentially to $p\left(\infty \right)$.