# The $L^2$ boundedness condition in nonamenable percolation

## TL;DR

This paper explores the implications of the $L^2$ boundedness conjecture for nonamenable transitive graphs in percolation theory, revealing new decay, geometric, and phase transition properties, and computing critical exponents in hyperbolic geometry.

## Contribution

It proves several new properties of percolation on nonamenable graphs assuming the $L^2$ boundedness conjecture, including decay rates, phase behavior, and critical exponents, extending understanding even in hyperbolic cases.

## Key findings

- Exponential decay of the two-point function for $p<p_{2 o 2}$
- Critical exponent for extrinsic diameter is 1 if $p_c<p_{2 o 2}$
- Percolation is ballistic below $p_{2 o 2}$

## Abstract

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. 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 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.