Diffusive persistence and the “sign-time” distribution

We present a method for extracting the persistence exponent θ for the diffusion equation, based on the distribution $P$ of “sign times.” With the aid of a numerically verified ansatz for $P,$ we derive an analytic formula for θ (in arbitrary spatial dimension $d)$ which we conjecture to be the exact result. Our results are in excellent agreement with previous numerical studies. Furthermore, our results indicate a qualitative change in $P$ above $d\simeq 36,$ signaling the existence of a sharp change in the ergodic properties of the diffusion field.