# Mixing Properties of Stable Random Fields Indexed by Amenable and Hyperbolic
Groups

Mahan Mj  Mahan Mj, School of Mathematics, Tata Institute of Fundamental
Research, 1 Homi Bhabha Road, Mumbai 400005, India<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Parthanil Roy  Parthanil Roy, Theoretical Statistics and
Mathematics Unit, Indian Statistical Institute, 8th Mile, Mysore Road, RVCE
Post, Bangalore 560059, India<EMAIL_ADDRESS>and  Sourav Sarkar
Sourav Sarkar, Centre for Mathematical Sciences, University of Cambridge,
Wilberforce Road, Cambridge CB3 0WB<EMAIL_ADDRESS>
###### Abstract.

We show that any stationary symmetric $\alpha$-stable (${S\alpha S}$) random
field indexed by a countable amenable group $G$ is weakly mixing if and only
if it is generated by a null action, extending the work of Samorodnitsky and
Wang-Roy-Stoev for abelian groups to all amenable groups. This enables us to
improve significantly the domain of a recently discovered connection to von
Neumann algebras. We also establish ergodicity of stationary ${S\alpha S}$
fields associated with boundary and double boundary actions of a hyperbolic
group $G$, where the boundary is equipped with either the Patterson-Sullivan
or the hitting measure of a random walk, and the double boundary is equipped
with the Bowen-Margulis-Sullivan measure.

###### Key words and phrases:

Stable process, random field, weak mixing, ergodicity, amenable group,
hyperbolic group, von Neumann algebra

###### 2010 Mathematics Subject Classification:

Primary 60G52, 60G60, 60G10; Secondary 20F67, 37A40, 37A50, 43A07, 46L10.

M.M. is partially supported by a DST J C Bose Fellowship, an endowment from
the Infosys Foundation, and by the Department of Atomic Energy, Government of
India, under project no.12-R&D-TFR-5.01-0500. M.M. also thanks CNRS for
support during a visit to Institut Henri Poincaré in the summer of 2022. P.R.
is partially supported by a DST SwarnaJayanti Fellowship, and a SERB grant
MTR/2017/000513.

##  1\. Introduction

This paper deals with various mixing properties of stationary symmetric stable
random fields indexed by countably infinite groups, particularly amenable and
hyperbolic ones. Random fields indexed by amenable groups (e.g., Heisenberg
groups, discrete matrix groups, group of permutations of $\mathbb{N}$ with
finite support, etc.) arise naturally in machine learning algorithms for
structured and dependent space-time data [3]. On the other hand, random fields
indexed by hyperbolic groups are useful in tree-indexed processes, branching
models, etc. [32, 46, 2]. Mixing properties of these fields are important
tools for estimation of ruin probabilities for stable processes [25],
investigation of asymptotic properties of algorithms in the context of space-
time statistical inference for max-stable random fields [11], analysis of
limiting behavior for functionals of Lévy driven processes [6], and so on.

Let $G$ be a countably infinite group and $\mathbf{Y}=\\{Y_{g}\\}_{g\in G}$ be
a random field (i.e., a collection of random variables defined on a common
probability space $(\Omega,\mathcal{A},\mathbb{P})$) indexed by $G$. Such a
random field is called _symmetric $\alpha$-stable_ (${S\alpha S}$) if each
finite linear combination of $Y_{g}$’s follows an ${S\alpha S}$ distribution.
A ${S\alpha S}$ random field is called left-stationary if its law is invariant
under the left-translation of its indices. Thanks to Rosiński [35, 36], the
joint distribution (and hence the dependence structure) of any left-stationary
${S\alpha S}$ field is uniquely determined by a non-singular $G$-action, an
$L^{\alpha}$ function and a $\pm 1$-valued cocycle (see Section 1.1 below).
Among these, the non-singular action, being an infinite-dimensional parameter
of a left-stationary ${S\alpha S}$ random field, contains a lot of information
about various probabilistic properties of the field.

