On large isolated regions in supercritical percolation (Q1863068)

Supercritical vertex percolation in \(Z^d\) with any non-degenerate uniform oriented pattern of connection is considered. A finite set of vertices of \(Z^d\) is called cut from \(\infty\) if all its elements cut from \(\infty\), i.e. the set of vertices reachable from this set is finite. For any finite \(S\subset Z^d\), by \(P_{\text{cut}}(S)\) it is denoted the probability that \(S\) is cut from \(\infty\). In the first part of the paper for all \(D\)-connected sets \(S\) and all \(\varepsilon\in(0,\varepsilon^*_d)\) the probability \(P_{\text{cut}}(S)\) is established as follows: \[ \varepsilon^{k_0(S)}\leq P_{\text{cut}}(S)\leq (C_0\cdot\varepsilon)^{k_0(S)}, \] here \(\varepsilon^*_d\) and \(C_0\) are some constants. In the second part it is proven that ``pancakes'' with a radius \(r\to\infty\) and constant thickness, parallel to a constant linear subspace \(L\), are isolated with probability whose logarithm grows asymptotically as \(r^{\text{dim}(L)}\) if percolation is possible across \(L\) and \(r^{\text{dim}(L)-1}\) otherwise. In the final third part, probabilities of large deviations in invariant measures of some cellular automata are established.