Open Access
2020 The $L^{2}$ boundedness condition in nonamenable percolation
Tom Hutchcroft
Electron. J. Probab. 25: 1-27 (2020). DOI: 10.1214/20-EJP525


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 $T_{p_{c}}(u,v)=\mathbb {P}_{p_{c}}(u\leftrightarrow v)$ is bounded as an operator $T_{p_{c}}:L^{2}(V)\to L^{2}(V)$ and proved that this conjecture holds for several classes of graphs, including all transitive, nonamenable, Gromov hyperbolic graphs. In notation, the conjecture states that $p_{c}<p_{2\to 2}$, where for each $q\in [1,\infty ]$ we define $p_{q\to q}$ to be the supremal value of $p$ for which the operator norm $\|T_{p}\|_{q\to q}$ 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 $L^{2}$ 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 $p<p_{2\to 2}$; ii) If $p_{c}<p_{2\to 2}$, then the critical exponent governing the extrinsic diameter of a critical cluster is $1$; iii) Below $p_{2\to 2}$, 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 $p_{c}<p_{2\to 2}$, then $\|T_{p_{c}}\|_{q\to q} \asymp (q-1)^{-1}$ and $p_{q\to q}-p_{c} \asymp q-1$ as $q\downarrow 1$; v) If $p_{c}<p_{2\to 2}$, 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 $p \downarrow p_{c}$. 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.


Download Citation

Tom Hutchcroft. "The $L^{2}$ boundedness condition in nonamenable percolation." Electron. J. Probab. 25 1 - 27, 2020.


Received: 12 April 2019; Accepted: 22 September 2020; Published: 2020
First available in Project Euclid: 16 October 2020

MathSciNet: MR4162843
Digital Object Identifier: 10.1214/20-EJP525

Primary: 60B99 , 60K35

Keywords: Critical exponents , nonamenable , percolation

Vol.25 • 2020
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