Markovian renormalisation for percolation in high-dimension: Semi-decidability of mean field behavior

Abstract: We develop a new approach to study Bernoulli percolation in dimensions $d>6$. The key idea is to approximate open paths at probability $p'>p$ by a Markov chain of pointed $p$-open clusters. This allows us to transfer sharp information on the two point function from $p$ to $p'$. This inductively gives good asymptotic estimates on the two point function as $p\uparrow p_c$. As a main application, we show that having critical meanfield behavior is a semi-decidable problem. Along the way, we also prove sharp asymptotics for the susceptibility and the average radius of gyration, and prove a local central limit theorem for the slightly subcritical two-point function.