# Haagerup property and Group-Invariant Percolation

By Chiranjib Mukherjee111Universität Münster, Einsteinstrasse 62, Münster
48149, Germany<EMAIL_ADDRESS>and Konstantin
Recke222Universität Münster, Einsteinstrasse 62, Münster 48149, Germany
<EMAIL_ADDRESS>
††AMS Subject Classification: 22D55, 60K35, 82B43, 20F16, 60D05, 60B99,
20F67.††Keywords: Haagerup property, percolation, group-invariance, Kazhdhan’s
Property (T), spaces with measured walls, two-point functions, amenability,
lamplighter groups, co-compact Fuchsian groups.

Universität Münster

March 30, 2023

> Abstract:  Let ${\mathcal{G}}$ be the Cayley graph of a finitely generated
> 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
> $\tau(g,h)=\mathbb{P}[g\leftrightarrow h]$ vanishing as $d(g,h)\to\infty$.
> Our result is inspired by the characterization of amenability via group-
> invariant percolation appearing in the work of Benjamini, Lyons, Peres and
> Schramm [7]. To derive our result, we use the characterization of the
> Haagerup property in terms of actions on spaces with measured walls in the
> sense of Cherix, Martin and Valette [11]. Our proof is based on a new
> construction using invariant point processes on such spaces with measured
> walls, which furthermore leads to quantitative bounds on the two-point
> functions in terms of the aforementioned group action. These bounds yield in
> particular exponential decay of the two-point function in several examples,
> including co-compact Fuchsian groups and lamplighters over free groups.
>
> Moreover, our method allows us to strengthen a consequence of Kazhdan’s
> property (T), due to Lyons and Schramm [44], to an equivalence. Namely, we
> show that $\Gamma$ has 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. We extend this result to the setting of
> relative property (T). We then use the corresponding threshold to give a new
> proof of the fact, already observed by Gaboriau and Tucker-Drob [21], that
> there is no unique infinite cluster at the uniqueness threshold for
> Bernoulli bond percolation on Cayley graphs of groups admitting an infinite
> normal subgroup with relative property (T).

##  1\. Introduction and main results

Let $\Gamma$ be a finitely generated group and let ${\mathcal{G}}=(V,E)$ be
the right Cayley graph with respect to a finite and symmetric set of
generators and equipped with the word metric $d(\cdot,\cdot)$. In this
article, all finitely generated groups are assumed to be countably infinite
and all appearing Cayley graphs are right Cayley graphs with respect to finite
and symmetric generating sets. Note that left multiplication by elements of
$\Gamma$ induces a left action on ${\mathcal{G}}$ via graph automorphisms.

A probability measure on subsets of $E$ is called a bond percolation on
${\mathcal{G}}$. This probability measure is called $\Gamma$-invariant if it
is invariant under the action of $\Gamma$. Denote the percolation
configurations by $\omega$ and the vertex corresponding to the identity in
$\Gamma$ by $o$. A cluster of $\omega$ is a connected component of the induced
subgraph, i.e. the subgraph consisting of all vertices and all edges which are
in $\omega$. We will write $\deg_{\omega}(o)$ for the degree of $o$ as a
vertex in this induced subgraph and $\deg(o)$ for the usual degree of $o$ in
the graph ${\mathcal{G}}$.

The most important example of a group-invariant percolation is given by
$p$-Bernoulli bond percolation in which edges are kept independently at random
with fixed probability $p\in[0,1]$. It is a classical result of Newman and
Schulman [49] that the number of infinite clusters in this model is constant
a.s. with this constant being equal to either $0,1$ or $\infty$. The critical
parameter $p_{c}=p_{c}({\mathcal{G}})$ is defined as the infimum over all
$p\in[0,1]$ such that in $p$-Bernoulli bond percolation, there exists an
infinite cluster almost surely. Similarly, the uniqueness threshold
$p_{u}=p_{u}({\mathcal{G}})$ is the infimum over all $p\in[0,1]$ such that in
$p$-Bernoulli bond percolation, there is a unique infinite cluster almost
surely.

The following fundamental result due to Benjamini, Lyons, Peres and Schramm
[7, Theorem 1.1] is a characterization of amenability of the underlying group
using only group-invariant percolations:

######  Theorem A.

Let ${\mathcal{G}}$ be the Cayley graph of a finitely generated group
$\Gamma$. Then $\Gamma$ is amenable if and only if for every $\alpha<1$, there
exists a $\Gamma$-invariant bond percolation $\mathbb{P}$ on ${\mathcal{G}}$
with no infinite clusters and such that

$\mathbb{E}[\deg_{\omega}(o)]>\alpha\deg(o).$ (1.1)

We note that the above result also follows from the Rokhlin lemma of Ornstein
and Weiss [50] and results of Adams and Lyons in [1], while [7, Theorem 5.1]
actually provides a more general statement.

###  1.1. Haagerup property and two-point functions of invariant percolations.

For a bond percolation $\mathbb{P}$, define the two-point function

$\tau\colon\Gamma\times\Gamma\to[0,1]\,,\,\tau(g,h):=\mathbb{P}\big{[}g\leftrightarrow
h\big{]}\coloneqq\mathbb{P}\big{[}\omega\colon
g\,\stackrel{{\scriptstyle\omega}}{{\longleftrightarrow}}\,h\big{]}$ (1.2)

to be the probability that $g$ and $h$ are in the same cluster of the
percolation configuration. Clearly, if a $\Gamma$-invariant bond percolation
has no infinite clusters, then the two-point function $\tau(o,\cdot)$ vanishes
at infinity, written $\tau(o,\cdot)\in C_{0}(\Gamma)$ and meaning that

$\lim_{r\to\infty}\sup_{\genfrac{}{}{0.0pt}{}{g,h\in\Gamma}{d(g,h)>r}}\,\tau(g,h)=0.$
(1.3)

However, this property does not imply having no infinite clusters. For
example, consider super-critical Bernoulli bond percolation on a regular tree
of degree at least three: its two-point function vanishes at infinity but it
has infinitely many infinite clusters (see Example 2.2.2). Thus (1.3) is a
weaker property than having no infinite clusters. In light of Theorem A our
first goal is to answer the following natural question:

######  Question 1.1.

Can we characterize all finitely generated groups whose Cayley graphs admit
invariant bond percolations with marginals arbitrarily close to $1$ (in the
sense of (1.1)) but two-point function vanishing at infinity?

The answer is provided by our first main result:

######  Theorem 1.2.

Let ${\mathcal{G}}$ be the Cayley graph of a finitely generated group
$\Gamma$. Then $\Gamma$ has the Haagerup property if and only if for every
$\alpha<1$, there exists a $\Gamma$-invariant bond percolation $\mathbb{P}$ on
${\mathcal{G}}$ such that $\mathbb{E}[\deg_{\omega}(o)]>\alpha\deg(o)$ and
$\tau(o,\cdot)\in C_{0}(\Gamma)$.

Recall that a locally compact, second countable group $G$ has the Haagerup
property, or is a-T-menable in the terminology of Gromov [24], if it admits a
continuous, affine, isometric action on a Hilbert space, which is metrically
proper, see [12]. The origins of this property can be traced back to the
seminal work of Haagerup [27] on group $C^{*}$-algebras of free groups. Groups
with the Haagerup property form a large class with classical examples provided
by compact groups, amenable groups, free groups, Coxeter groups, groups acting
properly on trees and isometry groups of real hyperbolic space. For more
examples and a survey of various consequences and characterizations of the
Haagerup property we refer the reader to [12]. Here we would like to emphasize
two salient features of this property: The first one is the outstanding result
of Higson and Kasparov [29] that groups with the Haagerup property satisfy the
Baum-Connes conjecture with coefficients. The second one is the fact that a
countable group has the Haagerup property if and only if each of its finitely
generated subgroups does, see [12, Proposition 6.1.1]. Therefore applications
of Theorem 1.2 are not restricted to the finitely generated case.

Two characterizations of the Haagerup property via probabilistic methods were
previously known. Jolissaint [12, Theorem 2.1.3] showed that a locally
compact, second countable group has the Haagerup property if and only if it
has a measure preserving action on a standard probability space which is
strongly mixing and contains a Følner sequence. This result is an analogue of
a characterization of Kazhdan’s property (T) due to Schmidt [54] and Connes
and Weiss [13]. Answering a question posed in [12], Glasner [22, Theorem
13.21] showed that a countable group has the Haagerup property if and only if
the closure of the set of $\Gamma$-invariant, mixing probability measures on
$\\{0,1\\}^{\Gamma}$ is convex. This characterization is again an analogue of
the well-known probabilistic characterization of Kazhdan’s property (T) due to
Glasner and Weiss [23], see the discussion below Theorem 1.5.

In comparison, perhaps a distinguishing feature of Theorem 1.2 is provided by
the following concrete insight about the geometry of the group: first note
that for any percolation on a connected graph the two properties appearing in
Question 1.1, namely having marginals arbitrarily close to $1$ and having two-
point function vanishing at infinity, are incongruous. Finitely generated
groups with the Haagerup property are then precisely the ones all of whose
Cayley graphs allow these two properties to be reconciled in an invariant way.
A related aspect worth mentioning is that the Haagerup property is a
generalization of amenability [5, 12], which, combined with Theorem A, is also
illustrated in this new geometric way by Theorem 1.2.

The Haagerup property naturally appears as a strong negation to property (T).
This is due to the fact that countable groups with both properties are
precisely the finite ones. In fact we use the insight from Theorem 1.2 to
provide a new characterization of property (T) in Theorem 1.5 below.

Let us now comment on the ideas behind the proof of Theorem 1.2 provided in
Section 3.3. The “if” direction is a direct consequence of a classical
observation by Aizenman and Newman [2] that two-point functions of invariant
percolations define positive definite functions. We prove the other direction
by providing an explicit construction using the characterization of the
Haagerup property via actions on spaces with measured walls introduced by
Cherix, Martin and Valette in [11]. In short, a space with measured walls is a
quadruplet $(X,{\mathcal{W}},{\mathcal{B}},\mu)$ where $X$ is a set,
${\mathcal{W}}$ is a set of partitions of $X$ into two classes (a partition is
called a wall), ${\mathcal{B}}$ is a $\sigma$-algebra on ${\mathcal{W}}$ and
$\mu$ is a measure on ${\mathcal{B}}$ such that for every two distinct points
$x$ and $y$ in $X$, the set ${\mathcal{W}}(x,y)$ of walls separating $x$ and
$y$ is ${\mathcal{B}}$-measurable and $\mu({\mathcal{W}}(x,y))<\infty$. We
will be interested mainly in invariant actions on spaces with measured walls,
which are actions on the set $X$ which extend to a measure-preserving action
on $({\mathcal{W}},{\mathcal{B}},\mu)$, see Section 3.3 for details. Our main
idea is to employ invariant (Poisson) point processes on such spaces with
measured walls, which can then be used to define invariant bond percolations
with suitable dependence structures, see Theorem 3.5. We also provide a
generalization of this construction in Theorem 3.6 that applies to many known
actions of groups on spaces with measured walls with useful geometric features
(see e.g. Section 8). This generalized result can thus be used to produce many
examples of positive definite functions witnessing the Haagerup property and
properties of specific actions can give us additional information about the
constructed percolations as will be illustrated by Theorem 1.8.

Besides considering the general case, we also demonstrate our result for two
sub-classes of groups with the Haagerup property: groups acting properly on
trees (cf. Proposition 3.1) and groups acting properly on spaces with walls in
the sense of Haglund and Paulin [28] (cf. Proposition 3.3). In these cases the
construction is more concrete and therefore they can serve as useful intuition
for the general result. Moreover, they provide interesting examples which are
illustrated along the way (cf. Example 3.1 and Example 3.2). Notably, in
Example 3.1 we give a very short probabilistic proof of Haagerup’s seminal
result that the word length with respect to the canonical generating set on a
free group is a conditionally negative definite function [27].

Moreover, we provide the following sufficient condition for the Haagerup
property in terms of Bernoulli percolation on invariant random spanning
subgraphs of ${\mathcal{G}}$ (see Theorem 3.7), which is inspired by [7,
Theorem 5.3]. Its statement requires the following definition: For a
configuration $\omega$, the parameter $p_{\scriptscriptstyle{\rm
conn}}(\omega)$ is defined to be the supremum over all $p\in[0,1]$ such that
the two-point function of $p$-Bernoulli bond percolation on $\omega$ vanishes
at infinity.

######  Theorem 1.3.

Let ${\mathcal{G}}$ be the Cayley graph of a finitely generated group
$\Gamma$. Then the following conditions are equivalent:

  1. (i)

there exists a $\Gamma$-invariant random spanning subgraph $\omega$ of
${\mathcal{G}}$ with $p_{\scriptscriptstyle{\rm conn}}(\omega)=1$ a.s.;

  2. (ii)

there exists a $\Gamma$-invariant random nonempty connected subgraph $\omega$
of ${\mathcal{G}}$ with $p_{\scriptscriptstyle{\rm conn}}(\omega)=1$ with
positive probability.

Moreover, if $\Gamma$ satisfies (i), or equivalently (ii), then $\Gamma$ has
the Haagerup property.

We remark that spanning subgraphs of ${\mathcal{G}}$ satisfying
$p_{\scriptscriptstyle{\rm conn}}=1$ may be understood as a generalization of
spanning trees since every tree satisfies $p_{\scriptscriptstyle{\rm conn}}=1$
trivially.

In Theorem 3.8, we also provide an analogue of Theorem 1.2 for invariant site
percolations – that is, invariant probability measures on subsets of vertices
and the corresponding random induced subgraph (the graph consisting of all
vertices in the configuration and all edges with both endpoints in the
configuration). There is a reason for treating bond percolations first, namely
that our construction alluded to above takes advantage of the additional
degrees of freedom provided by removing edges instead of vertices. This
construction then turns out to be applicable to several related questions,
which we now focus on.

###  1.2. Spaces with measured walls and exponential decay of two-point
functions.

As a first application, we obtain a quantitative version of Theorem 1.2. To
state it, we introduce the following terminology: Let $\Gamma$ be equipped
with a word length $|\cdot|$ and let $\rho\colon[0,\infty]\to[0,\infty]$.
Associated to an action of $\Gamma$ on a space with measured walls as defined
above, there is a pseudo-distance $D(\cdot,\cdot)$ on $\Gamma$ called the wall
distance (see Subsection 3.3). We will prove the following result (stated
formally in Theorem 4.1) in Section 4.

######  Theorem 1.4.

Let ${\mathcal{G}}$ be the Cayley graph of a finitely generated group $\Gamma$
and let $\rho\colon[0,\infty]\to[0,\infty]$ be a function with
$\\{\rho=0\\}=\\{0\\}$. Then each of the following conditions implies the next
one:

  1. (i)

There exists an invariant action of $\Gamma$ on a space with measured walls
with $\rho$-growth, meaning that $\rho$ is a compression function for the wall
distance $D$, that is to say

$D(o,g)\geq\rho(|g|).$

  2. (ii)

There exists $C>0$ such that for every $\alpha<1$, there exists a
$\Gamma$-invariant bond percolation $\mathbb{P}$ on ${\mathcal{G}}$ with
$\mathbb{E}[\deg_{\omega}(o)]>\alpha\deg(o)$ and with its two-points function
having exponential decay w.r.t. the pair $(C,\rho)$, meaning there exist
$\beta,\gamma>0$ with $\beta/\gamma\leq C$ and

${\rm e}^{-\beta|g|}\leq\tau(o,g)\leq{\rm e}^{-\gamma\rho(|g|)}\quad\mbox{for
all}\,\,g\in\Gamma.$

  3. (iii)

There exists a measure definite function on $\Gamma$ which admits $\rho$ as a
compression function.

Moreover, if $\Gamma$ is amenable or if the measure definite function in (iii)
can be assumed to be the square-root of a conditionally negative definite
function, then all three conditions are equivalent.

We refer to Section 2.1 for definitions of measure definite and conditionally
negative definite functions. We now comment on Theorem 1.4. The first
implication in Theorem 1.4 is very general and works for any invariant action
on a space with measured walls with a quantitative control of the wall
distance. The case where $\rho$ is the word length is particularly interesting
as it implies the existence of percolations with marginals arbitrarily close
to $1$ but exponential decay of the associated two-point functions. Condition
(i) in Theorem 1.4 is closely related to the equivariant $L^{1}$-compression
of $\Gamma$, see Remark 2 for details. The implication (iii)$\Rightarrow$(i)
was known under the additional amenability assumption from [16, Proposition
2.8] and under the other assumption from [11, 52]. We do not know whether it
holds in general and refer to Remark 1 for a short discussion. There are
prominent examples of non-amenable groups which satisfy all three conditions
listed in Theorem 1.4: To name two, these include
$\mathbb{Z}_{2}\wr\mathbb{F}_{2}$ (see Theorem 1.7 below) and co-compact
Fuchsian groups (see Theorem 1.8 below).

###  1.3. Kazhdan groups and two-point functions of invariant percolations.

A second application of our construction allows us to strengthen a known
consequence of Kazhdan’s property (T) about invariant percolations, which is
due to Lyons and Schramm [44], to an equivalence. This is our third main
result (see Theorem 5.3).

######  Theorem 1.5.

Let ${\mathcal{G}}$ be the Cayley graph of a finitely generated group
$\Gamma$. Then $\Gamma$ has 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 $\tau(o,\cdot)$
is bounded away from zero, i.e.

$\inf_{g,h\in\Gamma}\tau(g,h)>0.$

To the best of our knowledge, this is the first characterization of property
(T) involving only group invariant percolations, which answers a question
raised by Lyons in [41, p. 1123].

We remark that the threshold in Theorem 1.5 is not elusive. Estimates in terms
of Kazhdan pairs (see Section 5.2 for the definition) are available and we
refer to the proof of Theorem 5.3 and to Remark 7 for precise statements.
Moreover, we recall that every countable Kazhdan group is finitely generated
[6, Theorem 1.3.1] and hence Theorem 1.5 characterizes countable Kazhdan
groups.

Let us now recall the aforementioned probabilistic characterization of
property (T) due to Glasner and Weiss [23]: A countable group $\Gamma$ has
property (T) if and only if in the set of $\Gamma$-invariant probability
measures on $\\{0,1\\}^{\Gamma}$, the set of ergodic measures is closed and,
equivalently, not dense. In comparison, Theorem 1.5 again provides concrete
insight about the geometry of the group: namely, Cayley graphs of Kazhdan
groups are precisely the Cayley graphs with such high degree of connectivity
that as soon as the marginals of an invariant percolation are large enough,
any two points in the graph are connected with probability bounded away from
zero. Let us mention another interpretation: It is a result of Lyons and
Schramm [44, Theorem 4.1] that an invariant, ergodic, insertion-tolerant bond
percolation with two-point function bounded away from zero has a unique
infinite cluster. With this uniqueness-criterion in mind, Theorem 1.5 suggests
that Cayley graphs of Kazhdan groups are Cayley graphs with a high tendency
towards forming a unique infinite cluster. This intuition is complemented by a
result obtained by Hutchcroft and Pete [31] in a recent breakthrough article
on the cost of Kazhdan groups, which states that Cayley graphs of Kazhdan
groups admit invariant percolations with arbitrarily small expected degree but
with a unique infinite cluster.