Keeping the above discussion in mind, it is not surprising that various
aspects of probability theory (e.g., growth of maxima [43, 42, 28, 46, 2],
extremal point processes [34, 40, 46], mixing features [37, 44, 38, 39, 51],
path properties [29], large deviations [25, 14], functional central limit
theorem [27, 19], etc.) of a stationary ${S\alpha S}$ random field have been
linked to the ergodic theory of the underlying non-singular action. In this
paper, we extend this connection to the domain of geometric group theory,
particularly amenable and hyperbolic groups. We also significantly extend the
range of application of a recently discovered association with von Neumann
algebras resolving completely Problem 3 (and hence Problem 1 and Conjecture 2)
of [41]. We have also resolved Conjecture 5 (for hyperbolic groups, not just
for free groups) of this reference positively in Section 5.

We prove the two necessary and sufficient conditions for weak mixing of a
left-stationary ${S\alpha S}$ random field indexed by an amenable group - one
is in terms of the ergodic theory of the underlying non-singular group action
and the other is a von-Neumann algebraic characterization via the associated
crossed product construction.

######  Theorem 1.1.

Suppose that $G$ is a countable amenable group and
$\mathbf{Y}=\\{Y_{g}\\}_{g\in G}$ is a stationary ${S\alpha S}$ random field.
Suppose that $\mathbf{Y}$ admits a Rosiński representation such that the
underlying non-singular $G$-action $\\{\phi_{g}\\}$ is free. Then the
following are equivalent.

  1. (1)

$\\{Y_{g}\\}_{g\in G}$ is generated by a null action (in some and hence all
Rosiński representations).

  2. (2)

$\\{Y_{g}\\}_{g\in G}$ is weakly mixing;

  3. (3)

the group measure space construction corresponding to $\\{\phi_{g}\\}_{g\in
G}$ does not admit a $II_{1}$ factor in its central decomposition.

See Theorem 4.3 for the equivalence of (1) and (2) (which does not require the
assumption of freeness of the action), and Theorem 4.5 for the equivalence of
(2) and (3).

The main challenge in the proof of Theorem 1.1 is ergodic theoretic - more
precisely, the unavailability of an ergodic theorem for non-singular (but not
necessarily measure preserving) actions of amenable groups even along a
tempered Følner sequence. We remove this obstacle with the help of a
probabilistic truncation argument along with a theorem of Lindenstrauss [23]
refined by Tempelman [50] and finally by applying the Maharam skew-product.
This extends the main theorems of [44, 51, 41] for abelian groups to all
amenable groups. A rather different proof (more analytic in nature) of the
equivalence of (1) and (2) in Theorem 1.1 has recently been given by Avraham-
Re’em [4].

In Section 5, we turn to discrete groups $G$ acting on (Gromov-)hyperbolic
spaces $X$, giving rise to an action of $G$ on the boundary $\partial_{G}X$.
We shall be specifically interested in the following cases (See Example 5.4):

  1. (1)

$\partial_{G}X$ is the Gromov boundary of a (Gromov-)hyperbolic group $G$
equipped with the Patterson-Sullivan measure ${\mu^{PS}}$.

  2. (2)

More generally, we can let $\partial_{G}X$ be the limit set of a non-
elementary group $G$ acting properly by isometries on a hyperbolic space $X$,
and equip it with the Patterson-Sullivan measures ${\mu^{PS}}$.

  3. (3)

A non-elementary hyperbolic group $G$ acting on its Poisson boundary
$(\partial_{G}G,\mu_{p})$, where $\mu_{p}$ denotes the hitting measure of
random walks on the Poisson boundary.

  4. (4)

Let $\rho$ be an Anosov representation of a (Gromov-)hyperbolic group $G$ in
the sense of Labourie, in a semi-simple Lie group $L$. Equip the limit set
$\Lambda$ in the Furstenberg boundary with a ‘higher rank’ Patterson-Sullivan
measure ${\mu^{PS}}$.

We prove (see Theorem 5.6):

######  Theorem 1.2.

In all the above examples, the associated $G$-indexed ${S\alpha S}$ random
field is ergodic.

