Non-Markovian persistence and nonequilibrium critical dynamics

The persistence exponent $\theta$ for the global order parameter $M\left(t\right)\mathrm{}$ of a system quenched from the disordered phase to its critical point describes the probability, $p\left(t\right)\mathrm{}\sim t^{-\theta }$, that $M\left(t\right)\mathrm{}$ does not change sign in the time interval $t$ following the quench. We calculate $\theta$ to $O\left({\epsilon }^2\right)$ for model $A$ of Hohenberg and Halperin [Rev. Mod. Phys. 49, 435 (1977)] (and to order $\epsilon$ for model $C$) and show that at this order $M\left(t\right)\mathrm{}$ is a non-Markov process. Consequently, to our knowledge, $\theta$ is a new exponent. The calculation is performed by expanding around a Markov process, using a simplified version of the perturbation theory recently introduced by Majumdar and Sire [Phys. Rev. Lett. 77, 1420 (1996)].