In this context, we also discuss two weakenings of property (T). We consider
weakly Kazhdan groups, in the terminology of Glasner [22], which are groups
that do not have the Haagerup property. Clearly Theorem 1.2 fails for weakly
Kazhdan groups and in Proposition 5.1 we provide an estimate of the
corresponding threshold, which is similar to the aforementioned threshold for
Kazhdan groups. Moreover, we consider groups with relative property (T), for
the definition of which we refer to Section 5.3. In Theorem 5.7 we extend
Theorem 1.5 to this setting. We then provide an application of this threshold
to a well-known problem in Bernoulli percolation – namely, in Theorem 5.8 we
prove that the uniqueness threshold does not belong to the uniqueness phase in
Bernoulli bond percolation on any Cayley graph of an infinite group which has
property (T) relative to an infinite, normal subgroup. We note that a more
general result has appeared in the work of Gaboriau and Tucker-Drob [21]. The
argument we give here is quite different; it is based on the aforementioned
threshold and an approach to ruling out a unique infinite cluster at specific
values of the survival parameter due to Benjamini, Lyons, Peres and Schramm
[8]. We combine these two ingredients with ideas described by Hutchcroft and
Pete [31] to obtain the result (see Section 5.3). Due to the probabilistic
nature of our proof, we believe that it is of independent interest.

###  1.4. Amenable groups and spaces with measured walls.

We recall that amenable groups have the Haagerup property [5]. In fact Theorem
1.2 generalizes the amenability criterion Theorem A in Section 1 in the sense
that the constructive part of the proof of Theorem A from [7] can be recovered
using our construction in case the group is amenable. More precisely, we
obtain the following result (see Proposition 6.1).

######  Proposition 1.6.

Let ${\mathcal{G}}$ be the Cayley graph of a finitely generated amenable group
$\Gamma$. There exists an invariant action of $\Gamma$ on a space with
measured walls such that the construction used in the proof of Theorem 1.2
yields, for every $\alpha<1$, a $\Gamma$-invariant bond percolation
$\mathbb{P}$ on ${\mathcal{G}}$ with
$\mathbb{E}[\deg_{\omega}(o)]>\alpha\deg(o)$ and with no infinite clusters.

We note that information about the two-point function alone does not suffice
to characterize amenability. More precisely, our introductory example of
supercritical Bernoulli bond percolation on a regular tree of degree three
shows that even exponential decay of the two-point function does not imply
finiteness of percolation clusters. However, also in the amenable setting a
close connection with positive definite functions, which are not two-point
functions, exists. Essentially the idea is that for any percolation with only
finite clusters, there is a canonical way to associate a random positive
definite function to the cluster of the identity. This approach and a
characterization of amenable groups via such positive definite functions were
first described in [46, Theorem 1.6], which served as the motivation of the
current work. In the setting of [46], these positive definite functions
provided a large class of probabilistic examples of functions arising
naturally in the study of Schur multipliers of $C^{*}$-algebras associated to
$\Gamma$, which implement certain operator algebraic approximation properties.
Their specific form was then used to provide asymptotic decompositions and
establish limiting behavior for sequences of such multipliers by means of the
group-invariant compactification constructed in [47]; we refer the interested
reader to [46] for details.

###  1.5. Examples: Lamplighters over free groups and co-compact Fuchsian
groups.

The results described above in Section 1.1 \- Section 1.4 underline the wide
applicability of our construction. Moreover, it turns out to be very natural
in settings of particular interest. We stress this latter point in the final
part of the article by providing an in-depth discussion of two classes of
examples which have historically been of great relevance for both probability
theory and geometric group theory.

####  1.5.1. Lamplighters over free groups.

The first class of examples we treat are so-called lamplighter groups over
free groups. These are restricted wreath products of the form
$H\wr\mathbb{F}_{r}$, where the base group $\mathbb{F}_{r}$ is a free group of
rank $r\in\mathbb{N}$ and the lamp group $H$ is finite, see Section 7 for
details. It is a non-trivial fact due to de Cornulier, Stalder and Valette
[15] that these groups have the Haagerup property. More generally, stability
of the Haagerup property under wreath products was established by the same
authors in [16]. Lamplighter groups, especially ones of the form
$\mathbb{Z}_{2}\wr\mathbb{Z}^{d}$, have received substantial attention in the
theory of random walks since the work of Kaimanovich and Vershik [35]. Let us
mention the breakthrough results about identification of the Poisson boundary
proved by Erschler [19, 20] for $d\geq 5$ and by Lyons and Peres [43] for
$d=3,4$. The Poisson boundary of random walks on lamplighters over free groups
was investigated by Karlsson and Woess [36]. For Bernoulli percolation, Lyons
and Schramm [44] established that any Cayley graph of a lamplighter group over
some finitely generated base group with non-trivial, finite lamp group has
non-trivial uniqueness phase, i.e. $p_{u}<1$. Complementing this result, we
prove the following, perhaps surprising, result in the setting of invariant
percolations (see Theorem 7.1).

######  Theorem 1.7.

Let $H$ be a finite group, let $r\in\mathbb{N}$ and let ${\mathcal{G}}$ be any
Cayley graph of $H\wr\mathbb{F}_{r}$. Then there exists a constant $C>0$ such
that for every $\alpha<1$, there exists a $\Gamma$-invariant bond percolation
$\mathbb{P}$ on ${\mathcal{G}}$ with
$\mathbb{E}[\deg_{\omega}(o)]>\alpha\deg(o)$ and with two-point functions
having exponential decay, i.e.

${\rm e}^{-\beta|g|}\leq\tau(o,g)\leq{\rm e}^{-\gamma|g|}\quad\mbox{for
all}\,\,g\in\Gamma,$ (1.4)

where $\beta,\gamma>0$ such that $\beta/\gamma\leq C$.

####  1.5.2. Co-compact Fuchsian groups

The second class of examples we treat are planar Cayley graphs. We are
particularly interested in one-ended, nonamenable examples, i.e. Cayley graphs
which embed in a nice way into the hyperbolic plane $\mathbb{H}^{2}$, see [9].
Notably, these include the canonical Cayley graphs of co-compact Fuchsian
groups, an example considered in percolation theory first by Lalley [38]. For
important results about percolation in the hyperbolic plane setting, Bernoulli
as well as general invariant, we refer to [9, 38, 39, 42]. In this context,
let us mention that several fundamental conjectures about Bernoulli
percolation were recently established for all Cayley graphs of Gromov
hyperbolic groups by Hutchcroft [30]. In the planar hyperbolic setting (and
also for groups acting properly on $n$-dimensional real hyperbolic space), our
construction has a very natural interpretation in terms of removing edges
along boundaries of half-spaces, see Section 8 for details. As a consequence,
we derive a complete description of the number of infinite clusters in the
percolations guaranteed by Theorem 1.2 for many examples, including canonical
Cayley graphs of co-compact Fuchsian groups (see Theorem 8.2).

######  Theorem 1.8.

Let ${\mathcal{G}}$ be a planar, nonamenable Cayley graph with one end. Then
for every $\alpha<1$, there exists an ${\rm Aut}({\mathcal{G}})$-invariant
bond percolation $\mathbb{P}$ on ${\mathcal{G}}$ with
$\mathbb{E}[\deg_{\omega}(o)]>\alpha\deg(o)$, such that $\tau(o,\cdot)$ has
exponential decay (i.e. (1.4) holds) and such that it has infinitely many
infinite clusters.

Several other examples are described throughout the article (see Example 3.1,
Example 3.2, Example 8.1 and Example 8.2).

###  1.6. Organization of the rest of the article.

The rest of the article is organized as follows: Section 2 provides necessary
background about the Haagerup property, invariant percolations and point
processes. Section 3 contains the proof of Theorem 1.2 and auxiliary results
announced in Section 1.1. Section 4 is devoted to the proofs of the results
mentioned in Section 1.2. Section 5 contains the statement and proofs of the
results alluded to in Section 1.3 – more specifically, this section is split
into three subsections: In Section 5.1, we first present the quantitative
converse to Theorem 1.2. The proof of Theorem 1.5 and estimates of the
appearing threshold are the content of Section 5.2. Finally the extension of
Theorem 1.5 to groups with relative property (T) as well as the aforementioned
application to Bernoulli percolation are shown in Section 5.3. In Section 6,
we discuss spaces with measured walls naturally associated to amenable groups
and provide a proof of Proposition 1.6. Section 7 deals with the example of
lamplighter groups and constitutes the proof of Theorem 1.7. Finally, the
description of our construction for certain hyperbolic groups and the proof of
Theorem 1.8 are carried out in Section 8.

##  2\. Preliminaries

In this section provide some relevant facts pertinent to both geometric group
theory as well as probability.

###  2.1. The Haagerup property, positive definite and conditionally negative
definite functions.

We start with some background on the Haagerup property, which, by definition,
as mentioned in the introduction, is satisfied by a locally compact, second
countable group $G$ if it admits a continuous, affine, isometric action on a
Hilbert space, which is metrically proper, see [12]. In the current section,
we collect the most relevant characterizations of this property for finitely
generated groups $\Gamma$. We first recall the following standard definitions.

  * •

Positive definiteness: A function $k\colon\Gamma\times\Gamma\to\mathbb{C}$ is
called a positive definite kernel if the matrix $[k(g,h)]_{g,h\in F}$ is
positive for any finite subset $F\subset\Gamma$, i.e.

$\sum_{i,j=1}^{n}\overline{a_{i}}a_{j}k(g_{i},g_{j})\geq 0$

for every $g_{1},\ldots,g_{n}\in\Gamma$ and $a_{1},\ldots,a_{n}\in\mathbb{C}$.
It is called invariant if it is invariant under the diagonal action of
$\Gamma$. A function $\varphi\colon\Gamma\to \mathbb{C}$ is called positive
definite if $k_{\varphi}(g,h)\coloneqq\varphi(g^{-1}h)$ is a positive definite
kernel. It is called normalized if $\varphi(e)=1$.

  * •

Conditional negative definiteness: A function
$k\colon\Gamma\times\Gamma\to[0,\infty)$ is called a conditionally negative
definite kernel if $k(g,g)=0$ and $k(g,h)=k(h,g)$ for every $g,h\in\Gamma$ and

$\sum_{i,j=1}^{n}a_{i}a_{j}k(g_{i},g_{j})\leq 0$

for every $g_{1},\ldots,g_{n}\in\Gamma$ and $a_{1},\ldots,a_{n}\in\mathbb{R}$
with $\sum_{i=1}^{n}a_{i}=0$. It is called invariant if it is invariant under
the diagonal action of $\Gamma$. A function $\psi\colon\Gamma\to [0,\infty)$
is called conditionally negative definite if
$k_{\psi}(g,h)\coloneqq\psi(g^{-1}h)$ is a conditionally negative definite
kernel.

  * •

Measure definiteness: The notion of measure definiteness will be needed in
Section 4. Following [52, Definition 0.1.1], we say that a function
$k\colon\Gamma\times\Gamma\to[0,\infty)$ is a measure definite kernel if there
exists a measure space $(\Omega,{\mathcal{B}},\mu)$ and a map
$S\colon\Gamma\to{\mathcal{B}}$, $g\mapsto S_{g}$ such that

$k(g,h)=\mu(S_{g}\Delta S_{h})$

for every $g,h\in\Gamma$. Similarly, a function
$\psi\colon\Gamma\to[0,\infty)$ is called measure definite if the kernel
$k_{\psi}(g,h)=\psi(g^{-1}h)$ is measure definite.

According to a well-known result of Schoenberg, a function
$\psi:\Gamma\to\mathbb{C}$ is conditionally negative definite if and only if
for every $t\geq 0$, the map $g\mapsto\exp(-t\psi(g))$ is positive definite.
The origins of the Haagerup property can be traced back to the seminal work of
Haagerup [27] on group $C^{*}$-algebras of free groups, where, as a key
ingredient, it was shown that the word-length function $|\cdot|$ on the
finitely generated free group $\mathbb{F}_{r}$ (w.r.t. the standard set of
generators) is conditionally negative definite. Therefore, by Schoenberg’s
correspondence, Haagerup’s result can be restated by saying that, for each
$t\geq 0$, the map $\mathbb{F}_{r}\ni g\mapsto\exp(-t|g|)$ is positive
definite. Note that if

$\varphi_{j}(g):=\exp\Big{(}-\frac{1}{j}|g|\Big{)}\qquad j\geq 1,g\in\Gamma,$

then each $\varphi_{j}$ vanishes at infinity and the sequence converges
pointwise to $1$. In this vein, below we summarize the characterizations of
the Haagerup property in terms of positive and conditionally negative definite
functions that are relevant for our purposes:

######  Theorem 2.1 ([12, Theorem 2.1.1]).

Let $\Gamma$ be a finitely generated group. Then the following are equivalent:

  1. (i)

$\Gamma$ has the Haagerup property;

  2. (ii)

there exists a conditionally negative definite function $\psi$ on $\Gamma$
that is proper (the latter condition means that
$\lim_{|g|\to\infty}\psi(g)=\infty$).

  3. (iii)

there exists a sequence of normalized, positive definite functions on $\Gamma$
which vanish at infinity and converge pointwise to $1$.

There are further characterizations of the Haagerup property – for example, it
is also known that $\Gamma$ has the Haagerup property if and only if it admits
a mixing (or $C_{0}$) representation containing almost invariant vectors [12].
Since $\Gamma$ is amenable if and only if its left-regular representation
admits almost invariant vectors, it follows that amenable (e.g. abelian or
compact) groups satisfy the Haagerup property.

###  2.2. Group-invariant percolations.

We first introduce some graph-theoretic terminology. Let
${\mathcal{G}}=(V({\mathcal{G}}),E({\mathcal{G}}))$ be a graph with vertex set
$V({\mathcal{G}})$ and symmetric edge set $E({\mathcal{G}})\subset
V({\mathcal{G}})\times V({\mathcal{G}})$. We denote edges by $[u,v]$. Two
vertices $u$ and $v$ are called adjacent or neighbors, written $u\sim v$, if
$[u,v]$ is an edge. The degree $\deg(v)=\deg_{{\mathcal{G}}}(v)$ of a vertex
$v\in V({\mathcal{G}})$ is the number of vertices which are adjacent to it.
The graph distance between two vertices $u,v\in V({\mathcal{G}})$ is defined
to be the length of a shortest path in ${\mathcal{G}}$ which connects $u$ and
$v$ and is denoted by $d(u,v)=d_{{\mathcal{G}}}(u,v)$. A tree is a connected
graph without cycles and a forest is a graph all of whose connected components
are trees. The number of ends of ${\mathcal{G}}$ is defined to be the supremum
of the number of infinite components of ${\mathcal{G}}\setminus K$ over all
finite subsets $K$ of ${\mathcal{G}}$; in particular ${\mathcal{G}}$ has one
end if removing any finite set from ${\mathcal{G}}$ leaves exactly one
connected component.

An automorphism of ${\mathcal{G}}$ is a bijection of the set vertex set which
preserves adjacency and ${\rm Aut}({\mathcal{G}})$ denotes the group of all
automorphisms of ${\mathcal{G}}$ equipped with the topology of pointwise
convergence. A subgroup $G\subset{\rm Aut}({\mathcal{G}})$ is transitive if
$|V({\mathcal{G}})/G|=1$. It is quasi-transitive if
$|V({\mathcal{G}})/G|<\infty$. The graph ${\mathcal{G}}$ is called transitive,
resp. quasi-transitive, if ${\rm Aut}({\mathcal{G}})$ is transitive, resp.
quasi-transitive. A quasi-transitive graph is called amenable if
$\inf|\partial K|/|K|=0$, where $K$ runs over finite non-empty subsets of
$V({\mathcal{G}})$ and $\partial K$ denotes edge boundary of $K$, that is the
set of all edges with exactly one endpoint in $K$. If a quasi-transitive graph
satisfies $\inf|\partial K|/|K|>0$ it is called non-amenable.

Given the Cayley graph ${\mathcal{G}}=(\Gamma,E)$ of a finitely generated
group $\Gamma$, the action defined by left multiplication identifies $\Gamma$
with a transitive subgroup of ${\rm Aut}({\mathcal{G}})$. Recall from Section
1 that a $\Gamma$-invariant probability measure on subsets of $E$ is called a
$\Gamma$-invariant bond percolation. The percolation configurations are
denoted by $\omega$ and we will often identify $\omega$ with the corresponding
induced subgraph. A cluster of $\omega$ is a connected component of the
induced subgraph and we write $C(g)$ for the cluster containing $g\in\Gamma$.
We write $\deg_{\omega}(g)$ for the degree of $g\in\Gamma$ as a vertex in the
configuration $\omega$. For $g,h\in\Gamma$, the two-point function

$\tau(g,h):=\mathbb{P}\big{[}g\leftrightarrow h\big{]}$

is defined to be the probability that $g$ and $h$ are in the same cluster of
the percolation configuration.

####  2.2.1. Positive definite functions in invariant percolations.

The main link between the objects defined in Section 2.1 and invariant
percolation is provided by the following lemma.

######  Lemma 2.2.

Let ${\mathcal{G}}$ be the Cayley graph of a finitely generated group $\Gamma$
and let $\mathbb{P}$ be a $\Gamma$-invariant bond percolation on
${\mathcal{G}}$. Then $\tau(\cdot,\cdot)$ defines a positive definite
invariant kernel. In particular $\tau(o,\cdot)$ defines a positive definite
function.

###### Proof.

As pointed out in the introduction, this observation is due to Aizenman and
Newman [2]. For the convenience of the reader we include a proof. Let
$\mathbb{P}$ be a $\Gamma$-invariant bond percolation on ${\mathcal{G}}$ with
percolation configuration $\omega$. Invariance of the kernel
$\tau(\cdot,\cdot)$ follows from $\Gamma$-invariance of $\mathbb{P}$. Let
$g_{1},\ldots,g_{n}\in\Gamma$ and $a_{1},\ldots,a_{n}\in\mathbb{C}$ be
arbitrary and let ${\mathcal{C}}$ denote the set of open clusters in $\omega$.
Then

$\displaystyle\sum_{i,j=1}^{n}\overline{a_{i}}a_{j}\tau(g_{i},g_{j})$
$\displaystyle=\sum_{i,j}\overline{a_{i}}a_{j}\mathbb{E}\big{[}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{g_{i}\leftrightarrow
g_{j}\\}}\big{]}=\mathbb{E}\biggl{[}\
\sum_{i,j}\overline{a_{i}}a_{j}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{g_{i}\leftrightarrow
g_{j}\\}}\biggr{]}$ $\displaystyle=\mathbb{E}\biggl{[}\
\sum_{C\in{\mathcal{C}}}\,\sum_{\\{g_{i},g_{j}\\}\subset
C}\overline{a_{i}}a_{j}\biggr{]}=\mathbb{E}\biggl{[}\
\sum_{C\in{\mathcal{C}}}\,\Bigl{|}\sum_{g_{i}\in
C}a_{i}\Bigr{|}^{2}\biggr{]}\geq 0,$

which proves positive definiteness. ∎

####  2.2.2. $p$-Bernoulli percolation, and the critical parameters $p_{c}$,
$p_{u}$ and $p_{\scriptscriptstyle{\rm conn}}$.

