Sharp metastability transition for two-dimensional bootstrap percolation with symmetric isotropic threshold rules (Q6617189)

In two-dimensional \(\mathcal U\)-bootstrap, one fixes a set of update family \(\mathcal U\) (a family of subsets of \(\mathbb Z^2\setminus\{0\}\)) and defines iteratively the following process. Initially, each vertex in \(\mathbb Z^2\) is infected independently with probability \(p\). Then, at each step, every not-yet-infected vertex \(x\) becomes infected if \(x+U\) is completely infected for some \(U\in\mathcal U\). The main quantity of interest is \(\tau\), the \textit{infection time} of the origin.\N\NKey players in the behavior of \(\tau\) are the \textit{stable directions}, that define half-planes which remain non-infected if they are initially non-infected. The \textit{difficulty} of a stable direction is how many infections one has to add in that half-plane in order to infect infinitely many sites.\N\NThis paper considers so-called \textit{isotropic} update families, which have in particular a non-zero finite number of stable directions. It was shown in [\textit{B. Bollobás} et al., Proc. Lond. Math. Soc. (3) 126, No. 2, 620--703 (2023; Zbl 1538.60156)] that\N\[\N\lim_{p\rightarrow 0}\mathbb P_p\left(e^{c/p^\alpha}<\tau<e^{C/p^\alpha}\right)=1,\N\]\Nwhere \(\alpha\) is the maximal difficulty of the stable directions.\N\NFor a (large) subclass of isotropic update families, the authors here show a sharp scaling result for \(\tau\): for any \(\epsilon>0\)\N\[\N\lim_{p\rightarrow 0}\mathbb P_p\left(|p^\alpha\log\tau-\lambda|>\epsilon\right)=0,\N\]\Nwhere \(\lambda>0\) is defined through a variational problem and quantifies the probability of the optimal way for a small droplet of infections to spread throughout \(\mathbb Z^2\). The subclass considered includes threshold rules with two-dimensional convex symmetric neighbourhood.