Let \({\mathcal {G}}\) be the Cayley graph of a finitely generated, infinite group \(\Gamma \) . We show that \(\Gamma \) has the Haagerup property if and only if for every \(\alpha <1\) , there is a \(\Gamma \) -invariant bond percolation \(\mathbb {P}\) on \({\mathcal {G}}\) with \(\mathbb {E}[\deg _{\omega }(g)]>\alpha \deg _{{\mathcal {G}}}(g)\) for every vertex g and with the two-point function \(\tau (g,h)=\mathbb {P}\big [g\leftrightarrow h\big ]\) vanishing as \(d(g,h)\rightarrow \infty \) . As an upshot, we also obtain a sufficient condition for the Haagerup property using Bernoulli percolation and the associated threshold \(p_{\textrm{conn}}\) of random connected subgraphs of \(\mathcal G\) . On the other hand, we show that \(\Gamma \) has Kazhdan’s property (T) if and only if there exists a threshold \(\alpha ^*<1\) such that for every \(\Gamma \) -invariant bond percolation \(\mathbb {P}\) on \({\mathcal {G}}\) , \(\mathbb {E}[\deg _\omega (o)]>\alpha ^*\deg (o)\) implies that the two-point function is bounded away from zero. These results in particular answer questions about characterizations of properties of groups beyond amenability through group-invariant percolations, raised by Russell Lyons (J. Math. Phys. 41 1099-1126 (2000)). The method of proof is new and is based on a construction of percolations with suitable dependence structures built from invariant point processes on spaces with measured walls. In fact, we develop this new approach further to obtain quantitative estimates: we show that there is an explicit relationship between probabilistic quantities like large marginals and two-function decay of percolations and geometric features captured by growth of wall distances defined by invariant actions on spaces with measured walls. We apply this relationship to obtain quantitative bounds on the two-point functions, exhibiting in particular exponential decay of the two-point function in several prominent examples of Haagerup groups, including co-compact Fuchsian groups, co-compact discrete subgroups of \(\textrm{Isom}(\mathbb {H}^n)\) and lamplighters over free groups. This method also allows us to extend the aforementioned characterization of property (T) to the setting of relative property (T). As further applications of our methods, we give new proofs of the facts that for Bernoulli percolation at the uniqueness threshold \(p_u\) there is no unique infinite cluster for groups with property (T) as well as with relative property (T).
