Temporal Correlations and Persistence in the Kinetic Ising Model: the Role of Temperature

We study the statistical properties of the sum $S_t=\int_{0}^{t}dt’ \sigma_{t’}$, that is the difference of time spent positive or negative by the spin ${\sigma }_t$, located at a given site of a $D$-dimensional Ising model evolving under Glauber dynamics from a random initial configuration. We investigate the distribution of $S_t$ and the first-passage statistics (persistence) of this quantity. We discuss successively the three regimes of high temperature ($T>T_c$), criticality ($T=T_c$), and low temperature ($T<T_c$). We discuss in particular the question of the temperature dependence of the persistence exponent $\theta$, as well as that of the spectrum of exponents $\theta (x)$, in the low temperature phase. The probability that the temporal mean $S_t\mathrm{/}t$ was always larger than the equilibrium magnetization is found to decay as $t^{-\theta -\frac{1}{2}}$. This yields a numerical determination of the persistence exponent $\theta$ in the whole low temperature phase, in two dimensions, and above the roughening transition, in the low-temperature phase of the three-dimensional Ising model.