A similar ergodicity result (see Proposition 5.8) is proven for a
(Gromov-)hyperbolic group $G$ acting on the double boundary
$\big{(}(\partial_{G}X\times\partial_{G}X)\setminus\Delta\big{)}$ equipped
with the Bowen-Margulis-Sullivan measure, where $\Delta$ denotes the diagonal.

Acknowledgment: We are grateful to Nachi Avraham-Reem for kindly bringing his
recent work [4] to our notice as this paper was being written up. We are also
grateful to him for extremely helpful comments on an earlier version of this
paper. In addition to a different proof of the equivalence of (1) and (2) in
Theorem 1.1, he also establishes that these are equivalent to ergodicity of
the associated ${S\alpha S}$ field indexed by an amenable group. The second
and third authors would like to acknowledge the warm hospitality of Tata
Institute of Fundamental Research, Mumbai during their extremely productive
research visit in April 2022.

###  1.1. Stable random fields indexed by countable groups

We shall summarize briefly some basic material on stationary symmetric
$\alpha$-stable random fields indexed by a countable group. A (real-valued)
random variable $Y$ is said to follow a _s_ ymmetric $\alpha$-stable
(${S\alpha S}$) distribution with scale parameter $\sigma\in(0,\infty)$ and
tail parameter $\alpha\in(0,2]$ if its characteristic function (i.e., Fourier
transform) is of the form $\mathbb{E}(e^{i\theta
Y})=\exp{\\{-\sigma^{\alpha}|\theta|^{\alpha}\\}}$, $\theta\in\mathbb{R}$.
Here $\mathbb{E}$ denotes the expectation with respect to the probability
measure $\mathbb{P}$. Clearly, the distribution of $Y$ is the same as that of
$-Y$, i.e. it is _symmetric_. When $\alpha=2$, we get a centered normal random
variable with exponential tails. Here, our focus is on the non-Gaussian case
($0<\alpha<2$). Thus, $\mathbb{P}(Y>t)=\mathbb{P}(Y<-t)\sim ct^{-\alpha}$ as
$t\to\infty$.

Let $G$ be a countably infinite group with identity element $e$. A random
field $\mathbf{Y}=\\{Y_{g}\\}_{g\in G}$ indexed by $G$ is called an ${S\alpha
S}$ random field if for each $k\geq 1$, for all $g_{1},g_{2},\ldots,g_{k}\in
G$ and for all $c_{1},c_{2},\ldots,c_{k}\in\mathbb{R}$, the linear combination
$\sum_{i=1}^{k}c_{i}Y_{g_{i}}$ follows an ${S\alpha S}$ distribution, whose
scale parameter depends on the coefficients
$c_{1},c_{2},\ldots,c_{k}\in\mathbb{R}$ as follows (see [45] for further
details):

(1.1) $\sum_{i=1}^{k}c_{i}Y_{g_{i}}\sim{S\alpha
S}\bigg{(}\Big{\|}\sum_{i=1}^{k}c_{i}f_{g_{i}}\Big{\|}_{\alpha}\bigg{)}\,.$

Equivalently, any ${S\alpha S}$ random field $\mathbf{Y}=\\{Y_{g}\\}_{g\in G}$
admits an _integral representation_ (also known as a _spectral
representation_) of the form

(1.2) $Y_{g}\stackrel{{\scriptstyle d}}{{=}}\int_{S}f_{g}(x)M(dx),\mbox{ \ \
}g\in G,$

where $M$ is an S$\alpha$S random measure on some $\sigma$-finite standard
Borel space $(S,\mathcal{S},\mu)$, and $f_{g}\in L^{\alpha}(S,\mu)$, $g\in G$
are real valued functions [45, Theorem 13.1.2]. Conversely, given any
$\sigma$-finite standard Borel space $(S,\mathcal{S},\mu)$ and a collection of
real-valued functions $\\{f_{g}:g\in G\\}\subset L^{\alpha}(S,\mu)$, there
exists a stationary ${S\alpha S}$ random field $\mathbf{Y}=\\{Y_{g}\\}_{g\in
G}$ satisfying (1.1). Assume henceforth, without loss of generality, that
$\bigcup_{g\in G}\\{x\in S:f_{g}(x)\neq 0\\}=S$ holds modulo $\mu$-null sets.

