The L² boundedness condition in nonamenable percolation

Let $G=(V,E)$ be a connected, locally finite, transitive graph, and consider Bernoulli bond percolation on G. In recent work, we conjectured that if G is nonamenable then the matrix of critical connection probabilities T_pc(u,v)=P_pc(u↔ v) is bounded as an operator T_pc:L²(V)→ L²(V) and proved that this conjecture holds for several classes of graphs, including all transitive, nonamenable, Gromov hyperbolic graphs. In notation, the conjecture states that pc<p2→ 2, where for each q∈ [1,∞ ] we define pq→ q to be the supremal value of p for which the operator norm \|Tₚ\|q→ q is finite. We also noted in that work that the conjecture implies two older conjectures, namely that percolation on transitive nonamenable graphs always has a nontrivial nonuniqueness phase, and that critical percolation on the same class of graphs has mean-field critical behaviour. In this paper we further investigate the consequences of the L² boundedness conjecture. In particular, we prove that the following hold for all transitive graphs: i) The two-point function decays exponentially in the distance for all p<p2→ 2; ii) If pc<p2→ 2, then the critical exponent governing the extrinsic diameter of a critical cluster is $1$; iii) Below p2→ 2, percolation is “ballistic" in the sense that the intrinsic (a.k.a. chemical) distance between two points is exponentially unlikely to be much larger than their extrinsic distance; iv) If $pc