Multidimensional persistence behaviour in an Ising system

We consider a periodic Ising chain with nearest-neighbour and $r$-th neighbour interaction and quench it from infinite temperature to zero temperature. The persistence probability $P(t)$, measured as the probability that a spin remains unflipped upto time $t$, is studied by computer simulation for suitable values of $r$. We observe that as time progresses, $P(t)$ first decays as $t^{-0.22}$ (-the {\em first} regime), then the $P(t)-t$ curve has a small slope (in log-log scale) for some time (-the {\em second} regime) and at last it decays nearly as $t^{-3/8}$ (-the {\em third} regime). We argue that in the first regime, the persistence behaviour is the usual one for a two-dimensional system, in the second regime it is like that of a non-interacting (`zero-dimensional') system and in the third regime the persistence behaviour is like that of a one dimensional Ising model. We also provide explanations for such behaviour.