######  Definition 1.3.

The random field $\\{Y_{g}\\}_{g\in G}$ is called left-stationary if
$\\{Y_{g}\\}\overset{d}{=}\\{Y_{hg}\\}$ for all $h\in G$, i.e., for all
$k\in\mathbb{N}$, for all $g_{1},g_{2},\ldots,g_{k},h\in G$ and for all Borel
$B\subseteq\mathbb{R}^{k}$,

$\mathbb{P}\big{[}(Y_{g_{1}},Y_{g_{2}},\cdots,Y_{g_{k}})\in
B\big{]}=\mathbb{P}\big{[}(Y_{hg_{1}},Y_{hg_{2}},\cdots,Y_{hg_{k}})\in
B\big{]}.$

We will simply write stationary to mean left-stationary throughout this paper.
Note that stationarity of $\mathbf{Y}=\\{Y_{g}\\}_{g\in G}$ is equivalent to
saying that the left translation action $G\curvearrowright\mathbb{R}^{G}$
preserves the following probability measure

$\mathbb{P}_{\mathbf{Y}}=\mbox{\;law
of\;}\mathbf{Y}\,:=\mathbb{P}\Big{(}\big{\\{}\omega\in\Omega:\big{(}X_{g}(\omega):g\in
G\big{)}\in\cdot\big{\\}}\Big{)}.$

In particular, $\big{(}\mathbb{R}^{G},\,\mathcal{B}_{\mathbb{R}}^{\otimes
G},\,\mathbb{P}_{\mathbf{Y}},\,G\big{)}$ is a probability measure-preserving
dynamical system whenever $\mathbf{Y}=\\{Y_{g}\\}_{g\in G}$ is stationary.

By the pioneering work of Rosiński [35, 36], this induces a _non-singular_
$G$-action as follows. Let $(S,\mathcal{S},\mu)$ be a $\sigma$-finite standard
measure space and $G\curvearrowright(S,\mathcal{S},\mu)$ be a measurable
action, i.e., for all $g\in G$, the map $\phi_{g}:S\to S$ defined by
$\phi_{g}:x\mapsto g^{-1}.\,x$ is measurable. This $G$-action is said to be
_measure-preserving_ (or _invariant_) if for all $g\in G$ and for all
$A\in\mathcal{S}$, $g_{\ast}\mu(A):=\mu\circ\phi_{g}^{-1}(A)=\mu(A).$ The
action $G\curvearrowright(S,\mathcal{S},\mu)$ is called _non-singular_ (or
_quasi-invariant_ or _measure-class preserving_) if for all $g\in G$ and for
all $A\in\mathcal{S}$, $g_{\ast}\mu(A)=0\;\;\mbox{ if and only if
}\;\;\mu(A)=0$ (see [1]). Following [35], we shall refer to the collection
$\\{\phi_{g}:S\to S\\}_{g\in G}$ as a non-singular $G$-action throughout this
paper. For such an action, a $\pm 1$-valued measurable cocycle
$c_{g}:S\to\\{+1,-1\\}$, $g\in G$ is another of measurable maps (also indexed
by $G$) satisfying the cocycle relation
$c_{g_{1}g_{2}}(x)=c_{g_{2}}(x)c_{g_{1}}(\phi_{g_{2}}(x))$ for all
$g_{1},g_{2}\in G$ and for $\mu$-almost all $x\in S$. The following key
theorem forms the link between stationary ${S\alpha S}$ random fields and non-
singular actions.