As introduced in Section 1, $p$-Bernoulli bond percolation is the bond
percolation in which every edge is kept independently at random with fixed
probability $p\in[0,1]$. By a result of Newman and Schulman [49] (see also
[42, Theorem 7.5]), it is a consequence of insertion tolerance and ergodicity
that the number of infinite clusters in this model is constant a.s. with this
constant being equal to either $0,1$ or $\infty$. The critical parameter
$p_{c}=p_{c}({\mathcal{G}})$ is defined as the infimum over all $p\in[0,1]$
such that in $p$-Bernoulli bond percolation, there exists an infinite cluster
almost surely. Similarly, the uniqueness threshold
$p_{u}=p_{u}({\mathcal{G}})$ is the infimum over all $p\in[0,1]$ such that in
$p$-Bernoulli bond percolation, there is a unique infinite cluster almost
surely. By the well-known Harris-FKG-Inequality, increasing events are
positively correlated in any Bernoulli bond percolation [42, Section 5.8].

Example 2.3  As mentioned in the introduction, vanishing at infinity of the
two-point function does not necessarily imply having no infinite clusters.
Indeed, let $T$ denote the $3$-regular tree. It follows that $p_{c}(T)<1$, see
e.g. [42, Theorem 5.15]. Therefore supercritical $p$-Bernoulli bond
percolation $\mathbb{P}_{p}$, that is to say with $p\in(p_{c},1)$, produces an
infinite cluster a.s. To see that there are infinitely many infinite clusters,
it therefore suffices to rule out the possibility that there is a unique
infinite cluster. If there was a unique infinite cluster, then for every two
vertices $u$ and $v$ the Harris-FKG-Inequality and transitivity would imply
that $\mathbb{P}_{p}(u\leftrightarrow
v)\geq\mathbb{P}_{p}(|C(u)|=\infty,|C(v)|=\infty)\geq\mathbb{P}_{p}(|C(u)|=\infty)^{2}>0$.
This contradicts the fact that the two-point function of $\mathbb{P}_{p}$
clearly vanishes at infinity.

We now define the threshold $p_{\scriptscriptstyle{\rm conn}}$ that appeared
in the context of Theorem 1.3 pertinent to Bernoulli percolation. Let
${\mathcal{H}}$ be a countable, locally finite, connected graph
${\mathcal{H}}=(V,E)$. The need to go beyond Cayley graphs at this stage is
due to the fact that we want to relate the Haagerup property to phase
transitions in Bernoulli percolation performed on configurations of invariant
percolations, which clearly need not satisfy any form of transitivity. This
parameter was already introduced by Schonmann [55, Section 3.1] as a natural
critical point associated with Bernoulli percolation. Following the notation
in [55, Section 3.1], we define

$p_{\scriptscriptstyle{\rm conn}}\coloneqq p_{\scriptscriptstyle{\rm
conn}}({\mathcal{H}})\coloneqq\sup\bigg{\\{}p\in[0,1]\colon\lim_{r\to\infty}\sup_{\genfrac{}{}{0.0pt}{}{u,v\in
V}{d(u,v)>r}}\tau_{p}(u,v)=0\bigg{\\}},$

where $\tau_{p}$ denotes the two-point function of $p$-Bernoulli bond
percolation on ${\mathcal{H}}$. A related parameter, also introduced by
Schonmann [55, Section 3.1], is defined as

$\displaystyle\overline{p}_{\scriptscriptstyle{\rm
conn}}\coloneqq\overline{p}_{\scriptscriptstyle{\rm conn}}({\mathcal{H}})$
$\displaystyle\coloneqq\inf\bigg{\\{}p\in[0,1]\colon\inf_{u,v\in
V}\tau_{p}(u,v)>0\bigg{\\}}$
$\displaystyle=\sup\bigg{\\{}p\in[0,1]\colon\inf_{u,v\in
V}\tau_{p}(u,v)=0\bigg{\\}}.$

It is immediate from the definitions that

$p_{c}({\mathcal{H}})\leq p_{\scriptscriptstyle{\rm
conn}}({\mathcal{H}})\leq\overline{p}_{\scriptscriptstyle{\rm
conn}}({\mathcal{H}}).\vspace{1mm}$ (2.1)

If ${\mathcal{H}}$ is quasi-transitive, an application of the Harris-FKG-
Inequality as in Example 2.2.2 shows that
$\overline{p}_{\scriptscriptstyle{\rm conn}}\leq p_{u}$. Thus, for quasi-
transitive ${\mathcal{H}}$ we may extend (2.1) as follows

$p_{c}({\mathcal{H}})\leq p_{\scriptscriptstyle{\rm
conn}}({\mathcal{H}})\leq\overline{p}_{\scriptscriptstyle{\rm
conn}}({\mathcal{H}})\leq p_{u}({\mathcal{H}}).\vspace{1mm}$ (2.2)

By Lyons and Schramm [44, Theorem 4.1],
$p_{u}=\overline{p}_{\scriptscriptstyle{\rm conn}}$ for Cayley graphs. We note
that for every tree $p_{\scriptscriptstyle{\rm
conn}}=\overline{p}_{\scriptscriptstyle{\rm conn}}=p_{u}=1$. For a further
discussion of these parameters see [55].

###  2.3. Point processes.

Let $(X,{\mathcal{B}})$ be a measurable space. A measure on $X$ is called
$s$-finite if it is a countable sum of finite measures. Note that every
$\sigma$-finite measure is $s$-finite. Let $\mathbf{N}\coloneqq\mathbf{N}(X)$
denote the space of all measures on $X$ which can be written as countable sums
of finite, $\mathbb{N}_{0}$-valued measures. We equip $\mathbf{N}$ with the
$\sigma$-field $\mathcal{N}\coloneqq\mathcal{N}(X)$ generated by the
collection of all sets of the form

$\\{\mu\in\mathbf{N}(X)\colon\mu(B)=k\\} \quad
B\in{\mathcal{B}},k\in\mathbb{N}_{0}.$

A point process on $X$ is a random element of $(\mathbf{N},\mathcal{N})$. A
point process $\eta$ on $X$ is called proper if there exist random elements
$\xi_{1},\xi_{2},\ldots$ and an $\overline{\mathbb{N}_{0}}$-valued random
variable $M$ such that

$\eta=\sum_{i=1}^{M}\xi_{i}\quad\mbox{a.s.}$

Let $\lambda$ be an $s$-finite measure. A Poisson process with intensity
measure $\lambda$ is a point process $\eta$ on $X$ with the following two
properties:

  1. (i)

for every $B\in{\mathcal{B}}$ the distribution of $\eta(B)$ is Poisson with
parameter $\lambda(B)$, that is to say

$\mathbb{P}[\eta(B)=k]={\rm
e}^{-\lambda(B)}\frac{\lambda(B)^{k}}{k!}\quad\text{for every }
k\in\mathbb{N}_{0};$

  2. (ii)

for every $n\in\mathbb{N}$ and every collection of pairwise disjoint sets
$B_{1},\ldots,B_{n}\in{\mathcal{B}}$, the random variables
$\eta_{p}(B_{1}),\ldots,\eta_{p}(B_{n})$ are independent.

We need the standard fact that such processes exist.

######  Theorem 2.4 ([40, Theorem 3.6]).

Let $\lambda$ be an $s$-finite measure on a measurable space
$(X,{\mathcal{B}})$, then there exists a Poisson process on $X$ with intensity
measure $\lambda$.

A useful property of Poisson processes is that they satisfy a version of the
Harris-FKG-Inequality, which we now recall [40, Section 20.3]: Given
$B\in{\mathcal{B}}$, a measurable real-valued function $f$ on $\mathbf{N}(X)$
is called increasing on $B$ if

$f(\mu+\delta_{x})\geq f(\mu)\quad\mbox{for
all}\,\,\mu\in\mathbf{N}(X)\,\,\mbox{and all}\,\,x\in B,$

and is called decreasing on $B$ if $(-f)$ is increasing on $B$.

######  Theorem 2.5 ([40, Theorem 20.4]).

Let $\eta$ be a proper Poisson process on a measurable space
$(X,{\mathcal{B}})$ with $\sigma$-finite intensity measure $\lambda$ and law
$\mathbb{P}_{\eta}$. Let $B\in{\mathcal{B}}$ and let $f,f^{\prime}\in
L^{2}(\mathbb{P}_{\eta})$ be increasing on $B$ and decreasing on $X\setminus
B$, then

$\mathbb{E}\big{[}f(\eta)f^{\prime}(\eta)\big{]}\geq\mathbb{E}\big{[}f(\eta)\big{]}\,\mathbb{E}\big{[}f^{\prime}(\eta)\big{]}.$

##  3\. Characterization of the Haagerup property

In this section, we will prove Theorem 1.2 and the auxiliary results mentioned
in Section 1.1. Before considering the general case, we demonstrate our result
for two sub-classes of groups with the Haagerup property: groups acting
properly on trees and groups acting properly on spaces with walls. In these
cases the construction is more concrete. Therefore they can serve as useful
intuition for the general result. Moreover, they provide interesting examples
which are illustrated along the way.

###  3.1. Groups acting on trees.

Let $T$ be a locally finite tree. We say that $\Gamma$ acts properly on $T$ if
it acts on $T$ via automorphisms and such that $|\\{g\in\Gamma\colon gF\cap
F\neq\emptyset\\}|<\infty$ for every finite $F\subset T$.

######  Proposition 3.1.

Let ${\mathcal{G}}$ be the Cayley graph of a finitely generated group
$\Gamma$. If $\Gamma$ acts properly on a locally finite tree, then for every
$\alpha<1$, there exists a $\Gamma$-invariant bond percolation $\mathbb{P}$ on
${\mathcal{G}}$ such that $\mathbb{E}[\deg_{\omega}(o)]>\alpha\deg(o)$ and
$\tau(o,\cdot)\in C_{0}(\Gamma)$.

###### Proof.

Let $T=(V(T),E(T))$ be a locally finite tree on which $\Gamma$ acts properly
and fix a root $x_{0}\in V(T)$. For $p\in[0,1]$, let $\mathbb{P}_{p}$ denote
$p$-Bernoulli bond percolation on $T$. If $p<1$ we have

$\lim_{d(x_{0},x)\to\infty}\mathbb{P}_{p}\big{[}x_{0}\leftrightarrow
x\big{]}=0$ (3.1)

because the removal of any edge in the unique path between $x_{0}$ and $x$
disconnects the two vertices.

Let $\xi_{p}$ denote the configuration of a $p$-Bernoulli bond percolation on
$T$. We define the configuration $\omega_{p}$ of a bond percolation on
${\mathcal{G}}=(V,E)$ as follows: For each edge $e=[g,h]\in E$, let $e$ be in
$\omega_{p}$ if and only if one of the following holds:

  1. (i)

$gx_{0}=hx_{0}$,

  2. (ii)

$gx_{0}\neq hx_{0}$ but $gx_{0}$ is connected to $hx_{0}$ in $\xi_{p}$.

Let $\mathbf{P}_{p}$ denote the law of $\omega_{p}$. We first claim that
$\mathbf{P}_{p}$ defines a $\Gamma$-invariant bond percolation on
${\mathcal{G}}$. To see this, consider any cylinder event

$C=\big{\\{}\omega\in\\{0,1\\}^{E}\colon\omega(e_{1}),\ldots,\omega(e_{n})=1\,,\,\omega(e_{1}^{\prime}),\ldots,\omega(e_{m}^{\prime})=0\big{\\}},$

where $n,m\geq 1$ and
$e_{1}=[\gamma_{1},h_{1}],\ldots,e_{n}=[\gamma_{n},h_{n}],e_{1}^{\prime}=[\gamma_{1}^{\prime},h_{1}^{\prime}],\ldots,e_{m}^{\prime}=[\gamma_{m}^{\prime},h_{m}^{\prime}]\in
E$. Let $g\in\Gamma$, then

$gC=\big{\\{}\omega\in\\{0,1\\}^{E}\colon\omega(ge_{1}),\ldots,\omega(ge_{n})=1\,,\,\omega(ge_{1}^{\prime}),\ldots,\omega(ge_{m}^{\prime})=0\big{\\}}.$

It follows that

$\displaystyle\mathbf{P}_{p}[gC]$
$\displaystyle=\mathbb{P}_{p}\left[\begin{array}[]{cc}g\gamma_{i}x_{0}\leftrightarrow
gh_{i}x_{0}\text{ in }\xi_{p}\text{ for all }i\in\\{1,\ldots,n\\}\\\
g\gamma_{j}^{\prime}x_{0}\not\leftrightarrow gh_{j}^{\prime}x_{0}\text{ in
}\xi_{p}\text{ for all }j\in\\{1,\ldots,m\\}\end{array}\right]$
$\displaystyle=\mathbb{P}_{p}\circ
g\left[\begin{array}[]{cc}\gamma_{i}x_{0}\leftrightarrow h_{i}x_{0}\text{ in
}\xi_{p}\text{ for all }i\in\\{1,\ldots,n\\}\\\
\gamma_{j}^{\prime}x_{0}\not\leftrightarrow h_{j}^{\prime}x_{0}\text{ in
}\xi_{p}\text{ for all
}j\in\\{1,\ldots,m\\}\end{array}\right]=\mathbf{P}_{p}[C],$

where we have used the fact that $\Gamma$ acts on $T$ via automorphism and the
fact that the $\mathbb{P}_{p}$ is ${\rm Aut}(T)$-invariant. This proves
$\Gamma$-invariance of $\mathbf{P}_{p}$.

We note that $\mathbf{P}_{p}$ clearly has marginals arbitrarily close to $1$
for $p$ sufficiently close to $1$. This implies that
$\mathbf{E}_{p}[\deg_{\omega}(o)]>\alpha\deg(o)$ for $p$ sufficiently close to
$1$. Hence it suffices to prove that the two-point function
$\tau_{p}(o,g)=\mathbf{P}_{p}[o\leftrightarrow g]$ of $\mathbf{P}_{p}$
satisfies $\tau_{p}(o,\cdot)\in C_{0}(\Gamma)$ for every $p<1$. To show this,
simply observe that on the event $\\{o\leftrightarrow g\\}$ the unique path
which joins $x_{0}$ to $gx_{0}$ must be contained in $\xi_{p}$. Thus

$\tau_{p}(o,g)=\mathbf{P}_{p}\big{[}o\leftrightarrow
g\big{]}\leq\mathbb{P}_{p}\big{[}x_{0}\leftrightarrow gx_{0}\big{]}.$

Moerover, since the action of $\Gamma$ on $T$ is proper, we have that
$\lim_{g\to\infty}d(x_{0},gx_{0})=\infty$. Combined with Equation (3.1) we
therefore obtain

$\lim_{g\to\infty}\tau_{p}(o,g)\leq\lim_{g\to\infty}\mathbb{P}_{p}\big{[}x_{0}\leftrightarrow
gx_{0}\big{]}=0,$

which concludes the proof. ∎

Example 3.2  Let us comment on the particular example when $\Gamma$ is a
finitely generated free group acting on its canonical Cayley graph. In this
case it is easy to see that the two-point function of $p$-Bernoulli bond
percolation is given by

$\tau_{p}(g,h)=p^{d(g,h)}={\rm e}^{\log(p)d(g,h)}.$

Consider the distance function $|g|=d(o,g)$ on $\Gamma$. Since
$p\mapsto\log(p)$ is a bijection from $(0,1)$ to $(-\infty,0)$, we obtain that
the function $g \mapsto{\rm e}^{-\lambda|g|}$ is positive definite for every
$\lambda>0$. By Schoenberg’s theorem, the distance function is conditionally
negative definite, see e.g. [6, Theorem C.3.2]. As mentioned earlier, this
recovers a fundamental result due to Haagerup, see [27, Lemma 1.2]. We believe
that the proof presented here is significantly less involved and has the
benefit of being very natural from a probabilistic point of view.

###  3.2. Groups acting on spaces with walls.

The following terminology was introduced by Haglund and Paulin in [28]. Let
$X$ be a set and let ${\mathcal{W}}$ be a set of partitions of $X$ into two
classes. A partition $W\in{\mathcal{W}}$ is called a wall and we say that $W$
separates two distinct points $x$ and $y$ in $X$ if the points belong to
different classes in $W$. The pair $(X,{\mathcal{W}})$ is a space with walls
if, for every two distinct points $x$ and $y$ in $X$, the set

${\mathcal{W}}(x,y)\coloneqq\\{W\in{\mathcal{W}}\colon W\text{ separates
}x\text{ and }y\\}$

is finite. In this case $w(x,y)\coloneqq|{\mathcal{W}}(x,y)|$ is called the
number of walls separating $x$ and $y$. We say that a finitely generated group
$\Gamma$ acts properly on a space with walls $(X,{\mathcal{W}})$ if $\Gamma$
acts on $X$, preserves the family of walls and for some $x_{0}\in X$ (and
hence for all $x\in X$) the function $\Gamma\to\mathbb{N}_{0}\,,\,g\mapsto
w(x_{0},gx_{0})$ is proper, that is to say
$\lim_{g\to\infty}w(x_{0},gx_{0})=\infty.$

######  Proposition 3.3.

Let ${\mathcal{G}}$ be the Cayley graph of a finitely generated group
$\Gamma$. If $\Gamma$ acts properly on a space with walls, then for every
$\alpha<1$, there exists a $\Gamma$-invariant bond percolation $\mathbb{P}$ on
${\mathcal{G}}$ such that $\mathbb{E}[\deg_{\omega}(o)]>\alpha\deg(o)$ and
$\tau(o,\cdot)\in C_{0}(\Gamma)$.

###### Proof.

Let $\Gamma$ act properly on the space with walls $(X,{\mathcal{W}})$ and fix
a point $x_{0}\in X$. Let $p\in(0,1)$ and let
$\\{Z_{W}\\}_{W\in{\mathcal{W}}}$ be iid Bernoulli-$p$-distributed random
variables defined on some probability space
$(\Omega,{\mathcal{F}},\mathbb{P})$. We define the configuration $\omega_{p}$
of a bond percolation on ${\mathcal{G}}=(V,E)$ as follows: For each edge
$e=[g,h]\in E$, let $e$ be contained in $\omega_{p}$ if and only if one of the
following holds:

  1. (i)

$gx_{0}=hx_{0}$,

  2. (ii)

$gx_{0}\neq hx_{0}$ and for every $W\in{\mathcal{W}}(gx_{0},hx_{0})$ we have
$Z_{W}=1$.

