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

Abstract

We study the statistical properties of the sum $S_t={\int }_0^{t}d{t}^{\prime }{\sigma }_{t^{\prime }}$, 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 $S_t/t$ 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.

21 pages, 11 PostScript figures included (1 color figure)

Cited by in corpus (9)

• Aging Wiener-Khinchin Theorem and Critical Exponents of $1/f$ Noise
• Persistence in the two dimensional ferromagnetic Ising model
• Statistics of the occupation time for a random walk in the presence of a moving boundary
• Virtual walks inspired by a mean field kinetic exchange model of opinion dynamics
• Dynamics of the two-dimensional directed Ising model: zero-temperature coarsening
• Record statistics of integrated random walks and the random acceleration process
• Virtual walks in spin space: a study in a family of two-parameter models
• Thermalization with a multibath: an investigation in simple models
• Occupation time of a renewal process coupled to a discrete Markov chain