This paper presents a new way to study Bernoulli bond percolation in dimensions above six. The authors build an approximation that treats long open paths at a higher bond probability p' as a Markov chain made of pointed clusters at a lower probability p. That approximation lets them transfer precise information about the two-point function — the probability that two sites are connected — from p to p'. Using this inductively as p approaches the critical value p_c, they obtain sharp asymptotic estimates and a striking application: the question of whether the model has the so-called mean-field critical behaviour is semi-decidable. In plain terms, when the property holds it can be checked in finite time by a computable procedure, though the procedure need not certify false cases in finite time.

A few words about the model and the goal. In Bernoulli bond percolation each edge of a lattice is independently open with probability p and closed otherwise. Clusters are connected groups of open edges. There is a phase transition at a critical probability p_c: below p_c all clusters are finite, above p_c an infinite cluster appears. Mean-field behaviour means that certain key quantities follow the same power laws as on a tree; for example, the two-point function at criticality should decay like a constant times |x|^{2-d}. Historically, the lace expansion has been the main tool to prove mean-field behaviour in very high dimensions. The new method aims to get the same conclusions with a different, inductive approach.

How the new idea works at a high level. The authors take a path that is open at probability p' > p and cut it into segments that are open at probability p, separated by single edges that are p'-open but p-closed. Instead of ignoring the correlations between those p-open clusters, they keep the correlation between adjacent clusters by using Aizenman’s “off” method. This lets them interpret the sequence of pointed clusters (a cluster together with a marked vertex at its boundary) as the states of a Markov chain. They build an explicit transition kernel, analyze its spectral properties, and apply a Doob transform to control the chain. Under a modest initialization assumption — an explicit bound on the two-point function at some fixed probability p_0 with sufficiently large susceptibility — they show this Markov chain mixes exponentially fast and use it to propagate sharp bounds up to p_c.