Let $\mathbf{P}_{p}$ denote the law of $\omega_{p}$. We again claim that
$\mathbf{P}_{p}$ defines a $\Gamma$-invariant bond percolation: First observe
that the maps $\\{B,B^{c}\\}=W\mapsto gW=\\{gB,gB^{c}\\}$ define an action of
$\Gamma$ on the set of walls and note that
${\mathcal{W}}(gx,gy)=g{\mathcal{W}}(x,y)$ for every $g\in\Gamma,x,y\in X$.
Consider any cylinder event

$C=\big{\\{}\omega\in\\{0,1\\}^{E}\colon\omega(e_{1}),\ldots,\omega(e_{n})=1\,,\,\omega(e_{1}^{\prime}),\ldots,\omega(e_{m}^{\prime})=0\big{\\}},$

where $n,m\geq 1$ and
$e_{1}=[\gamma_{1},h_{1}],\ldots,e_{n}=[\gamma_{n},h_{n}],e_{1}^{\prime}=[\gamma_{1}^{\prime},h_{1}^{\prime}],\ldots,e_{m}^{\prime}=[\gamma_{m}^{\prime},h_{m}^{\prime}]\in
E$. Then

$\mathbf{P}_{p}[C]=\mathbb{P}\left[\begin{array}[]{cc}Z_{W}=1\text{ for all
}W\in{\mathcal{W}}(\gamma_{i}x_{0},h_{i}x_{0})\text{ and all
}i\in\\{1,\ldots,n\\}\\\ \text{ for all }j\in\\{1,\ldots,m\\}\text{ there
exists }W\in{\mathcal{W}}(\gamma_{j}^{\prime}x_{0},h_{j}^{\prime}x_{0})\text{
such that }Z_{W}=0\end{array}\right].$

We define

${\mathcal{V}}=\bigcup_{i=1}^{n}{\mathcal{W}}(\gamma_{i}x_{0},h_{i}x_{0}),$

and recursively for $j\in\\{1,\ldots,m\\}$

${\mathcal{V}}_{j}={\mathcal{W}}(\gamma_{j}^{\prime}x_{0},h_{j}^{\prime}x_{0})\setminus\big{(}{\mathcal{V}}\cup{\mathcal{V}}_{1}\cup\ldots\cup{\mathcal{V}}_{j-1}\big{)}.$

It follows that

$\mathbf{P}_{p}[C]=p^{|{\mathcal{V}}|}\prod_{j=1}^{m}\bigl{(}1-p^{|{\mathcal{V}}_{j}|}\bigr{)}.$

Similarly, for every $g\in\Gamma$, we have that

$\mathbf{P}_{p}[gC]=\mathbb{P}\left[\begin{array}[]{cc}Z_{W}=1\text{ for all
}W\in{\mathcal{W}}(g\gamma_{i}x_{0},gh_{i}x_{0})\text{ and all
}i\in\\{1,\ldots,n\\}\\\ \text{ for all }j\in\\{1,\ldots,m\\}\text{ there
exists
}W\in{\mathcal{W}}(g\gamma_{j}^{\prime}x_{0},gh_{j}^{\prime}x_{0})\text{ such
that }Z_{W}=0\end{array}\right]$

and, by the same argument as above and the preceding observation, that

$\mathbf{P}_{p}[gC]=p^{|g{\mathcal{V}}|}\prod_{j=1}^{m}\bigl{(}1-p^{|g{\mathcal{V}}_{j}|}\bigr{)}=p^{|{\mathcal{V}}|}\prod_{j=1}^{m}\bigl{(}1-p^{|{\mathcal{V}}_{j}|}\bigr{)}=\mathbf{P}_{p}[C].$

This proves that $\mathbf{P}_{p}$ is indeed $\Gamma$-invariant.

We observe that $\mathbf{P}_{p}$ has marginals arbitrarily close to $1$ for
$p$ sufficiently close to $1$: Consider any edge $e=[g,h]\in E$, then

$\displaystyle\mathbf{P}_{p}\big{[}[g,h]\in\omega_{p}\big{]}$
$\displaystyle={\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{gx_{0}=hx_{0}\\}}+{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{gx_{0}\neq
hx_{0}\\}}\mathbb{P}\big{[}Z_{W}=1\text{ for all
}W\in{\mathcal{W}}(gx_{0},hx_{0})\big{]}$
$\displaystyle={\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{gx_{0}=hx_{0}\\}}+{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{\\{gx_{0}\neq
hx_{0}\\}}p^{w(gx_{0},hx_{0})},$

which converges to $1$ as $p\to 1$. Hence
$\mathbf{E}_{p}[\deg_{\omega}(o)]>\alpha\deg(o)$ for $p$ sufficiently close to
$1$. It remains to show that the two-point function
$\tau_{p}(o,g)=\mathbf{P}_{p}[o\leftrightarrow g]$ of $\mathbf{P}_{p}$
satisfies $\tau_{p}(o,\cdot)\in C_{0}(\Gamma)$. To see this let $g\in\Gamma$
be arbitrary, let $o=h_{0},h_{1},h_{2},\ldots,h_{n}=g$ be any path of vertices
joining $o$ to $g$ in ${\mathcal{G}}$ and let $W$ be any wall separating
$x_{0}$ and $gx_{0}$. Let $B\sqcup B^{c}=X$ be the partition determined by $W$
and assume without loss of generality that $x_{0}\in B$. By assumption
$gx_{0}\in B^{c}$ and thus

$i\coloneqq\max\\{j\colon 0\leq j\leq n-1\text{ and }h_{j}x_{0}\in B\\}$

is well defined and satisfies $h_{i}x_{0}\in B$ as well as $h_{i+1}x_{0}\in
B^{c}$. In other words the wall $W$ also separates $h_{i}x_{0}$ and
$h_{i+1}x_{0}$. With this observation, we now prove the claim: Assume that $o$
is connected to $g$ in $\omega_{p}$, then there exists some path of open edges
$e_{1},\ldots,e_{n}$ joining $o$ to $g$ in ${\mathcal{G}}$. The above argument
shows that for every wall $W\in{\mathcal{W}}(x_{0},gx_{0})$ there exists an
edge $e=[h,\gamma]\in\\{e_{1},\ldots,e_{n}\\}$ such that
$W\in{\mathcal{W}}(hx_{0},\gamma x_{0})$. By definition, $e$ being open
implies that $Z_{W}=1$. Therefore we obtain that on the event
$\\{o\leftrightarrow g\\}$, we must have $Z_{W}=1$ for every
$W\in{\mathcal{W}}(x_{0},gx_{0})$. Thus

$\tau_{p}(o,g)=\mathbf{P}_{p}\big{[}o\leftrightarrow
g\big{]}\leq\mathbb{P}\big{[}Z_{W}=1\text{ for every
}W\in{\mathcal{W}}(x_{0},gx_{0})\big{]}=p^{w(x_{0},gx_{0})}.$

Since $\Gamma$ acts properly, it follows that

$\lim_{g\to\infty}\tau_{p}(o,g)\leq\lim_{g\to\infty}p^{w(x_{0},gx_{0})}=0,$

which concludes the proof. ∎

Example 3.4  We give two familiar examples of spaces with walls.

  1. (i)

A locally finite tree $T$ has a natural structure of space with walls by
declaring ${\mathcal{W}}$ to consist of partitions into two connected
components which are obtained by removing an edge. A group acts properly on
$(T,{\mathcal{W}})$ if and only if it acts properly on the tree $T$.

  2. (ii)

The graph $\mathbb{Z}^{2}$ can be split into two discrete half-spaces
orthogonal to the horizontal axis, resp. vertical axis, by cutting for fixed
$x\in\mathbb{Z}$ along all edges of the form $[(x,y),(x+1,y)]$, resp. by
cutting for fixed $y\in\mathbb{Z}$ along all edges of the form
$[(x,y),(x,y+1)]$. This defines a family of walls on $\mathbb{Z}^{2}$ such
that the number of walls separating two points is equal to their graph
distance. Clearly the same applies in the case of $\mathbb{Z}^{d}$ for some
$d\in\mathbb{N}$.

###  3.3. Groups acting on spaces with measured walls and proof of Theorem
1.2.

As mentioned in the introduction, the following concept was introduced by
Cherix, Martin and Valette in [11]. Let $X$ be a set, ${\mathcal{W}}$ a set of
walls on $X$, ${\mathcal{B}}$ a $\sigma$-algebra on ${\mathcal{W}}$ and $\mu$
a measure on ${\mathcal{B}}$. The $4$-tuple
$(X,{\mathcal{W}},{\mathcal{B}},\mu)$ is a space with measured walls if, for
every two distinct points $x$ and $y$ in $X$, the set ${\mathcal{W}}(x,y)$ of
walls separating $x$ and $y$ is ${\mathcal{B}}$-measurable and
$w(x,y)\coloneqq\mu({\mathcal{W}}(x,y))<\infty$. Note that if $\mu$ is a
counting measure, then we recover a space with walls. We say that a finitely
generated group $\Gamma$ acts properly on a space with measured walls
$(X,{\mathcal{W}},{\mathcal{B}},\mu)$ if $\Gamma$ acts on $X$, preserves the
family of walls, preserves the function $w(\cdot,\cdot)$ and for some
$x_{0}\in X$ (and hence for all $x\in X$) the function
$\Gamma\to[0,\infty)\,,\,g\mapsto w(x_{0},gx_{0})$ is proper. This again
extends the definition of a proper action on a space with walls in the case
that $\mu$ is a counting measure. Note that

$\Gamma\times\Gamma\to[0,\infty)\,,\,D(g,h)\coloneqq w(gx_{0},hx_{0})$ (3.2)

defines a pseudo-distance on $\Gamma$ which we call the wall distance.

It was proved in [11, Proposition 1] that a locally compact group which acts
properly on a space with measured walls has the Haagerup property. Moreover it
was proved in [11, Theorem 1.(1)] that a countable discrete group with the
Haagerup property admits a proper action on a space with measured walls. In
fact, building on the ideas developed in [52], the proof given in [11]
provides explicitly a space with measured walls and a corresponding proper
action, see also [12, Proposition 7.5.1]. Namely, for a finitely generated
group $\Gamma$, we may consider the following:

  * •

Let $X=\Gamma$ on which $\Gamma$ acts via left-multiplication and let

$\Omega=\\{0,1\\}^{\Gamma}\setminus\\{(0,0,\ldots),(1,1,\ldots)\\},$

the set of all non-empty and non-full subsets of $\Gamma$, equipped with the
Borel $\sigma$-algebra ${\mathcal{B}}(\Omega)$. Then $\Gamma$ has a natural
measurable action on $\Omega$ defined by $g\omega(x)=\omega(g^{-1}x)$.

  * •

For $\omega\in\Omega$ define $W_{\omega}=\\{\omega,\omega^{c}\\}$ and let
${\mathcal{W}}=\\{W_{\omega}\colon\omega\in\Omega\\}$. Clearly ${\mathcal{W}}$
defines a family of walls on $X$ which is preserved by the action of $\Gamma$.

  * •

The set ${\mathcal{W}}$ can be identified with the quotient of $\Omega$ by the
fixed-point free involution $\omega\mapsto\omega^{c}$. Let ${\mathcal{B}}$ be
the direct image of ${\mathcal{B}}(\Omega)$.

  * •

For $g\in\Gamma$ define
$S_{g}=\\{\omega\in\Omega\colon\omega(g)=1\\}\in{\mathcal{B}}(\Omega)$. Let
$\psi$ be a conditionally negative definite function on $\Gamma$. By [52,
Proposition 1.4], there exists a regular Borel measure $\nu$ on $\Omega$ such
that

$\sqrt{\psi(g^{-1}h)}=\nu(S_{g}\Delta S_{h})\quad\text{for all }g,h\in\Gamma.$

Moreover, it follows from the proof of [52, Theorem 2.1] that $\nu$ is
$\Gamma$-invariant. Let $\mu$ denote the direct image of $\nu$ on
${\mathcal{W}}$. In particular

$\mu({\mathcal{W}}(g,h))=\nu(S_{g}\Delta
S_{h})=\sqrt{\psi(g^{-1}h)}<\infty\quad\text{for all }g,h\in\Gamma.$

Then $(\Gamma,{\mathcal{W}},{\mathcal{B}},\mu)$ defines a space with measured
walls. Moreover, the induced $\Gamma$ action on the set of walls is
measurable, leaves $\mu$ invariant and satisfies ${\mathcal{W}}(\gamma
g,\gamma h)=\gamma{\mathcal{W}}(g,h)$. In particular the function
$w(\cdot,\cdot)$ is preserved. Now if $\Gamma$ has the Haagerup property then
there exists a proper, conditionally negative definite function $\psi$ on
$\Gamma$. In this case, $\Gamma$ clearly acts properly on
$(\Gamma,{\mathcal{W}},{\mathcal{B}},\mu)$.

######  Theorem 3.5.

Let ${\mathcal{G}}$ be the Cayley graph of a finitely generated group
$\Gamma$. If $\Gamma$ has the Haagerup property, then for every $\alpha<1$,
there exists a $\Gamma$-invariant bond percolation $\mathbb{P}$ on
${\mathcal{G}}$ such that $\mathbb{E}[\deg_{\omega}(o)]>\alpha\deg(o)$ and
$\tau(o,\cdot)\in C_{0}(\Gamma)$.

###### Proof.

Let $\psi$ be a proper, conditionally negative definite function on $\Gamma$
and consider the proper action of $\Gamma$ on the space with measured walls
$(\Gamma,{\mathcal{W}},{\mathcal{B}},\mu)$ described above. Let $p\in(0,1)$
and let $\eta_{p}$ be a Poisson process on ${\mathcal{W}}$ with intensity
measure $(1-p)\mu$ defined on some probability space
$(\Omega,{\mathcal{F}},\mathbb{P})$. Existence follows from Theorem 2.4
because the measure $\mu$ is $\sigma$-finite. We define the configuration
$\omega_{p}$ of a bond percolation on ${\mathcal{G}}=(V,E)$ as follows: For
each edge $e=[g,h]\in E$, let $e$ be contained in $\omega_{p}$ if and only if
$\eta_{p}({\mathcal{W}}(g,h))=0$.

Let $\mathbf{P}_{p}$ denote the law of $\omega_{p}$. We claim that
$\mathbf{P}_{p}$ defines a $\Gamma$-invariant bond percolation. We first show
that, for every $g\in\Gamma$, the process $\eta_{p}\circ g$ has the same law
under $\mathbb{P}$ as $\eta_{p}$. To shorten notation, denote the intensity
measure by $\lambda\coloneqq(1-p)\mu$ and recall that $\lambda$ is
$\Gamma$-invariant. Now let $n\geq 1$ and consider pairwise disjoint
$B_{1},\ldots,B_{n}\in{\mathcal{B}}$. Then

$\big{(}\eta_{p}(B_{1}),\ldots,\eta_{p}(B_{n})\big{)}\overset{d}{=}{\rm
Poi}\big{(}\lambda(B_{1})\big{)}\otimes\ldots\otimes{\rm
Poi}\big{(}\lambda(B_{n})\big{)}.$

Moreover, $gB_{1},\ldots,gB_{n}\in{\mathcal{B}}$ are pairwise disjoint with
$\lambda(gB_{i})=\lambda(B_{i})$ for all $i\in\\{1,\ldots,n\\}$ by
$\Gamma$-invariance of $\lambda$. It follows that

$\big{(}\eta_{p}\circ g(B_{1}),\ldots,\eta_{p}\circ
g(B_{n})\big{)}=\big{(}\eta_{p}(gB_{1}),\ldots,\eta_{p}(gB_{n})\big{)}\overset{d}{=}{\rm
Poi}\big{(}\lambda(B_{1})\big{)}\otimes\ldots\otimes{\rm
Poi}\big{(}\lambda(B_{n})\big{)}.$

By [40, Proposition 2.10 (i) $\Leftrightarrow$ (ii)] the point process
$\eta_{p}\circ g$ has the same law as $\eta_{p}$. Now consider any cylinder
event

$C=\big{\\{}\omega\in\\{0,1\\}^{E}\colon\omega(e_{1}),\ldots,\omega(e_{n})=1\,,\,\omega(e_{1}^{\prime}),\ldots,\omega(e_{m}^{\prime})=0\big{\\}},$

where $n,m\geq 1$ and
$e_{1}=[\gamma_{1},h_{1}],\ldots,e_{n}=[\gamma_{n},h_{n}],e_{1}^{\prime}=[\gamma_{1}^{\prime},h_{1}^{\prime}],\ldots,e_{m}^{\prime}=[\gamma_{m}^{\prime},h_{m}^{\prime}]\in
E$. Then

$\mathbf{P}_{p}[C]=\mathbb{P}\left[\begin{array}[]{cc}\eta_{p}({\mathcal{W}}(\gamma_{i},h_{i}))=0\text{
for all }i\in\\{1,\ldots,n\\}\\\
\eta_{p}({\mathcal{W}}(\gamma_{j}^{\prime},h_{j}^{\prime}))>0\text{ for all
}j\in\\{1,\ldots,m\\}\end{array}\right]$

and

$\displaystyle\mathbf{P}_{p}[gC]$
$\displaystyle=\mathbb{P}\left[\begin{array}[]{cc}\eta_{p}({\mathcal{W}}(g\gamma_{i},gh_{i}))=0\text{
for all }i\in\\{1,\ldots,n\\}\\\
\eta_{p}({\mathcal{W}}(g\gamma_{j}^{\prime},gh_{j}^{\prime}))>0\text{ for all
}j\in\\{1,\ldots,m\\}\end{array}\right]$
$\displaystyle=\mathbb{P}\left[\begin{array}[]{cc}\eta_{p}\circ
g({\mathcal{W}}(\gamma_{i},h_{i}))=0\text{ for all }i\in\\{1,\ldots,n\\}\\\
\eta_{p}\circ g({\mathcal{W}}(\gamma_{j}^{\prime},h_{j}^{\prime}))>0\text{ for
all }j\in\\{1,\ldots,m\\}\end{array}\right],$

since ${\mathcal{W}}(g\gamma,gh)=g{\mathcal{W}}(\gamma,h)$ for any
$\gamma,h\in\Gamma$. This proves $\Gamma$-invariance of $\mathbf{P}_{p}$.