######  Theorem 1.4 (_Rosiński Representation_ [35, 36]).

(a) For any stationary ${S\alpha S}$ random field
$\mathbf{Y}=\\{Y_{g}\\}_{g\in G}$, there exist a standard measure space
$(S,\mathcal{S},\mu)$ equipped with a non-singular group action
$\\{\phi_{g}\\}_{g\in G}$, a $\pm 1$-valued measurable cocycle
$\\{c_{g}\\}_{g\in G}$ and a real-valued function
$f=f_{e}\in{L}^{\alpha}(S,\mu)$ such that $\mathbf{Y}$ admits an integral
representation of the form

(1.3)
$f_{g}(x)=c_{g}(x)\left(\frac{d(\mu\circ\phi_{g})}{d\mu}(x)\right)^{1/\alpha}\left(f\circ\phi_{g}\right)(x),\mbox{\
\ }g\in G.$

(b) Conversely, given any standard measure space $(S,\mathcal{S},\mu)$ endowed
with a non-singular group action $\\{\phi_{g}\\}_{g\in G}$, a $\pm 1$-valued
measurable cocycle $\\{c_{g}\\}_{g\in G}$ for $\\{\phi_{g}\\}$, and a real-
valued function $f=f_{e}\in{L}^{\alpha}(S,\mu)$, there exists a stationary
${S\alpha S}$ random field indexed by $G$ admitting a Rosiński representation
(1.3).

##  2\. Conditions for weak mixing

As discussed in Section 1.1, a stationary random field $\\{Y_{g}\\}_{g\in G}$
(not necessarily ${S\alpha S}$) induces a measure-preserving dynamical system
$\big{(}\mathbb{R}^{G},\,\mathcal{B}_{\mathbb{R}}^{\otimes
G},\,\mathbb{P}_{\mathbf{Y}},\,G\big{)}$. We say that $\\{Y_{g}\\}_{g\in G}$
is ergodic if the induced dynamical system is so, i.e., the (left) translation
invariant measurable subsets of $\mathbb{R}^{G}$ have
$\mathbb{P}_{\mathbf{Y}}$ measure $0$ or $1$. Roughly speaking, ergodicity is
a very weak form of asymptotic independence of the random field. A slightly
stronger notion is that of weak mixing, which is defined below.

######  Definition 2.1.

A stationary random field $\\{Y_{g}\\}_{g\in G}$ (not necessarily ${S\alpha
S}$) is called weakly mixing if the dynamical system induced by the diagonal
action
$G\curvearrowright(\mathbb{R}^{G}\times\mathbb{R}^{G},\,\mathcal{B}_{\mathbb{R}}^{\otimes
G}\,\otimes\,\mathcal{B}_{\mathbb{R}}^{\otimes
G},\,\mathbb{P}_{\mathbf{Y}}\otimes\mathbb{P}_{\mathbf{Y}})$ defined by

$g.(x_{1},x_{2})=(g.x_{1},g.x_{2}),\;\;\;\;\;\;\;\;\;\;\;\;\;\;g\in
G,\,(x_{1},x_{2})\in\mathbb{R}^{G}\times\mathbb{R}^{G}$

is ergodic.

In this paper, we introduce a new notion of mixing that depends on a sequence
$\mathcal{F}=\\{F_{n}:n\geq 1\\}$ of increasing exhaustive finite subsets of
$G$.

######  Definition 2.2.

Suppose $G$ is a countable group and $\mathcal{F}=\\{F_{n}:n\geq 1\\}$ is a
sequence of finite subsets $F_{n}\uparrow G$. We say that a stationary random
field $\\{Y_{g}\\}_{g\in G}$ (not necessarily ${S\alpha S}$) is $\mathcal{F}$
mixing if for all $k\geq 1$, for all $g_{1},g_{2},\ldots,g_{k},h\in G$ and for
all Borel $A,B\subset\mathbb{R}^{k}$,

