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Published October 16, 2020 | Published + Accepted Version
Journal Article Open

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_(p_c) (u,v) = ℙ_(p_c) (u↔v) is bounded as an operator T_(p_c) : L²(V)→L²(V) and proved that this conjecture holds for several classes of graphs. 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 < p_(2→2); ii) If p_c < p_(2→2), then the critical exponent governing the extrinsic diameter of a critical cluster is 1; iii) Below p_(2→2), percolation is "ballistic" in the sense that the intrinsic distance between two points is exponentially unlikely to be much larger than their extrinsic distance; iv) If p_c < p_(2→2), then ‖T_(p_c) ‖_(q→q) ≍ (q−1)−1 and p_(q→q) − p_c ≍ q − 1 as q↓1. v) If p_c < p_(2→2), then various 'multiple-arm' events have probabilities comparable to the upper bound given by the BK inequality. In particular, the probability that the origin is a trifurcation point is of order (p − p_c)³ as p ↓ p_c. All of these results are new even in the Gromov hyperbolic case. Finally, we apply these results together with duality arguments to compute the critical exponents governing the geometry of intrinsic geodesics at the uniqueness threshold of percolation in the hyperbolic plane.

Additional Information

© 2020 The author(s). Creative Commons Attribution 4.0 International License. Received: 12 April 2019; Accepted: 22 September 2020; Published: 2020. First available in Project Euclid: 16 October 2020. We thank Gady Kozma for helpful discussions on interpolation theory of operators, and thank Asaf Nachmias for comments on a draft. We also thank the anonymous referees for their careful reading and helpful comments.

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Published - 20-EJP525.pdf

Accepted Version - 1904.05804.pdf


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