We again observe that $\mathbf{P}_{p}$ has marginals arbitrarily close to $1$
for $p$ sufficiently close to $1$: Let $e=[g,h]\in E$, then

$\mathbf{P}_{p}\big{[}[g,h]\in\omega_{p}\big{]}=\mathbb{P}\big{[}\eta_{p}({\mathcal{W}}(g,h))=0\big{]}={\rm
e}^{-(1-p)\mu({\mathcal{W}}(g,h))}$

which, since $0\leq\mu({\mathcal{W}}(g,h))<\infty$, converges to $1$ as $p\to
1$. To conclude the proof, it remains to show that the two-point function
$\tau_{p}(o,g)=\mathbf{P}_{p}[o\leftrightarrow g]$ of $\mathbf{P}_{p}$
satisfies $\tau_{p}(o,\cdot)\in C_{0}(\Gamma)$. Let $g\in\Gamma$ and recall
from the proof of Proposition 3.3 that, given any path of edges
$e_{1},\ldots,e_{n}$ joining $o$ to $g$ in ${\mathcal{G}}$ and any wall
$W\in{\mathcal{W}}(o,g)$, there exists $e\in\\{e_{1},\ldots,e_{n}\\}$ such
that $W$ also separates the endpoints of the edge $e$. Writing
${\mathcal{W}}(e)$ for the set of walls separating the endpoints of an edge
$e$, it follows that

${\mathcal{W}}(o,g)\subset\bigcup_{i=1}^{n}{\mathcal{W}}(e_{i}).$

We obtain that on the event that $o$ and $g$ are in the same cluster of
$\omega_{p}$, we have $\eta_{p}({\mathcal{W}}(o,g))=0$. Thus

$\tau_{p}(o,g)=\mathbf{P}_{p}\big{[}o\leftrightarrow
g\big{]}\leq\mathbb{P}\big{[}\,\eta_{p}({\mathcal{W}}(o,g))=0\big{]}={\rm
e}^{-(1-p)\mu({\mathcal{W}}(o,g))}.$

Since the action of $\Gamma$ is proper, it follows that

$\lim_{g\to\infty}\tau_{p}(o,g)\leq\lim_{g\to\infty}{\rm
e}^{-(1-p)\mu({\mathcal{W}}(o,g))}=0,$

which concludes the proof. ∎

Proof of Theorem 1.2. Suppose that for every $\alpha<1$, there exists a
$\Gamma$-invariant bond percolation $\mathbb{P}$ on ${\mathcal{G}}$ such that
$\mathbb{E}[\deg_{\omega}(o)]>\alpha\deg(o)$ and $\tau(o,\cdot)\in
C_{0}(\Gamma)$. Choose a sequence $(\mathbb{P}_{i})_{i\in\mathbb{N}}$ of
$\Gamma$-invariant bond percolations on ${\mathcal{G}}$ such that the two
point function $\tau_{i}(o,\cdot)$ of each $\mathbb{P}_{i}$ vanishes at
infinity and such that

$\lim_{i\to\infty}\mathbb{E}_{i}[\deg_{\omega}(o)]=\deg(o).$

Then $\tau_{i}(o,\cdot)$ is a sequence of positive definite functions, see
Lemma 2.2, which vanish at infinity and which converge pointwise to $1$. That
is to say $\Gamma$ has the Haagerup property. The converse direction is
exactly Theorem 3.5 above. ∎

###  3.4. A generalized construction with invariant point processes.

Our next result is a generalization of the construction used in the proof of
Theorem 3.5. We are primarily interested in such a generalization because
there exist many known actions of groups on spaces with measured walls which
are well understood or have useful geometric features. Our general
construction applies to all of these and will be exemplified in Section 8. It
can thus be used to produce many examples of positive definite functions
witnessing the Haagerup property. Likewise, properties of specific actions can
give us additional information about the constructed percolations as
illustrated by Theorem 1.8.

We first need to introduce the concept of invariance for actions on spaces
with measured walls. We say that $\Gamma$ has an invariant action on a space
with measured walls $(X,{\mathcal{W}},{\mathcal{B}},\mu)$, if $\Gamma$ acts on
$X$ and the maps

${\mathcal{W}}\to{\mathcal{W}}\,,\,\\{B,B^{c}\\}=W\mapsto gW=\\{gB,gB^{c}\\}$

define a ${\mathcal{B}}$-measurable $\Gamma$-action on ${\mathcal{W}}$ for
which the measure $\mu$ is invariant.

######  Theorem 3.6.

Let ${\mathcal{G}}$ be the Cayley graph of a finitely generated group
$\Gamma$. Suppose that $\Gamma$ has an invariant action on a space with
measured walls $(X,{\mathcal{W}},{\mathcal{B}},\mu)$ and fix $x_{0}\in X$. For
$p\in(0,1)$, define

$\alpha_{p}\coloneqq\exp\Bigl{(}-(1-p)\max_{g\sim
o}\,w(x_{0},gx_{0})\Bigr{)}\in(0,1).$

Then there exists a $\Gamma$-invariant bond percolation $\mathbb{P}$ on
${\mathcal{G}}$ such that $\mathbb{E}[\deg_{\omega}(o)]\geq\alpha_{p}\deg(o)$
and

$\exp\Bigl{(}-(1-p)\max_{g\sim
o}\,w(x_{0},gx_{0})|g|\Bigr{)}\leq\tau(o,g)\leq\exp\bigl{(}-(1-p)w(x_{0},gx_{0})\bigr{)}\quad\text{for
every }g\in\Gamma.$

###### Proof.

We first show that $\Gamma$ admits an invariant action on a space with
measured walls $(X_{0},{\mathcal{W}}_{0},{\mathcal{B}}_{0},\mu_{0})$ such that
the measure $\mu_{0}$ is $s$-finite and satisfies
$\mu_{0}({\mathcal{W}}_{0}(gx_{0},hx_{0}))=\mu({\mathcal{W}}(gx_{0},hx_{0}))$
for all $g,h\in\Gamma$. Consider the set of walls

${\mathcal{W}}_{0}\coloneqq\bigcup_{g,h\in\Gamma}{\mathcal{W}}(gx_{0},hx_{0})\subset{\mathcal{W}}.$

Note that ${\mathcal{W}}_{0}\in\mathcal{B}$ as a countable union of
${\mathcal{B}}$-measurable sets. Let ${\mathcal{B}}_{0}$ be the restriction of
${\mathcal{B}}$ to ${\mathcal{W}}_{0}$ and let $\mu_{0}$ be the restriction of
$\mu$ to ${\mathcal{B}}_{0}$. Then
$(X,{\mathcal{W}}_{0},{\mathcal{B}}_{0},\mu_{0})$ is a space with measured
walls because for every two distinct points $x$ and $y$ in $X$ the set
${\mathcal{W}}_{0}(x,y)$ of walls separating $x$ and $y$ is given by

${\mathcal{W}}_{0}(x,y)={\mathcal{W}}(x,y)\cap{\mathcal{W}}_{0}\in{\mathcal{B}}_{0}$

and satisfies
$\mu_{0}({\mathcal{W}}_{0}(x,y))=\mu({\mathcal{W}}(x,y)\cap{\mathcal{W}}_{0})\leq\mu({\mathcal{W}}(x,y))<\infty$.
Moreover, the measure $\mu_{0}$ is $s$-finite because
$\mu_{0}({\mathcal{W}}(gx_{0},hx_{0}))<\infty$ for every $g,h\in\Gamma$.
Finally, for every $\gamma,g,h\in\Gamma$ and every
$W\in{\mathcal{W}}(gx_{0},hx_{0})$ we have that $\gamma
W\in{\mathcal{W}}(\gamma gx_{0},\gamma hx_{0})$, thus ${\mathcal{W}}_{0}$ is a
$\Gamma$-invariant set and therefore the action of $\Gamma$ restricts to an
invariant action of $\Gamma$ on
$(X,{\mathcal{W}}_{0},{\mathcal{B}}_{0},\mu_{0})$ as claimed.

Let $p\in(0,1)$. Let $\eta$ be a Poisson process on ${\mathcal{W}}_{0}$ with
intensity measure $(1-p)\mu_{0}$ defined on some probability space
$(\Omega,{\mathcal{F}},\mathbf{P})$, see Theorem 2.4. Define the configuration
$\omega$ of a bond percolation on ${\mathcal{G}}=(V,E)$ as follows: For each
edge $e=[g,h]\in E$, let $e$ be contained in $\omega$ if and only if
$\eta({\mathcal{W}}(gx_{0},hx_{0}))=0$. Let $\mathbb{P}$ denote the law of
$\omega$. Since the intensity measure of $\eta$ is $\Gamma$-invariant, exactly
the same argument as in the proof of Theorem 3.5 shows that $\mathbb{P}$
defines a $\Gamma$-invariant bond percolation. Now for every edge $e=[g,h]\in
E$, we have that

$\mathbb{P}[e\in\omega]=\mathbf{P}\big{[}\eta_{p}({\mathcal{W}}(gx_{0},hx_{0}))=0\big{]}={\rm
e}^{-(1-p)\mu_{0}({\mathcal{W}}(gx_{0},hx_{0}))}={\rm
e}^{-(1-p)w(gx_{0},hx_{0})}$

and thus

$\displaystyle\mathbb{E}[\deg_{\omega}(o)]=\sum_{g\sim
o}\mathbb{P}\big{[}[o,g]\in\omega\big{]}$ $\displaystyle\geq\deg(o)\min_{o\sim
g} \mathbb{P}\big{[}[o,g]\in\omega\big{]}$
$\displaystyle=\deg(o)\exp\Bigl{(}-(1-p)\max_{g\sim
o}\,w(x_{0},gx_{0})\Bigr{)}.$

Moreover, again by the same argument as in the proof of Theorem 3.5, it
follows that

$\tau(o,g)\leq\mathbf{P}\big{[}\eta({\mathcal{W}}(o,g))=0\big{]}=\exp\bigl{(}-(1-p)w(x_{0},gx_{0})\bigr{)}\quad\text{for
every }g\in\Gamma.$

For the lower bound, let $g\in\Gamma$ and choose a shortest path
$e_{1},\ldots,e_{|g|}$ of edges joining $o$ to $g$ in ${\mathcal{G}}$. Recall
that ${\mathcal{W}}(e_{i})$ denotes the set of walls separating the endpoints
of the edge $e_{i}$ and define

${\mathcal{V}}\coloneqq\bigcup_{i=1}^{|g|}{\mathcal{W}}(e_{i}).$

By the union bound and invariance

$\mu_{0}({\mathcal{V}})\leq\sum_{i=1}^{|g|}\mu_{0}({\mathcal{W}}(e_{i}))\leq|g|
\max_{e\in E}\mu({\mathcal{W}}(e))= |g| \max_{g\sim o}\,w(x_{0},gx_{0}).$

Finally, note that $\\{o\leftrightarrow
g\\}\supset\\{e_{1},\ldots,e_{|g|}\in\omega\\}=\\{\eta({\mathcal{V}})=0\\}$
and thus

$\tau(o,g)=\mathbb{P}\big{[}o\leftrightarrow
g\big{]}\geq\mathbf{P}\big{[}\eta({\mathcal{V}})=0\big{]}={\rm
e}^{-(1-p)\mu({\mathcal{V}})}\geq\exp\Bigl{(}-(1-p)|g| \max_{g\sim
o}\,w(x_{0},gx_{0})\Bigr{)}.$

The proof of Lemma 3.6 is thus complete. ∎

###  3.5. A sufficient condition for the Haagerup property via Bernoulli
percolation.

In this section, we prove the sufficient criterion for the Haagerup property
stated in Theorem 1.3. For the definition of $p_{\scriptscriptstyle{\rm
conn}}$ we refer to Section 2.

######  Theorem 3.7.

Let ${\mathcal{G}}$ be the Cayley graph of a finitely generated group
$\Gamma$. The following conditions are equivalent:

  1. (i)

there exists a $\Gamma$-invariant random spanning subgraph of ${\mathcal{G}}$
with $p_{\scriptscriptstyle{\rm conn}}=1$ a.s.;

  2. (ii)

there exists a $\Gamma$-invariant random nonempty connected subgraph of
${\mathcal{G}}$ with $p_{\scriptscriptstyle{\rm conn}}=1$ w.p.p.

Moreover, if $\Gamma$ satisfies (i), or equivalently (ii), then $\Gamma$ has
the Haagerup property.

###### Proof.

The implication (i) $\Rightarrow$ (ii) is obvious.

(ii) $\Rightarrow$ (i): We use a standard way of efficiently augmenting an
invariant percolation with a unique (necessarily infinite) cluster to an
invariant random spanning subgraph, see e.g. [31, Proposition 2.1] for another
application of this strategy. Let $\omega$ be a $\Gamma$-invariant random
nonempty connected subgraph of ${\mathcal{G}}$ with $p_{\scriptscriptstyle{\rm
conn}}=1$ with positive probability. Conditioning on this event, we may assume
that $p_{\scriptscriptstyle{\rm conn}}=1$ a.s. Now we augment the
configuration $\omega$ as follows: For every vertex which is not already in
$\omega$, choose an edge uniformly among those edges connecting it to a vertex
closer to $\omega$ in the graph distance of the underlying Cayley graph
${\mathcal{G}}$. The resulting configuration, denoted by $\xi$, is clearly
spanning and invariant. Moreover, $\xi\setminus\omega$ is a forest. Therefore
the property ”$p_{\scriptscriptstyle{\rm conn}}(\xi)=1$ a.s.” is trivially
inherited from the assumption that $p_{\scriptscriptstyle{\rm
conn}}(\omega)=1$ a.s.

Finally, let us prove that (i) (and therefore (ii)) implies the Haagerup
property. Therefore, let us assume that condition (i) holds. Let $\omega$ be a
$\Gamma$-invariant random spanning subgraph of ${\mathcal{G}}$ with
$p_{\scriptscriptstyle{\rm conn}}=1$ a.s. Let $p\in(0,1)$ and let $\tau_{p}$
denote the two-point function of the bond percolation $\xi_{p}$ obtained by
performing $p$-Bernoulli bond percolation on $\omega$. As the two-point
function of a $\Gamma$-invariant bond percolation, $\tau_{p}(o,\cdot)$ defines
a positive definite function. Recalling Theorem 2.1, it is therefore
sufficient to show that $\tau_{p}(o,\cdot)$ vanishes at infinity and converges
pointwise to $1$ as $p\to 1$. To that end, fix $g\in\Gamma$. Fixing $\omega$,
we see that

$\lim_{p\to
1}\,\mathbb{P}\Big{[}o\xleftrightarrow{\xi_{p}}g\,\big{|}\,\omega\Big{]}=1$

because $o$ and $g$ are joined by some finite path in the spanning
configuration $\omega$, which, with probability converging to one is fully
contained in $p$-Bernoulli bond percolation as $p$ converges to $1$. The
bounded convergence theorem yields that indeed $\lim_{p\to 1}\tau_{p}(o,g)=1$.
We now verify that $\tau_{p}(o,\cdot)$ vanishes at infinity. To see this, it
suffices to show that for every sequence $(g_{n})_{n=1}^{\infty}\subset\Gamma$
with $g_{n}\to\infty$, we have that $\tau_{p}(o,g_{n})\to 0$. Fix such a
sequence. Then conditional on the configuration $\omega$, the probability that
$o$ and $g_{n}$ are in the same cluster of $\xi_{p}$ converges to zero a.s. as
$n\to\infty$ by assumption. The claim follows again from the bounded
convergence theorem. ∎

###  3.6. Haagerup property and site percolation.

######  Theorem 3.8.

Let ${\mathcal{G}}$ be the Cayley graph of a finitely generated group
$\Gamma$. Then $\Gamma$ has the Haagerup property if and only if for every
$\alpha<1$, there exists a $\Gamma$-invariant site percolation $\mathbb{P}$ on
${\mathcal{G}}$ such that $\mathbb{P}[o\in\omega]>\alpha$ and
$\tau(o,\cdot)\in C_{0}(\Gamma)$.

###### Proof.

The ”if” direction follows exactly as in the proof of Theorem 1.2 above.

To prove the converse, assume that $\Gamma$ has the Haagerup property. By
Theorem 1.2 there exists a sequence $(\mathbb{P}_{i})_{i=1}^{\infty}$ of
$\Gamma$-invariant bond percolations such that the corresponding two-point
functions $\tau_{i}(o,\cdot)$ vanish at infinity for every $i\in\mathbb{N}$
and such that

$\lim_{i\to\infty}\mathbb{E}_{i}[\deg_{\omega}(o)]=\deg(o).$

Let $\omega_{1},\omega_{2},\ldots$ denote configurations with law
$\mathbb{P}_{i}$ defined on some common probability space
$(\Omega,{\mathcal{F}},\mathbb{P})$. For each $i\in\mathbb{N}$, define the
configuration $\xi_{i}$ of a site percolation as follows: Given the
configuration $\omega_{i}$, remove all vertices which are elements of the
vertex boundary of some cluster of $\omega_{i}$. More formally, for a subset
$U\subset V$, let

$\partial_{V}(U)=\\{x\in V\setminus U\colon\text{there exists }y\in U\text{
with }x\sim y\\}$

denote the vertex boundary of $U$. Let $\mathcal{C}$ be the collection of all
clusters of $\omega_{i}$ and define, for $x\in V$, that $x\in\xi_{i}$ if and
only if

$x\in V\setminus\bigcup_{C\in\mathcal{C}}\partial_{V}(C).$

Let $\mathbf{P}_{i}$ denote the law of $\xi_{i}$. Because the law of
$\omega_{i}$ is $\Gamma$-invariant, $\mathbf{P}_{i}$ is a $\Gamma$-invariant
site percolation. Now note that

$\\{o\in\xi_{i}\\}\supset\bigl{\\{}[o,g]\in\omega_{i}\text{ for all }g\sim
o\bigr{\\}}$

because if all edges incident to $o$ are contained in $\omega_{i}$, then $o$
and all its neighbors belong to the same cluster and therefore $o$ is not an
element of the vertex boundary of any cluster. This implies that

$\lim_{i\to\infty}\mathbf{P}_{i}[o\in\omega]=\lim_{i\to\infty}\mathbb{P}[o\in\xi_{i}]\geq\lim_{i\to\infty}\mathbb{P}\bigl{[}\,[o,g]\in\omega_{i}\text{
for all }g\sim o\bigr{]}=1.$

On the other hand, note that if $o$ and $g$ are not in the same cluster of
$\omega_{i}$, then the vertex boundary of the cluster of $o$ is not contained
in $\xi_{i}$ and therefore separates $o$ and $g$ in $\xi_{i}$. Taking
complements yields that

$\\{o\leftrightarrow g\text{ in }\xi_{i}\\}\subset\\{o\leftrightarrow g\text{
in }\omega_{i}\\}.$

Hence the two-point function of $\mathbf{P}_{i}$ is bounded by $\tau_{i}$,
thus also decaying at infinity. It follows that $\mathbf{P}_{i}$, for $i$
large enough, has the desired properties. ∎

##  4\. Spaces with measured walls and exponential decay of two-point
functions

With our construction established in the previous section, we now turn to the
quantitative version of our main result, namely Theorem 1.4, which describes
the relation between the growth of wall distances and the decay of the
associated percolations. We first set up the relevant terminology: Let
$\Gamma$ be a finitely generated group equipped with a word length $|\cdot|$
and let $\rho\colon[0,\infty]\to[0,\infty]$ be a function.

  * •

Let $D$ be a left-invariant pseudo-distance on $\Gamma$. Following [16,
Definition 7.1], we say that $\rho$ is a compression function for $D$ if
$D(o,g)\geq\rho(|g|)$ for all $g\in\Gamma$.