$\displaystyle\frac{1}{|F_{n}|}\sum_{h\in
F_{n}}\Big{|}\mathbb{P}\big{[}(Y_{hg_{1}},Y_{hg_{2}},\ldots,Y_{hg_{k}})\in
A,\;(Y_{g_{1}},Y_{g_{2}},\ldots,Y_{g_{k}})\in B\big{]}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-\mathbb{P}\big{[}(Y_{g_{1}},Y_{g_{2}},\ldots,Y_{g_{k}})\in
A\big{]}\,\mathbb{P}\big{[}(Y_{g_{1}},Y_{g_{2}},\ldots,Y_{g_{k}})\in
B\big{]}\Big{|}\to 0$

as $n\to\infty$

Clearly, $\mathcal{F}$-mixing (for some $\mathcal{F}$) implies ergodicity;
further, it is equivalent to weak mixing when $G$ is amenable and
$\mathcal{F}$ is a Følner sequence; see Theorem 2.3 below.

Recall that a countable group $G$ is amenable if and only if it admits an
increasing Følner sequence $F_{n}\uparrow G$, i.e., an increasing sequence of
exhausting finite subsets $F_{n}\subset G$ such that for all $g\in G$,

$\lim_{n\to\infty}\frac{|gF_{n}\,\Delta\,F_{n}|}{|F_{n}|}=0.$

Examples of amenable groups include finite groups, abelian groups, groups of
sub-exponential growth, solvable groups, lamplighter groups, etc. As mentioned
in the introduction, random fields indexed by amenable groups are important in
machine learning algorithms for structured and dependent data [3].

In order to state the characterization result for weak mixing of stationary
${S\alpha S}$ random fields indexed by amenable groups, we need to introduce
the Neveu decomposition. Let $(S,\mathcal{S},\mu)$ be as in Theorem 1.4,
equipped with a non-singular group action $\\{\phi_{g}\\}_{g\in G}$, where $G$
is any countable group. By applying [51, Lemma 2.2, Theorem 2.3 (i)] in this
general setup, we get a partition (known as the Neveu decomposition)
$S=\mathcal{P}\cup\mathcal{N}$, where the set $\mathcal{P}$ is the largest
(modulo $\mu$) $G$-invariant set where one can have a finite $G$-invariant
measure equivalent to $\mu$, and $\mathcal{N}=\mathcal{P}^{c}$. The subsets
$\mathcal{N}$ and $\mathcal{P}$ of $S$ are known as the _null part_ and the
_positive part_ of $\\{\phi_{g}\\}_{g\in G}$, respectively. A measurable set
$W\subset S$ is called _weakly wandering_ if there exists an infinite subset
$L\subseteq G$ such that $\\{\phi_{g}(W):g\in L\\}$ is a pairwise disjoint
collection. It can be shown that $\mathcal{N}$ is the measurable union (and
hence a countable union) of weakly wandering subsets of $S$ while
$\mathcal{P}$ has no weakly wandering subset of positive $\mu$-measure; see
[1]. It was shown in [44] that for $G=\mathbb{Z}$, a stationary ${S\alpha S}$
process is weakly mixing if and only if the underlying non-singular
$\mathbb{Z}$-action has no nontrivial positive part. In [51], this result was
generalized to $G=\mathbb{Z}^{d}$ for any $d\in\mathbb{N}$ with the help of
[49]. In Theorem 4.3 below, we shall establish this criterion for a stationary
${S\alpha S}$ random field indexed by an arbitrary amenable group.

Thanks to [13], random fields indexed by amenable groups enjoy the following
simpler and useful characterization of weak mixing based on Definition 2.2.

######  Theorem 2.3 ([7], Theorem 1.6).

Let $G$ be an amenable group with an increasing Følner sequence
$\mathcal{F}=\\{F_{n}:n\geq 1\\}$. Then a $G$-indexed stationary random field
(not necessarily ${S\alpha S}$) $\\{Y_{g}\\}_{g\in G}$ is weakly mixing if and
only if it is $\mathcal{F}$-mixing.

######  Definition 2.4.

Assume that $\mathcal{F}=\\{F_{n}:n\geq 1\\}$ is an increasing sequence of
finite subsets $F_{n}\uparrow G$ (not necessarily satisfying the Følner
condition). We say that a set $E\subset G$ has $\mathcal{F}$-density zero if

$\lim_{n\to\infty}\frac{|E\cap F_{n}|}{|F_{n}|}=0.$

Then the proof of [33, Lemma 6.2] gives the following. (In the statement,
‘$g\to\infty,\,g\notin E$’ simply means that $g$ stays away from $E$ and
eventually escapes any finite subset.)

