Scaling exponent conjecture for the mixed threshold FA model

Let $\tau_{0}$ be the relaxation time at the origin in the mixed threshold FA model, let $q$ be the vacancy density, let $\nu$ denote the law of the random environment, and let $\pi$ be the density of easy sites. Scaling exponent conjecture. For $\nu$-almost every environment, the limit

$$\lim_{q\rightarrow0}\frac{\log\tau_{0}}{\log\nicefrac{1}{q}}$$

exists. Its value $\alpha$ is a random variable whose law depends on $\pi$. Moreover, the law of $\pi\alpha$ converges, in some sense, to a non-trivial law as $\pi$ tends to $0$. This conjecture describes the random-environment scaling exponent of the relaxation time and predicts a non-trivial limiting distribution after rescaling by the density of easy sites; the source leaves the mode of convergence unspecified and does not establish the claim.