  * •

Recall the definition of wall distance from Section 3.3. If $\Gamma$ has an
invariant action on a space with measured walls
$(X,{\mathcal{W}},{\mathcal{B}},\mu)$ such that for some $x_{0}\in X$ the wall
distance $D(g,h)=w(gx_{0},hx_{0})$ admits $\rho$ as a compression function, we
say that the action has $\rho$-growth. If $\rho$ is linear, we say that the
action has linear growth.

  * •

Let $C>0$ be a constant and let $\tau$ be the two-point function of a
$\Gamma$-invariant bond percolation. We say that $\tau$ has
$(C,\rho)$-exponential decay if there exist $\beta,\gamma>0$ such that
$\beta/\gamma\leq C$ and

${\rm e}^{-\beta|g|}\leq\tau(o,g)\leq{\rm e}^{-\gamma\rho(|g|)}\quad\mbox{for
all}\,\,g\in\Gamma.$

If $\rho$ is linear, we say that $\tau$ has exponential decay.

  * •

From Section 2.1 we also recall that $k\colon\Gamma\times\Gamma\to[0,\infty)$
is a measure definite kernel if there exists a measure space
$(\Omega,{\mathcal{B}},\mu)$ and a map $S\colon\Gamma\to{\mathcal{B}}$,
$g\mapsto S_{g}$ such that $k(g,h)=\mu(S_{g}\Delta S_{h})$. Similarly, a
function $\psi\colon\Gamma\to[0,\infty)$ is called measure definite if the
kernel $k_{\psi}(g,h)=\psi(g^{-1}h)$ is measure definite.

Note that wall distances associated to an invariant action of $\Gamma$ on a
space with measured walls and measure definite kernels on $\Gamma$ can have at
most linear growth. By Example 3.1, resp. Example 3.2(ii), finitely generated
free groups, resp. the groups $\mathbb{Z}^{d}$ with $d\in\mathbb{N}$, admit
wall distances with linear growth. With this terminology, Theorem 1.4 now
reads as follows:

######  Theorem 4.1.

Let ${\mathcal{G}}$ be the Cayley graph of a finitely generated group $\Gamma$
and let $\rho\colon[0,\infty]\to[0,\infty]$ be a function with
$\\{\rho=0\\}=\\{0\\}$. Then each of the following conditions implies the next
one:

  1. (i)

There exists an invariant action of $\Gamma$ on a space with measured walls
with $\rho$-growth;

  2. (ii)

There exists $C>0$ such that for every $\alpha<1$, there exists a
$\Gamma$-invariant bond percolation $\mathbb{P}$ on ${\mathcal{G}}$ with
$\mathbb{E}[\deg_{\omega}(o)]>\alpha\deg(o)$ and with $(C,\rho)$-exponential
decay;

  3. (iii)

There exists a measure definite function on $\Gamma$ which admits $\rho$ as a
compression function.

Moreover, if $\Gamma$ is amenable or if the measure definite function in (iii)
can be assumed to be the square-root of a conditionally negative definite
function, then all three conditions are equivalent.

###### Remark 1

Let us comment on the question of equivalence between the conditions in
Theorem 4.1. The square-root of a conditionally negative definite kernel is
measure definite by [52, Proposition 1.4]. As described in Section 3.3 above,
it is also the wall distance associated to an invariant action of $\Gamma$ on
a space with measured walls [52, 11]. Hence under this additional assumption,
condition (iii) implies condition (i). Under the additional assumption that
the group is amenable, the fact that every measure definite function is the
wall distance associated to an invariant action on a space with measured walls
was obtained by de Cornulier, Stalder and Valette [16, Proposition 2.8]. We do
not know whether this fact (which would imply equivalence between the
conditions in Theorem 4.1) holds in general and refer to [10, 16] for more
details on this question.

###### Remark 2

Condition (i) in Theorem 4.1 is related to the equivariant $L^{1}$-compression
exponent $\alpha_{1}^{\scriptscriptstyle{\\#}}(\Gamma)$ of the group $\Gamma$,
which is defined to be the supremum over all $\alpha\in[0,1]$, such that there
exists an isometric action of $\Gamma$ on an $L^{1}$-space $E$ together with a
$\Gamma$-equivariant map $f\colon\Gamma\to E$ with

$\|f(g)-f(h)\|_{E}\geq|g^{-1}h|^{\alpha}\quad\mbox{for all} \,\,g,h\in\Gamma.$

For background on this exponent and related quantities we refer the reader to
[25, 48]. As observed in [16, Section 7], it follows from the results in [10]
that $\|f(g)-f(h)\|_{E}$ is a wall distance associated to an invariant action
of $\Gamma$ on a space with measured walls. Thus
$\alpha_{1}^{\scriptscriptstyle{\\#}}(\Gamma)$ is also the supremum over all
$\alpha\in[0,1]$ such that there exists an invariant action of $\Gamma$ on a
space with measured walls with $\rho_{\alpha}$-growth, where
$\rho_{\alpha}(t)=t^{\alpha}$. When $\rho=\rho_{\alpha}$ in Theorem 4.1,
condition (i) is therefore equivalent to the statement
$\alpha_{1}^{\scriptscriptstyle{\\#}}(\Gamma)\geq\alpha$.

We now turn to the

Proof of Theorem 4.1. (i) $\Rightarrow$ (ii): Suppose that $\Gamma$ has an
invariant action on a space with measured walls
$(X,{\mathcal{W}},{\mathcal{B}},\mu)$ with $\rho$-growth, i.e. for some
$x_{0}\in X$ we have that

$w(x_{0},gx_{0})\geq\rho(|g|)\quad\mbox{for all}\,\,g\in\Gamma.$

Note that

$C\coloneqq\max_{g\sim o}\,w(x_{0},gx_{0})\in(0,\infty).$

Now let $p\in(0,1)$. By Lemma 3.6, there exists a $\Gamma$-invariant bond
percolation $\mathbb{P}$ on ${\mathcal{G}}$ such that

$\mathbb{E}[\deg_{\omega}(o)]\geq{\rm e}^{-(1-p)C}\deg(o)$

and

$\exp\bigl{(}-(1-p)C|g|\bigr{)}\leq\tau(o,g)\leq\exp\bigl{(}-(1-p)w(x_{0},gx_{0})\bigr{)}\quad\text{for
every }g\in\Gamma.\vspace{1mm}$

Define $\beta_{p}\coloneqq(1-p)C>0$ and $\gamma_{p} \coloneqq(1-p)>0$. It
follows that

${\rm e}^{-\beta_{p}|g|}\leq\tau(o,g)\leq{\rm
e}^{-\gamma_{p}\rho(|g|)}\quad\mbox{for all}\,\,g\in\Gamma,$

and $\beta_{p}/\gamma_{p}=C$. Finally note that

$\lim_{p\to 1}{\rm e}^{-(1-p)C}=1.$

Therefore choosing $p$ close enough to $1$ yields a percolation with the
desired properties.

(ii) $\Rightarrow$ (iii): The proof is reminiscent of the proof of
Schoenberg’s Theorem in [6, Theorem C.3.2]. The main difference is that we are
working with two-point functions instead of merely positive definite
functions; we can therefore combine the proof idea with results about measured
definite kernels due to Chatterji, Drutu and Haglund [10] to actually produce
a measure definite function instead of a merely conditionally negative
definite one.

To start off, note that our assumption implies that there is a constant $C>0$
and a sequence $(\mathbb{P}_{n})_{n=1}^{\infty}$ of $\Gamma$-invariant bond
percolations such that
$\lim_{n\to\infty}\mathbb{E}_{n}[\deg_{\omega}(o)]=\deg(o)$ and such that

${\rm
e}^{-\beta_{n}|g|}\leq\tau_{n}(o,g)\coloneqq\mathbb{P}_{n}\big{[}o\leftrightarrow
g\big{]}\leq{\rm e}^{-\gamma_{n}\rho(|g|)}\quad\mbox{for all}\,\,g\in\Gamma,$

where $\beta_{n},\gamma_{n}>0$ such that $\beta_{n}/\gamma_{n}\leq C$. Also
recall that $\lim_{n\to\infty}\mathbb{E}_{n}[\deg_{\omega}(o)]=\deg(o)$
implies that the functions $\tau_{n}$ converge pointwise to $1$ and therefore
$\gamma_{n}\to 0$, as well as $\beta_{n}\to 0$, as $n\to\infty$. For each
$n\in\mathbb{N}$, define a function

$\psi_{n}\colon\Gamma\to[0,\infty)\,,\,\psi_{n}(g)\coloneqq\frac{1-\tau_{n}(o,g)}{\gamma_{n}}.$
(4.1)

Note that $\psi_{n}$ is conditionally negative definite [6, Proposition
C.2.4]. Now for every $g\in\Gamma$, we have that

$\displaystyle\limsup_{n\to\infty}\ \psi_{n}(g)=\limsup_{n\to\infty}\
\frac{1-\tau_{n}(o,g)}{\gamma_{n}}$ $\displaystyle\leq\limsup_{n\to\infty}\
\frac{1-{\rm e}^{-\beta_{n}|g|}}{\gamma_{n}}$
$\displaystyle=\limsup_{n\to\infty}\ \frac{\beta_{n}}{\gamma_{n}}\frac{1-{\rm
e}^{-\beta_{n}|g|}}{\beta_{n}}$ $\displaystyle\leq C|g|.$

By going over to a subsequence if necessary, we may assume that
$\lim_{n\to\infty}\psi_{n}(g)$ exists. Since $\Gamma$ is countable, we may go
over to another subsequence such that the limit exists for every $g\in\Gamma$.
Therefore we may assume that the sequence of percolations
$(\mathbb{P}_{n})_{n=1}^{\infty}$ is such that the corresponding sequence
$(\psi_{n})_{n=1}^{\infty}$ defined above converges pointwise. Define

$\psi\colon\Gamma\to[0,\infty)\,,\,\psi(g)\coloneqq\lim_{n\to\infty}\psi_{n}(g).$

Note that $\psi$ is conditionally negative definite as a pointwise limit of
conditionally negative definite functions [6, Proposition C.2.4] and satisfies

$\psi(g)=\lim_{n\to\infty}\psi_{n}(g)\geq\liminf_{n\to\infty}\ \frac{1-{\rm
e}^{-\gamma_{n}\rho(|g|)}}{\gamma_{n}}=\rho(|g|)\quad\mbox{for
all}\,\,g\in\Gamma.$

It remains to show that $\psi$ is in fact measure definite. By [52,
Proposition 1.3], the class of measure definite kernels on $\Gamma$ is closed
under pointwise convergence. Hence it suffices to show that $\psi_{n}$ is
measure definite for all $n\in\mathbb{N}$.

Verifying that $\psi_{n}$ is measure definite: We use the following
characterization of measure definite kernels defined on arbitrary sets due to
Chatterji, Drutu and Haglund [10]. Let $X$ be a set. A map $k:X\times
X\to[0,\infty)$ is called a measure definite kernel if there exists a measure
space $(\Omega,{\mathcal{B}},\mu)$ and a map $S\colon X\to{\mathcal{B}}$,
$x\mapsto S_{x}$ such that $k(x,y)=\mu(S_{x}\Delta S_{y})$. In the terminology
of [10, Section 6], $k$ is called a kernel if $k(x,x)=0$ and $k(x,y)=k(y,x)$
for all $x,y\in X$. The characterization obtained in [10, Corollary 6.17]
states that a kernel $k$ on a set $X$ is measure definite if and only if the
following two conditions hold:

  1. (a)

$k(\cdot,\cdot)$ satisfies the triangle inequality.

  2. (b)

For every finite subset $F\subset X$,

$k\big{|}_{F\times F}(\cdot,\cdot)=\sum_{S\subset
F}\lambda_{S}\delta_{S}(\cdot,\cdot)$

for some $\lambda_{S}\geq 0$ and

$\displaystyle\delta_{S}(x,y)=\begin{cases}1\quad&\mbox{if}\quad
k(x,y)>0\,\,\mbox{and}\,\,\big{|}S\cap\\{x,y\\}\big{|}=1\\\
0\quad&\mbox{else.}\end{cases}$

To see that each $\psi_{n}$ defined in (4.1) is in fact measure definite, we
now show the following more general statement:

######  Lemma 4.2.

For any bond percolation $\mathbb{P}$ on ${\mathcal{G}}$, the map

$k\colon\Gamma\times\Gamma\to[0,1]\,,\,k(g,h)\coloneqq\mathbb{P}\big{[}g\not\leftrightarrow
h\big{]}=1-\tau(g,h)$

is a measure definite kernel.

###### Proof.

First observe that $k(g,g)=0$ and $k(g,h)=k(h,g)$ for all $g,h\in\Gamma$. Thus
$k$ is a kernel in the sense of [10]. It therefore suffices to verify
conditions (a) and (b) stated above.

Verifying condition (a) for $k$. Let us fix $g,h,\gamma\in\Gamma$. Then

$\displaystyle k(g,h)$ $\displaystyle=\mathbb{P}\big{[}g\not\leftrightarrow
h\big{]}$
$\displaystyle\leq\mathbb{P}\bigl{[}\bigl{(}\\{g\leftrightarrow\gamma\\}\cap\\{\gamma\leftrightarrow
h\\}\bigr{)}^{c}\bigr{]}$
$\displaystyle=\mathbb{P}\bigl{[}\\{g\not\leftrightarrow\gamma\\}\cup\\{\gamma\not\leftrightarrow
h\\}\bigr{]}$
$\displaystyle\leq\mathbb{P}\bigl{[}g\not\leftrightarrow\gamma\bigr{]}+\mathbb{P}\bigl{[}\gamma\not\leftrightarrow
h\bigr{]}$ $\displaystyle=k(g,\gamma)+k(\gamma,h).$

Verifying condition (b) for $k$. Let us fix a finite subset $F$ of $\Gamma$
and write $F=\\{g_{1},\ldots,g_{N}\\}$. Given the configuration $\omega$ of
$\mathbb{P}$, write $C(g)$ for the cluster of $g$ viewed as a subset of
$\Gamma$. We define $C_{1}\coloneqq C(g_{1})\cap F$ and, iteratively for
$i\in\\{2,\ldots,N\\}$,

$\displaystyle C_{i}\coloneqq\begin{cases}C(g_{i})\cap F\quad&\mbox{if}\quad
g_{i}\notin\bigsqcup_{j=1}^{i-1}C_{j}\\\
\emptyset\quad&\mbox{else.}\end{cases}$

Here ‘$\bigsqcup$’ denotes disjoint union. We obtain a finite sequence
$C_{1},\ldots,C_{N}$ of random, pairwise disjoint subsets of $F$ which are
either empty or are the restriction of one of the clusters of $\omega$ to the
set $F$. Let $\mu_{i}$ denote the law of $C_{i}$ viewed as a probability
measure on $\\{0,1\\}^{F}$ and let
$S_{g}=\\{\xi\in\\{0,1\\}^{F}\colon\xi(g)=1\\}$ for $g\in F$. It follows
immediately from the definition of $C_{1},\ldots,C_{N}$ that for all pairs
$g,h\in F$, the vertices $g$ and $h$ are not connected in $\omega$ if and only
if there exists a unique $i\in\\{1,\ldots,N\\}$ such that $C_{i}$ contains $g$
and does not contain $h$ (the $C_{i}$ which coincides with the restriction of
$C(g)$ to $F$). It follows that, for all $g,h\in F$,

$\displaystyle k(g,h)$ $\displaystyle=\mathbb{P}\big{[}g\not\leftrightarrow
h\big{]}=\mathbb{P}\Big{[}\,g\in C_{i}\,\,\text{and}\,\,h\notin
C_{i}\,\,\mbox{for a unique}\,\,i\in\\{1,\ldots,N\\}\,\Big{]}$
$\displaystyle=\mathbb{P}\bigg{[}\,\bigsqcup_{i=1}^{N}\\{g\in C_{i},h\notin
C_{i}\\}\,\biggr{]}$ $\displaystyle=\sum_{i=1}^{N}\mathbb{P}\big{[}g\in
C_{i},h\notin C_{i}\big{]}$
$\displaystyle=\sum_{i=1}^{N}\mu_{i}(S_{g}\setminus S_{h}).$ (4.2)

On the other hand, $g$ and $h$ are not connected in $\omega$ if and only if
there exists a unique $i\in\\{1,\ldots,N\\}$ such that $C_{i}$ contains $h$
and does not contain $g$. By the same argument we obtain that

$k(g,h)=\sum_{i=1}^{N}\mu_{i}(S_{h}\setminus S_{g})\quad\mbox{for all}
\,\,g,h\in F.$ (4.3)

Summing up Equations (4.2) and (4.3), we obtain that

$k(g,h)=\frac{1}{2}\sum_{i=1}^{N}\mu_{i}(S_{g}\Delta S_{h})\quad\mbox{for all}
\,\,g,h\in F.$ (4.4)

Now for every $i\in\\{1,\ldots,N\\}$, the map

$k_{i}\colon F\times F\to[0,1]\,,\,k_{i}(g,h)\coloneqq\mu_{i}(S_{g}\Delta
S_{h})$ (4.5)

is a measure definite kernel by definition.1††1The definition of $k_{i}$ in
(4.5) justifies why we work with a kernel on a general set, and not
necessarily on a group. By [10, Corollary 6.17] stated above, it may therefore
be written as $\sum_{S\subset F}\lambda_{S}\delta_{S}$ for some
$\lambda_{S}\geq 0$ (which can also be seen using elementary arguments). Since
$k_{|F\times F}$ is a non-negative linear combination of the kernels $k_{i}$
by Equation (4.4), it follows that $k$ can also be written in the desired
form. The proof of Lemma 4.2 and therefore that of the remaining implication
(ii) $\Rightarrow$ (iii) in Theorem 4.1, is thus complete. ∎

As remarked earlier, the equivalence (i)$\Leftrightarrow$ (ii)
$\Leftrightarrow$ (iii) under the additional assumptions is shown as follows:
The implication (iii)$\Rightarrow$(i) follows from from [16, Proposition 2.8]
under the amenability assumption, and from [11, 52] if the measure definite
function is assumed to be the square root a conditionally negative definite
function. Thus, the proof of Theorem 4.1 is complete. ∎

##  5\. Characterization of Kazhdan’s property (T)

The main theme of this section is the appearance of thresholds such that
invariant bond percolations with expected degree exceeding the threshold do
not exhibit decay of the two-point functions. We first provide such a
threshold for groups which do not have the Haagerup property, see Section 5.1.
Developing these ideas further, we then derive the characterizations of
property (T) and relative property (T) in Section 5.2 and Section 5.3
respectively. The consequences of these thresholds for the uniqueness
parameter in Bernoulli bond percolation are explained in the general set-up of
a group with relative property (T) at the end of Section 5.3.

