Universality conjecture for critical update families with infinitely many stable directions

Let ${\mathcal U}$ be a critical update family with an infinite number of stable directions. Let $q$ denote the vacancy probability, let $\tau_0$ be the infection time at the origin, and let $T_{\mathrm{rel}}$ be the corresponding relaxation time. An update family is balanced or unbalanced according to the distinction introduced in the paper. Universality conjecture. If ${\mathcal U}$ is balanced, then

$$\mathbb{E}(\tau_0)=\exp\left(\frac{\Theta(1)}{q^{2\alpha}}\right).$$

If ${\mathcal U}$ is unbalanced, then

$$\mathbb{E}(\tau_0)=\exp\left(\frac{\Theta\left(\left(\log q\right)^4\right)}{q^{2\alpha}}\right).$$

The same asymptotics hold for $T_{\mathrm{rel}}$. This conjecture seeks to determine the polylogarithmic correction to the infection and relaxation-time scaling beyond the bounds currently known. The balanced and unbalanced cases are motivated by the different correction factors found in bootstrap percolation, while the Duarte model is the only critical model with infinitely many stable directions for which the correction is known.