######  Lemma 2.5 (Koopman-von Neumann).

Let $\psi:G\to[0,\infty)$ be a bounded function. Then
$\displaystyle{\lim_{n\to\infty}\frac{1}{|F_{n}|}\sum_{g\in F_{n}}\psi(g)=0}$
if and only if there exists $E\subset G$ of $\mathcal{F}$-density zero such
that $\displaystyle{\lim_{g\to\infty,g\notin E}\psi(g)=0}$.

In order to use the ergodic theorem of [50], we shall further need to choose
the Følner sequence to be tempered (in the sense of [23]) as defined below.

######  Definition 2.6.

A sequence of finite sets $F_{n}\subset G$ is called tempered if there exists
$C\in(0,\infty)$ such that for all $n\geq 2$, the Shulman condition

$\left|\bigcup_{k=1}^{n-1}F_{k}^{-1}F_{n}\right|\leq C|F_{n}|.$

holds.

It was shown in [23, Proposition 1.4, Theorem 1.2] that every amenable group
admits an increasing tempered Følner sequence along which pointwise ergodic
theorem holds. In addition to the work of [23] (and its refinement [50])
mentioned above, we also need an extension of a result from [17]. The
statement is essentially the countable groups version of [44, Equation (3.1)].

######  Lemma 2.7.

Let $G$ be any countable group and $\mathcal{F}=\\{F_{n}:n\geq 1\\}$ be a
sequence of finite subsets $F_{n}\uparrow G$. Then a $G$-indexed stationary
${S\alpha S}$ random field $\mathbf{Y}=\\{Y_{g}\\}_{g\in G}$, with a Rosiński
representation of the form (1.3), is $\mathcal{F}$-mixing if and only if for
all $\delta>0$ and for all $\epsilon>0$,

(2.1) $\lim_{n\to\infty}\frac{1}{|F_{n}|}\sum_{g\in F_{n}}\mu\big{(}x\in
S:\delta\leq|f_{e}(x)|\leq\delta^{-1},\,|f_{g}(x)|\geq\epsilon\big{)}=0.$

In particular, if $G$ is a countable amenable group and
$\mathcal{F}=\\{F_{n}:n\geq 1\\}$ is an increasing Følner sequence, then (2.1)
is equivalent to weak mixing of $\\{Y_{g}\\}_{g\in G}$.

###### Proof.

The first part follows from the techniques of [51] (see the proof of
Proposition $4.2$ in the appendix therein) with the help of Lemma 2.5 above.
The second part follows from the first one using Theorem 2.3. ∎

##  3\. Truncation and Maharam Extension

In this section, we apply a probabilistic truncation argument along with
Maharam skew product to establish that it is enough to check condition (2.1)
for indicator functions $f$ and measure-preserving $G$-actions
$\\{\phi_{g}\\}$ (see Theorem 3.4 for a precise statement) in (1.3).
Throughout, we assume that $G$ is any countably infinite group and
$\mathcal{F}=\\{F_{n}:n\geq 1\\}$ is a sequence of finite subsets
$F_{n}\uparrow G$. The results of this section do not depend on the
amenability of the group $G$ and hold for all countable groups.

Before we can move to the main theorem of this section, we shall need a series
of lemmas. We assume that the stationary ${S\alpha S}$ random field
$\mathbf{Y}=\\{Y_{g}\\}_{g\in G}$ has an integral representation of the form

$Y_{g}\stackrel{{\scriptstyle d}}{{=}}\int_{S}f_{g}(x)M(dx),\mbox{ \ \ }g\in
G,$

where $M$ is an ${S\alpha S}$ random measure on some $\sigma$-finite standard
Borel space $(S,\mathcal{S},\mu)$ and $f_{g}$ is given by (1.3), for some non-
singular group action $\\{\phi_{g}\\}_{g\in G}$, a $\pm 1$-valued measurable
cocycle $\\{c_{g}\\}_{g\in G}$ and a real-valued function
$f=f_{e}\in{L}^{\alpha}(S,\mu)$.