###  5.1. Weak Kazhdan property and a quantitative converse to Theorem 1.2.

In this subsection, we consider groups which do not have the Haagerup
property. Following Glasner [22, Definition 13.16], we call such groups weakly
Kazhdan or say they have property $(T_{w})$. Consider a Cayley graph
${\mathcal{G}}$ of a group $\Gamma$ with property $(T_{w})$, then Theorem 3.5
implies the existence of a threshold
$\alpha^{*}_{w}\coloneqq\alpha^{*}_{w}({\mathcal{G}})<1$ such that whenever
$\mathbb{P}$ is a $\Gamma$-invariant bond percolation with
$\mathbb{E}[\deg_{\omega}(o)]>\alpha^{*}_{w}\deg(o)$, then $\tau(o,\cdot)$
does not vanish at infinity. We now give an upper bound on $\alpha^{*}_{w}$ in
terms of another family of parameters associated with weakly Kazhdan groups.

Since $\Gamma$ does not have the Haagerup property, there exist
$\varepsilon>0$ and $K\subset\Gamma$ finite with the following property:

$\displaystyle\text{every normalized positive definite function}\ \varphi\
\text{on}\ \Gamma\ \text{satisfying}$ (5.1) $\displaystyle\qquad\min_{g\in K}
\varphi(g)\geq 1-\varepsilon\ \text{does not vanish at infinity}.$

Define the parameters

$\displaystyle d_{K,{\mathcal{G}}}\coloneqq\max\big{\\{}d(o,g)\colon g\in
K\big{\\}},$ $\displaystyle\alpha_{\varepsilon,K,{\mathcal{G}}}\coloneqq
1-\frac{\varepsilon}{\deg(o)d_{K,{\mathcal{G}}}},\qquad\mbox{and}$
$\displaystyle\kappa_{w}({\mathcal{G}})\coloneqq\inf\bigg{\\{}\alpha_{\varepsilon,K,{\mathcal{G}}}\colon\mbox{the
pair $(\varepsilon,K)$ satisfies \eqref{eq-WeakKazhdan}}\bigg{\\}}.$

######  Proposition 5.1.

Let ${\mathcal{G}}$ be the Cayley graph of a finitely generated group $\Gamma$
which has property $(T_{w})$. Then
$\alpha^{*}_{w}({\mathcal{G}})\leq\kappa_{w}({\mathcal{G}})$.

###### Proof.

Let $\varepsilon>0$ and $K\subset\Gamma$ finite such that every normalized
positive definite function $\varphi\colon\Gamma\to[0,1]$ which satisfies

$\min_{g\in K} \varphi(g)\geq 1-\varepsilon,$ (5.2)

does not vanish at infinity. Let $\mathbb{P}$ be a $\Gamma$-invariant bond
percolation on ${\mathcal{G}}=(V,E)$ with two-point function $\tau$ and define

$m\coloneqq\inf_{e\in E}\mathbb{P}[e\in\omega]=\min_{o\sim g}\tau(o,g).$

For every $g\in\Gamma$, we then have the lower bound $\tau(o,g)\geq
1-d(o,g)(1-m)$: To see this, let $(o,g)$ denote a shortest path of edges
joining $o$ to $g$, then

$\displaystyle\tau(o,g)$ $\displaystyle=\mathbb{P}\big{[}o\leftrightarrow
g\big{]}\geq\mathbb{P}\big{[}(o,g)\subset\omega\big{]}$
$\displaystyle=\mathbb{P}\bigl{[}e\in\omega\text{ for all }e\in(o,g)\bigr{]}$
$\displaystyle\geq 1-\sum_{e\in(o,g)}\mathbb{P}[e\notin\omega]$
$\displaystyle\geq 1-d(o,g)(1-m).$

In particular, it follows that

$\min_{g\in K}\tau(o,g)\geq 1-d_{K,{\mathcal{G}}}(1-m).$

Note that if $m>1-\varepsilon/d_{K,{\mathcal{G}}}$, then $\tau(o,\cdot)$
satisfies Inequality (5.2) and thus does not vanish at infinity. It remains to
show that
$\mathbb{E}[\deg_{\omega}(o)]>\alpha_{\varepsilon,K,{\mathcal{G}}}\deg(o)$
implies $m>1-\varepsilon/d_{K,{\mathcal{G}}}$. To see this, we rewrite

$\mathbb{E}[\deg_{\omega}(o)]=\sum_{o\sim g}\tau(o,g)\leq m+(\deg(o)-1).$

From the lower bound on the expected degree we therefore obtain

$\displaystyle m$
$\displaystyle>\alpha_{\varepsilon,K,{\mathcal{G}}}\deg(o)-\deg(o)+1$
$\displaystyle=\biggl{(}1-\frac{\varepsilon}{\deg(o)d_{K,{\mathcal{G}}}}\biggr{)}\deg(o)-\deg(o)+1=1-\frac{\varepsilon}{d_{K,{\mathcal{G}}}},$

as desired. ∎

###### Remark 3

The proof of Proposition 5.1 shows that the lower bound on the expected degree
may be replaced by the assumption that each individual edge is present with
probability strictly above
$\inf_{(\varepsilon,K)}1-\varepsilon/d_{K,{\mathcal{G}}}$, where the infimum
is taken over all pairs $(\varepsilon,K)$ satisfying (5.1). We chose to give a
threshold for the expected degree to keep the exposition consistent.

###### Remark 4

In the setting of site percolation, Corollary 3.8 implies the existence of a
possibly different threshold $\alpha^{*}_{w,{\rm
site}}\coloneqq\alpha^{*}_{w,{\rm site}}({\mathcal{G}})<1$ such that whenever
$\mathbb{P}$ is a $\Gamma$-invariant site percolation satisfying
$\mathbb{P}[o\in\omega]>\alpha^{*}_{w,{\rm site}}$, then $\tau(o,\cdot)$ does
not vanish at infinity. It follows from essentially the same argument as in
the proof of Proposition 5.1 above that $\alpha^{*}_{w,{\rm
site}}\leq\inf_{(\varepsilon,K)}1-\varepsilon/d_{K,{\mathcal{G}}}$ with the
infimum taken over all pairs $(\varepsilon,K)$ satisfying (5.1).

At this point, let us make the important observation that a two-point function
$\tau(o,\cdot)$ which fails to vanish at infinity, can do so in two distinct
ways, as described by Lyons and Schramm in [44]: It is said to exhibit
connectivity decay if

$\inf_{g,h\in\Gamma}\tau(g,h)=0,$

otherwise it is said to exhibit long-range order. If $\tau$ fails to vanish at
infinity, connectivity decay is a particularly interesting case as it implies
that $\tau(o,g)$ decays to zero in at least one direction $g\to\infty$, while
it is bounded away from zero in some other direction; see [44, Remark 1.3] for
an example of this phenomenon in Bernoulli bond percolation. As pointed out by
Lyons [41], this behavior reveals a lack of spherical symmetry in Cayley
graphs (where a Cayley graph is called spherically symmetric if for all
vertices with $d(o,g)=d(o,h)$, there exists an automorphism which fixes $o$
and maps $g$ to $h$). For Cayley graphs which satisfy a weak form of spherical
symmetry, Proposition 5.1 implies the following extension of a well-known
result about Kazhdan groups [44, Corollary 6.6] to the weak Kazhdan case.

######  Corollary 5.2.

Let ${\mathcal{G}}$ be the Cayley graph of a finitely generated group $\Gamma$
which has property $(T_{w})$. Suppose that ${\mathcal{G}}$ satisfies the
following spherical symmetry assumption:

  * (Sym)

For every $p\in(0,1)$, the two-point function $\tau_{p}(\cdot,\cdot)$ of
$p$-Bernoulli bond percolation vanishes at infinity if and only if it exhibits
connectivity decay.

Then $p_{u}({\mathcal{G}})<1$.

###### Proof.

By Proposition 5.1, $\tau_{p}(o,\cdot)$ does not vanish at infinity for $p$
sufficiently close to $1$. Assumption (Sym) implies that it has long-range
order, i.e. is bounded away from zero. By [44, Theorem 4.1], this implies that
the corresponding $p$-Bernoulli bond percolation has a unique infinite cluster
a.s. ∎

###### Remark 5

In the setting of Corollary 5.2, it follows from Remark 3 that
$p_{u}({\mathcal{G}})\leq\inf_{(\varepsilon,K)}1-\varepsilon/d_{K,{\mathcal{G}}}$
with the infimum taken over all pairs $(\varepsilon,K)$ with the property
described after Equation (5.1).

###  5.2. Kazhdan’s property (T).

We start by recalling the definition of Kazhdan’s property (T): If $(\pi,H)$
is a unitary representation of $\Gamma$, $\varphi\in H$ is an invariant vector
if $\pi(s)\varphi=\varphi$ for every $s\in\Gamma$. Let $\varepsilon>0$ and
$K\subset\Gamma$ finite, then a unit vector $\varphi\in H$ is called
$(\varepsilon,K)$-invariant if

$\sup_{s\in K}\|\pi(s)\varphi-\varphi\|<\varepsilon.$

The pair $(\varepsilon,K)$ is called a Kazhdan pair for $\Gamma$ if every
unitary representation of $\Gamma$ which has an $(\varepsilon,K)$-invariant
vector also has a non-zero invariant vector. The group $\Gamma$ has property
(T) if there exist $\varepsilon>0$ and a finite set $K\subset\Gamma$ such that
$(\varepsilon,K)$ is a Kazhdan pair. The goal of this subsection is to prove
the following characterization of property (T) in terms of invariant
percolations.

######  Theorem 5.3.

Let ${\mathcal{G}}$ be the Cayley graph of a finitely generated group
$\Gamma$. Then $\Gamma$ has 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 $\tau$ is
bounded away from zero, i.e.

$\inf_{g,h\in\Gamma}\tau(g,h)>0.$

###### Remark 6

The direct implication of Theorem 5.3 was known. It can be obtained by the
argument described by Lyons and Schramm in [44, Remark 6.7]. This consequence
of property (T), combined with the result that Bernoulli bond percolation
produces a unique infinite cluster if and only if the two-point function is
bounded away from zero [44, Theorem 4.1], was applied by Lyons and Schramm
[44, Corollary 6.6] to show that $p_{u}<1$ for Cayley graphs of Kazhdan groups
and that there is non-uniqueness at $p_{u}$. A quantitative version of the
first result involving Kazhdan pairs was proved by Ionana, Kechris and Tsankov
[32, Theorem 5.6].

Our proof makes use of well-known characterizations of Kazhdan groups in terms
of positive and conditionally negative definite functions, which are similar
to the characterization of the Haagerup property stated in Theorem 2.1. We
recall here the version for finitely generated groups.

######  Theorem 5.4 ([17, 26, 3]).

Let $\Gamma$ be a finitely generated group. Then the following are equivalent:

  1. (i)

$\Gamma$ has property (T);

  2. (ii)

every conditionally negative definite function on $\Gamma$ is bounded;

  3. (iii)

every sequence of normalized, positive definite functions on $\Gamma$ which
which converges pointwise to $1$, converges to $1$ uniformly on $\Gamma$.

Theorem 5.3 suffices for the purpose of establishing Theorem 5.4. However, a
quantitative version involving an estimate for the threshold $\alpha^{*}$ can
be proved following the strategy in [32]. For the convenience of the reader,
we include a proof of this quantitative version of the direct implication,
which is based on [32, Section 5]. There is a minor difference, namely that we
consider an a-priori fixed Cayley graph of $\Gamma$ and not one built using
the set $K$, thereby getting slightly different constants. The proof requires
the following lemma, which follows from results of Deutsch and Robertson [18].

######  Lemma 5.5 ([32, Proposition 5.3]).

Let $\Gamma$ be a group with property (T), let $(\varepsilon,K)$ be a Kazhdan
pair for $\Gamma$, let $\delta>0$ and let $\varphi$ be a normalized, positive
definite function. Then

$\min_{g\in K}\,{\rm Re}(\varphi(g))\geq
1-\frac{\delta^{2}\varepsilon^{2}}{2},$

where ${\rm Re}(z)$ for $z\in\mathbb{C}$ denotes the real part, implies that

$\inf_{g\in\Gamma}\,{\rm Re}(\varphi(g))\geq 1-2\delta^{2}.$

###### Proof of Theorem 5.3.

Suppose first that $\Gamma$ has property (T) and let $(\varepsilon,K)$ be a
Kazhdan pair for $\Gamma$. As previously, define
$d_{K,{\mathcal{G}}}\coloneqq\max\\{d(o,g)\colon g\in K\\}.$ Fix
$\delta\in(0,1/\sqrt{2})$. It follows exactly as in the proof of Proposition
5.1 that for every $\Gamma$-invariant bond percolation $\mathbb{P}$ on
${\mathcal{G}}$ such that

$\mathbb{E}[\deg_{\omega}(o)]>\biggl{(}1-\frac{\delta^{2}\varepsilon^{2}}{2\deg(o)d_{K,{\mathcal{G}}}}\biggr{)}\deg(o)=\deg(o)-\frac{\delta^{2}\varepsilon^{2}}{2d_{K,{\mathcal{G}}}},$

we have that

$\min_{g\in K}\tau(o,g)\geq 1-\frac{\delta^{2}\varepsilon^{2}}{2}.$

For such a percolation, Lemma 5.5 then yields that

$\inf_{g,h}\tau(g,h)=\inf_{g\in\Gamma}\tau(o,g)>1-2\delta^{2}>0.$

This shows that $\alpha^{*}$ can be taken to be

$1-\frac{\delta^{2}\varepsilon^{2}}{2\deg(o)d_{K,{\mathcal{G}}}}<1,$

as claimed. Letting $\delta\to 1/\sqrt{2}$, we in particular see that the
optimal threshold is less than or equal to

$\inf\biggl{\\{}1-\frac{\varepsilon^{2}}{4\deg(o)d_{K,{\mathcal{G}}}}\colon(\varepsilon,K)\,\,\mbox{is
a Kazhdan pair for}\,\,\Gamma\biggr{\\}}.$ (5.3)

This concludes the proof of the direct implication.

We now prove the converse implication. To that end, assume that there exists a
threshold $\alpha^{*}<1$ such that for every $\Gamma$-invariant bond
percolation $\mathbb{P}$ on ${\mathcal{G}}$ with
$\mathbb{E}[\deg_{\omega}(o)]>\alpha^{*}\deg(o)$, the two-point function is
bounded away from zero. By Theorem 5.4, $\Gamma$ has property (T) if and only
if every conditionally negative definite function $\psi$ on $\Gamma$ is
bounded. Let $\psi$ be any conditionally negative definite function. As in the
proof of Theorem 3.5, we may construct $\Gamma$-invariant bond percolations
$\mathbf{P}_{p}$ on ${\mathcal{G}}$, for $p\in(0,1)$, such that

$\lim_{p\to 1}\mathbf{E}_{p}[\deg_{\omega}(o)]=\deg(o)$

and such that

$\tau_{p}(o,g) \leq{\rm e}^{-(1-p)\sqrt{\psi(g)}}\quad\mbox{for
every}\,\,g\in\Gamma,p\in(0,1).$

Then, for a fixed $p$ sufficiently close to $1$, the two-point function
$\tau_{p}$ of $\mathbf{P}_{p}$ is bounded away from zero. Hence $\sqrt{\psi}$,
and thus $\psi$, must be bounded. It follows that $\Gamma$ has property (T). ∎

###### Remark 7

In the proof of the direct implication, we again chose to give a threshold for
the expected degree to keep the exposition consistent. As pointed out in
Remark 3, the lower bound on the expected degree may be replaced by the
assumption that each individual edge is present with probability at least
$1-(\delta^{2}\varepsilon^{2})/(2d_{K,{\mathcal{G}}})$. If $S$ denotes the
generating set used to define ${\mathcal{G}}$, then $d_{S,{\mathcal{G}}}=1$
and it is a standard fact that $(\varepsilon_{S},S)$ is a Kazhdan pair for
$\Gamma$ for a suitable $\varepsilon_{S}>0$. Letting $\delta\to 1/\sqrt{2}$,
it follows that for every $\Gamma$-invariant bond percolation which contains
each individual edge with probability at least $1-\varepsilon_{S}^{2}/4$, the
two-point function is bounded away from zero. This is the bound obtained in
[32, Corollary 5.4 (ii)]. Moreover, we conclude that the optimal threshold for
the expected degree is less than or equal to
$1-\varepsilon_{S}^{2}/(4\deg(o))$.

###  5.3. Relative property (T) and non-uniqueness at $p_{u}$.

In this section, we find an extension of Theorem 5.3 to groups which have
relative property (T) and provide an application to the question of uniqueness
in Bernoulli bond percolation. To start off, recall the definition of relative
property (T): Let $\Gamma$ be a finitely generated group and let $H$ be a
subgroup. The pair $(\Gamma,H)$ has relative property (T) if every unitary
representation of $\Gamma$ on a Hilbert space which almost has invariant
vectors, has a non-zero $H$-invariant vector, see [6, Definition 1.4.3]. The
notion of relative property (T) for a pair was considered implicitly in
Kazhdan’s work [37] and explicitly by Margulis [45]. A well-known example is
the pair $(\mathbb{Z}^{2}\rtimes{\rm SL}_{2}(\mathbb{Z}),\mathbb{Z}^{2})$,
which has relative property (T) but $\mathbb{Z}^{2}\rtimes{\rm
SL}_{2}(\mathbb{Z})$ does not have property (T). Many characterizations of
property (T) have analogous formulations for relative property (T), see e.g.
[4, 34, 14]. In particular, the pair $(\Gamma,H)$ has relative property (T) if
and only if every sequence of positive definite functions on $\Gamma$ which
converges pointwise to $1$, converges to $1$ uniformly on $H$, see e.g. [14,
Theorem 1.1]. As observed by Jaudon [33], the methods of [52, 11] also yield
the following characterization of relative property (T) in terms of invariant
actions on spaces with measured walls.

######  Theorem 5.6 ([33, Theorem 3.3.6]).

Let $\Gamma$ be a countable group and let $H$ be a subgroup. Then the pair
$(\Gamma,H)$ has relative property (T) if and only if for every invariant
action of $\Gamma$ on a space with measured walls
$(X,{\mathcal{W}},{\mathcal{B}},\mu)$, the $H$-orbits are bounded in the sense
that for every $x_{0}\in X$

$\sup_{h\in H}w(x_{0},hx_{0})=\sup_{h\in
H}\mu({\mathcal{W}}(x_{0},hx_{0}))<\infty.$

We can now extend Theorem 5.3 to the situation of relative property (T).

######  Theorem 5.7.

Let $\Gamma$ be a finitely generated group, let $H$ be a subgroup and let
${\mathcal{G}}$ be a Cayley graph of $\Gamma$. Then the pair $(\Gamma,H)$ has
relative 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
$\tau$ is bounded away from zero on each left $H$-coset, i.e. for every
$g\in\Gamma$

$\inf_{h\in H}\tau(g,gh)>0.$

###### Proof.

Suppose that 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 $\tau$ is
bounded away from zero on each left $H$-coset. We show that the pair
$(\Gamma,H)$ has relative property (T) by verifying that every invariant
action of $\Gamma$ on a space with measured walls has bounded $H$-orbits and
then applying Theorem 5.6 above. To that end, suppose that $\Gamma$ has an
invariant action on a space with measured walls
$(X,{\mathcal{W}},{\mathcal{B}},\mu)$ and let $x_{0}\in X$. By Lemma 3.6, for
suitable $p\in(0,1)$, there exists a $\Gamma$-invariant bond percolation
$\mathbb{P}$ on ${\mathcal{G}}$ such that
$\mathbb{E}[\deg_{\omega}(o)]>\alpha^{*}\deg(o)$ and

$\tau(o,g)\leq\exp\bigl{(}-(1-p)w(x_{0},gx_{0}) \bigr{)}\quad\mbox{for
every}\,\,g\in\Gamma.$ (5.4)

It follows from our assumption that

$\inf_{h\in H}\tau(o,h)>0.$ (5.5)

Combining Inequality (5.4) and Inequality (5.5) shows that

$\sup_{h\in H}w(x_{0},hx_{0})<\infty,$

i.e. the $H$-orbit of $x_{0}$ is bounded. Since $x_{0}\in X$ was arbitrary,
all $H$-orbits are bounded and it follows from Theorem 5.6 that the pair
$(\Gamma,H)$ has relative property (T).

Conversely, suppose there exists a sequence $\mathbb{P}_{n}$ of
$\Gamma$-invariant bond percolations on ${\mathcal{G}}$ such that

$\lim_{n\to\infty}\mathbb{E}_{n}[\deg_{\omega}(o)]=\deg(o)$

and such that for every $n\in\mathbb{N}$ there exists $g_{n}\in\Gamma$ with

$\inf_{h\in H}\tau_{n}(g_{n},g_{n}h)=0.$

Then the sequence $\tau_{n}(o,\cdot)=\tau_{n}(g_{n},g_{n}\cdot)$ is a sequence
of positive definite functions converging pointwise to $1$, but not uniformly
on $H$. Therefore the pair $(\Gamma,H)$ does not have relative property (T). ∎

Recall that for Cayley graphs of groups with property (T), Bernoulli
percolation at the uniqueness threshold does not have a unique infinite
cluster. This is due to Peres, see Lyons and Schramm [44, Corollary 6.6]. The
fact that this also holds for groups $\Gamma$ which admit an infinite normal
subgroup $H$ such that the pair $(\Gamma,H)$ has relative property (T) has
appeared in the work of Gaboriau and Tucker-Drob [21]. Note that this in
particular implies $p_{u}<1$ for these Cayley graphs. We will give alternative
proofs of these results using Theorem 5.3 and Theorem 5.7. Due to the fact
that these proofs only use invariant percolations, we believe them to be of
interest and in this vein, we will include a proof of the strongest result.

######  Theorem 5.8.

Let $\Gamma$ be a finitely generated group and let $H$ be an infinite, normal
subgroup such that the pair $(\Gamma,H)$ has relative property (T). Let
${\mathcal{G}}$ be any Cayley graph of $\Gamma$. Then
$p_{u}({\mathcal{G}})$-Bernoulli bond percolation does not have a unique
infinite cluster a.s.

###### Proof.

Let $\omega_{p_{u}}$ denote the configuration of $p_{u}\coloneqq
p_{u}({\mathcal{G}})$-Bernoulli bond percolation on ${\mathcal{G}}$. Suppose
that there is a unique infinite cluster $C_{\infty}$ in $\omega_{p_{u}}$ a.s.
We use a construction of Benjamini, Lyons, Peres and Schramm [8]: For each
$g\in\Gamma$ let $C(g)$ denote the set of vertices in $C_{\infty}$ which are
closest to $g$ in the graph distance. Let $\gamma_{\varepsilon}$ denote the
configuration of an $\varepsilon$-Bernoulli bond percolation on
${\mathcal{G}}$ which is independent of $\omega_{p_{u}}$. Now define a bond
percolation $\xi_{\varepsilon}$ as follows: Let the edge $[g,h]$ be contained
in $\xi_{\varepsilon}$ if and only if
$\max\\{d(g,C_{\infty}),d(h,C_{\infty})\\}<1/\varepsilon$ holds and
additionally $C(g)$ and $C(h)$ are contained in the same connected component
of $\omega\setminus\gamma_{\varepsilon}$. It follows from the bounded
convergence theorem that $\xi_{\varepsilon}$ has marginals arbitrarily close
to $1$ when $\varepsilon>0$ is sufficiently small, see [8, Section 3]. Now
Theorem 5.7 implies that for some fixed, sufficiently small $\varepsilon>0$,
we have that $\tau_{\xi_{\varepsilon}}$ is bounded away from zero on (left)
$H$-cosets. We claim that there exists a constant $c>0$ such that

$c\,\tau_{\xi_{\varepsilon}}(g,h)\leq\tau_{\omega_{p_{u}}\setminus\gamma_{\varepsilon}}(g,h)\quad\mbox{for
all}\,\,g,h\in\Gamma.$ (5.6)

To see this, observe that $\omega_{p_{u}}\setminus\gamma_{\varepsilon}$ is a
Bernoulli bond percolation with parameter $(1-\varepsilon)p_{u}>0$. In
particular

$c_{1}\coloneqq\inf_{u\in\partial
B_{1/\varepsilon}(o)}\tau_{\omega_{p_{u}}\setminus\gamma_{\varepsilon}}(o,u)>0.$

By transitivity, for every $g\in\Gamma$,

$c_{1}=\inf_{u\in\partial
B_{1/\varepsilon}(o)}\tau_{\omega_{p_{u}}\setminus\gamma_{\varepsilon}}(g,u).$

Let $g,h\in\Gamma$, $u\in\partial B_{1/\varepsilon}(g)$ and $v\in\partial
B_{1/\varepsilon}(h)$. Applying the Harris-FKG-Inequality, we see that

$\displaystyle\tau_{\omega_{p_{u}}\setminus\gamma_{\varepsilon}}(g,h)$
$\displaystyle\geq\mathbb{P}\Bigl{(}g\xleftrightarrow{\omega_{p_{u}}\setminus\gamma_{\varepsilon}}u,u\xleftrightarrow{\omega_{p_{u}}\setminus\gamma_{\varepsilon}}v,v\xleftrightarrow{\omega_{p_{u}}\setminus\gamma_{\varepsilon}}y\Bigr{)}$
$\displaystyle\geq
c_{1}^{2}\tau_{\omega_{p_{u}}\setminus\gamma_{\varepsilon}}(u,v).$ (5.7)

Now by the definition of $\xi_{\varepsilon}$ we have that

$\Bigl{\\{}g\xleftrightarrow{\xi_{\varepsilon}}h
\Bigr{\\}}\subset\Bigl{\\{}B_{1/\varepsilon}(g)\xleftrightarrow{\omega_{p_{u}}\setminus\gamma_{\varepsilon}}B_{1/\varepsilon}(h)\Bigr{\\}}.$

Applying monotonicity, the union bound and Inequality (5.7), we obtain that

$\displaystyle\tau_{\xi_{\varepsilon}}(g,h)$
$\displaystyle\leq\mathbb{P}\Bigl{[}B_{1/\varepsilon}(g)\xleftrightarrow{\omega_{p_{u}}\setminus\gamma_{\varepsilon}}B_{1/\varepsilon}(h)\Bigr{]}$
$\displaystyle\leq\sum_{\begin{subarray}{c}u\in\partial
B_{1/\varepsilon}(g)\\\ v\in\partial
B_{1/\varepsilon}(h)\end{subarray}}\tau_{\omega_{p_{u}}\setminus\gamma_{\varepsilon}}(u,v)$
$\displaystyle\leq\frac{|\partial
B_{1/\varepsilon}(o)|^{2}}{c_{1}^{2}}\tau_{\omega_{p_{u}}\setminus\gamma_{\varepsilon}}(g,h).$

This proves the inequality (5.7) with the constant $c=(|\partial
B_{1/\varepsilon}(o)|/c_{1})^{2}$. In particular, the fact that
$\tau_{\xi_{\varepsilon}}$ is bounded away from zero on $H$-cosets implies
that the same holds for $\tau_{\omega_{p_{u}}\setminus\gamma_{\varepsilon}}$.
To finish the proof, we adapt the proof strategy of Hutchcroft and Pete in
[31] to our setting: First observe that the $H$-cosets define a partition of
the Cayley graph ${\mathcal{G}}$ into isomorphic subgraphs, say all isomorphic
to ${\mathcal{G}}_{H}$ which is infinite with transitive, unimodular and
closed automorphism subgroup $H$. Since $H$ is infinite normal, the graphs of
neighboring $H$-cosets are connected by infinitely many edges, see [31]. Now
the percolation which $\omega_{p_{u}}\setminus\gamma_{\varepsilon}$ induces on
the graph ${\mathcal{G}}_{H}$, is insertion tolerant, $H$-invariant and
ergodic. By [44, Theorem 4.1], there is a unique infinite cluster a.s. in each
$H$-coset. We can now apply sprinkling as in [31, Section 3]: For every
$\varepsilon^{\prime}>0$, adding an independent
$\varepsilon^{\prime}$-Bernoulli bond percolation to
$\omega_{p_{u}}\setminus\gamma_{\varepsilon}$ a.s. connects the clusters in
neighboring $H$-cosets. Hence the resulting $\Gamma$-invariant percolation has
a unique infinite cluster a.s. However, it is also
$\bigl{(}(1-\varepsilon)p_{u}+(1-(1-\varepsilon)p_{u})\varepsilon^{\prime}\bigr{)}$-Bernoulli
bond percolation. For $\varepsilon^{\prime}$ sufficiently small, this
contradicts the definition of $p_{u}$. Therefore $\omega_{p_{u}}$ cannot have
a unique infinite cluster and the proof of Theorem 5.8 is complete. ∎

##  6\. Spaces with measured walls for amenable groups

Recall that amenable groups have the Haagerup property [5]. In this section,
we show that under the additional assumption that $\Gamma$ is amenable, there
exists a natural space with measured walls which admits a proper, invariant
$\Gamma$-action, such that our constructive proof of Theorem 3.5 recovers the
corresponding statement of Theorem A stated in Section 1.

######  Proposition 6.1.

Let ${\mathcal{G}}$ be the Cayley graph of a finitely generated amenable group
$\Gamma$. Consider the proper, invariant action of $\Gamma$ on the space with
measured walls $(X,{\mathcal{W}},{\mathcal{B}},\mu)$ described in [11, Proof
of Theorem I(5)]. For $p\in(0,1)$, let $\eta_{p}$ be a Poisson process on
${\mathcal{W}}$ with intensity $(1-p)\mu$ and let $\omega_{p}$ denote the
associated invariant bond percolation on ${\mathcal{G}}$ as in the proof of
Theorem 3.5. Then $\omega_{p}$ has no infinite cluster.

###### Proof.

We first need to introduce the space with measured walls from [11]. To that
end, we recall a notion of product for spaces with measured walls from [11,
Definition 5]: Let
$(X_{n},{\mathcal{W}}_{n},{\mathcal{B}}_{n},\mu_{n})_{n=1}^{\infty}$ be a
sequence of spaces with measured walls. Distinguish a point $x_{n}^{0}\in
X_{n}$ for every $n\in\mathbb{N}$. Now define the set

$X\coloneqq\Biggl{\\{}x=(x_{n})_{n=1}^{\infty}\in\prod_{n=1}^{\infty}X_{n}\colon\sum_{n=1}^{\infty}w_{n}(x_{0}^{(n)},x_{n})<\infty\Biggr{\\}}.$

For each $n\in\mathbb{N}$, let $p_{n}\colon X\to X_{n}$ denote the projection
onto the $n$’th coordinate. Note that if $n\in\mathbb{N}$ and
$\\{A,A^{c}\\}\in{\mathcal{W}}_{n}$, then

$W(\\{A,A^{c}\\}) \coloneqq p_{n}^{-1}\bigl{(}\\{A,A^{c}\\}\bigr{)}=\Bigl{\\{}
\bigl{\\{} x\in X\colon x_{n}\in A\bigr{\\}},\bigl{\\{} x\in X\colon x_{n}\in
A^{c}\bigr{\\}}\Bigr{\\}}$

defines a wall on $X$. The (disjoint) union of such walls can be identified
with the coproduct of ${\mathcal{W}}_{n}$. Let ${\mathcal{B}}$ be the
$\sigma$-algebra generated by $p_{n}^{-1}({\mathcal{B}}_{n})$ and let $\mu$ be
the unique measure on ${\mathcal{W}}$ whose restriction to
$p_{n}^{-1}({\mathcal{B}}_{n})$ coincides with $\mu_{n}$. Then
$(X,{\mathcal{W}},{\mathcal{B}},\mu)$ is a space with measured walls which is
called the product of the pointed spaces
$(X_{n},{\mathcal{W}}_{n},{\mathcal{B}}_{n},\mu_{n},x_{n}^{0})$.

Now let $(K_{n})_{n=1}^{\infty}$ be a sequence of non-empty finite subsets of
$\Gamma$ increasing to $\Gamma$. Assuming amenability, there exists a Følner
sequence for $(K_{n})$, i.e. a sequence $(A_{n})_{n=1}^{\infty}$ of non-empty
finite subsets of $\Gamma$ such that, for every $n\in\mathbb{N}$ and $g\in
K_{n}$,

$\frac{|gA_{n}\Delta A_{n}|}{|A_{n}|}<2^{-n}.$

For each $n\in\mathbb{N}$, form the space with measured walls
$(\Gamma,{\mathcal{W}}_{n},{\mathcal{B}}_{n},\mu_{n})$, where the set of walls
is

${\mathcal{W}}_{n}=\Bigl{\\{}
\bigl{\\{}gA_{n}^{-1},(gA_{n}^{-1})^{c}\bigr{\\}}\colon g\in\Gamma
\Bigr{\\}},$

the $\sigma$-field ${\mathcal{B}}_{n}$ is the power set and $\mu$ is counting
measure normalized by $n/|A_{n}|$, see [11, Example 1]. Observe that the
corresponding wall distance is given by

$w_{n}(g,h)=\frac{n}{|A_{n}|}|gA_{n}\Delta hA_{n}|,$

which is clearly bounded by $n$. In order to obtain an action on a space with
unbounded wall distance we therefore need to pack up (a formulation due to
[11]) the sequence of spaces described above. To that end, distinguish the
point $x_{0}^{n}=o$ for each $n\in\mathbb{N}$. Let
$(X,{\mathcal{W}},{\mathcal{B}},\mu)$ be the product of the pointed spaces
with measured walls $(\Gamma,{\mathcal{W}}_{n},{\mathcal{B}}_{n},\mu_{n},o)$.
It is then not difficult to check that $\Gamma$ acts on $X$ by the diagonal
action, see [11, Proof of Theorem I(5)]. Distinguish the point
$x_{0}=(o,o,\ldots)\in X$.

With these technical preliminaries established, we are now ready to conclude
the proof: Since $\omega_{p}$ is a $\Gamma$-invariant bond percolation, it
suffices to prove that the cluster of the identity is finite a.s. Consider the
set of walls

${\mathcal{V}}\coloneqq\bigsqcup_{n=1}^{\infty}\Bigl{\\{}p_{n}^{-1}\bigl{(}\bigl{\\{}gA_{n}^{-1},(gA_{n}^{-1})^{c}\bigr{\\}}\bigr{)}\colon
g\in\Gamma\,\,\mbox{such that}\,\,o\in gA_{n}^{-1}\ \Bigr{\\}}.$

In the particular setting described here, we claim that ${\mathcal{V}}$ may be
interpreted as a particular set of all walls which separate $o$ from $\infty$
in ${\mathcal{G}}$: Let $n\in\mathbb{N}$ and $g\in\Gamma$ such that $o\in
gA_{n}^{-1}$. If the wall

$W=p_{n}^{-1}\bigl{(}\bigl{\\{}gA_{n}^{-1},(gA_{n}^{-1})^{c}\bigr{\\}}\bigr{)}$

satisfies $W\in{\rm supp}(\eta_{p})$, then, in the configuration $\omega_{p}$,
the edge boundary of a finite neighborhood of $o$ in ${\mathcal{G}}$ is
completely removed. Therefore on the event that $\eta_{p}({\mathcal{V}})>0$,
the cluster of $o$ in $\omega_{p}$ is finite. But now, since $\eta_{p}$ is a
Poisson process with intensity $(1-p)\mu$, the fact that

$\displaystyle\mu({\mathcal{V}})$
$\displaystyle=\sum_{n=1}^{\infty}\mu_{n}\Bigl{(}\Bigl{\\{}\bigl{\\{}gA_{n}^{-1},(gA_{n}^{-1})^{c}\bigr{\\}}\colon
g\in\Gamma\,\,\mbox{such that}\,\,o\in gA_{n}^{-1}\Bigr{\\}}\Bigr{)}$
$\displaystyle=\sum_{n=1}^{\infty}\frac{n}{|A_{n}|}|A_{n}|=\infty,$

already implies that $\mathbb{P}(\eta_{p}({\mathcal{V}})>0)=1$. Thus
$\omega_{p}$ has no infinite cluster a.s. ∎

##  7\. Lamplighters over free groups: Proof of Theorem 1.7

The goal of this section is to prove the following result (stated earlier in
Theorem 1.7):

######  Theorem 7.1.

Let $H$ be a finite group, let $r\in\mathbb{N}$ and let ${\mathcal{G}}$ be any
Cayley graph of $H\wr\mathbb{F}_{r}$. Then there exists a constant $C>0$ such
that for every $\alpha<1$, there exists a $\Gamma$-invariant bond percolation
$\mathbb{P}$ on ${\mathcal{G}}$ with
$\mathbb{E}[\deg_{\omega}(o)]>\alpha\deg(o)$ and with

${\rm e}^{-\beta|g|}\leq\tau(o,g)\leq{\rm e}^{-\gamma|g|}\quad\mbox{for
all}\,\,g\in\Gamma,$

where $\beta,\gamma>0$ such that $\beta/\gamma\leq C$.

Theorem 7.1 is somewhat surprising as the geometry of the wreath product
$H\wr\mathbb{F}_{r}$ is considerably distinct from the geometry of the base
free group: any Cayley graph of $H\wr\mathbb{F}_{r}$ has one end and even
$p_{u}<1$ by [44]. We proceed with the proof of Theorem 7.1 in the following
two steps: We first recall the wreath product construction (see Section 7.1)
and then recall from [15] a way of defining a wall structure on wreath
products over free groups (see Section 7.2). It was observed in the follow-up
article [16], that the corresponding wall-distance can be quantified. In our
probabilistically motivated set-up, we give a slightly different, self-
contained estimate for the convenience of the reader and then apply Theorem
4.1 to obtain the desired result.

###  7.1. Wreath products of finitely generated groups.

Let $H$ and $G$ be any two groups. The (standard, restricted) wreath product
$H\wr G$ is the semidirect product $H^{(G)}\rtimes G$, where

$H^{(G)}=\bigotimes_{x\in G}H$

denotes the direct sum of copies of $H$ on which $G$ acts by left translation.
In other words, the elements of $H\wr G$ are pairs $(\varphi,x)$ where $x\in
G$ and where $\varphi \colon G\to H$ is a function with $f(y)$ equal to the