First observe that, without loss of generality, we can assume that $\mu$ is a
probability measure. For if not, we can always get a probability measure $\nu$
equivalent to the sigma-finite measure $\mu$ and then the random field
$\mathbf{Y}=\\{Y_{g}\\}_{g\in G}$ has an integral representation of the form

$Y_{g}\stackrel{{\scriptstyle d}}{{=}}\int_{S}h_{g}(x)N(dx),\mbox{ \ \ }g\in
G,$

where $N$ is an S$\alpha$S random measure on $(S,\mathcal{S},\nu)$ and

$h(s)=f(s)\left(\frac{d\mu}{d\nu}(s)\right)^{1/\alpha}\in{L}^{\alpha}(S,\nu),$

and $h_{g}$ is as given in (1.3) with $h,\nu$ in place of $f,\mu$. Henceforth,
we assume in this section that $\mu$ is a probability measure. Throughout, we
fix $\delta>0$ and for simplicity of notation, we let

$A:=\\{x\in S:\delta\leq|f_{e}(x)|\leq\delta^{-1}\\}\,.$

Then, we have the following lemma that shows that it is enough to check (2.1)
for $f$ bounded.

######  Lemma 3.1.

Let $\mu$ be a probability measure. Then for any $\epsilon>0$,

$\displaystyle\limsup_{n}\frac{1}{|F_{n}|}\sum_{g\in F_{n}}\mu\left(x\in
A:|f\circ\phi_{g}(x)|^{\alpha}\frac{d(\mu\circ\phi_{g})}{d\mu}(x)>\epsilon\right)$
$\displaystyle\leq$
$\displaystyle\limsup_{L\to\infty}\limsup_{n\to\infty}\frac{1}{|F_{n}|}\sum_{g\in
F_{n}}\mu\left(x\in A:|f\mathbf{1}_{|f|\leq
L}\circ\phi_{g}(x)|^{\alpha}\frac{d(\mu\circ\phi_{g})}{d\mu}(x)>\epsilon/2\right).$

In particular, if (2.1) holds for all $f$ bounded, then (2.1) holds for all
$f\in{L}^{\alpha}(S,\mu)$.

###### Proof.

For any fixed $L>0$, we can write $f=f\mathbf{1}_{|f|\leq
L}+f\mathbf{1}_{|f|>L}$. So, for any fixed $\epsilon>0$,

$\displaystyle\mu(x\in A:|f_{g}(x)|^{\alpha}>\epsilon)$ $\displaystyle\leq$
$\displaystyle\mu\left(x\in A:|f\mathbf{1}_{|f|\leq
L}\circ\phi_{g}(x)|^{\alpha}\frac{d(\mu\circ\phi_{g})}{d\mu}(x)>\epsilon/2\right)$
$\displaystyle+\mu\left(x\in
A:|f\mathbf{1}_{|f|>L}\circ\phi_{g}(x)|^{\alpha}\frac{d(\mu\circ\phi_{g})}{d\mu}(x)>\epsilon/2\right)$

Now, we can approximate the second term on the right-hand side as follows.

$\displaystyle\mu\left(x\in
A:|f\mathbf{1}_{|f|>L}\circ\phi_{g}(x)|^{\alpha}\frac{d(\mu\circ\phi_{g})}{d\mu}(x)>\epsilon/2\right)$
$\displaystyle\leq$
$\displaystyle\int\mathbf{1}\left(|f\circ\phi_{g}(x)|>L,|f\circ\phi_{g}(x)|^{\alpha}\frac{d(\mu\circ\phi_{g})}{d\mu}(x)>\epsilon/2\right)d\mu$
$\displaystyle\leq$
$\displaystyle\frac{2}{\epsilon}\int|f\circ\phi_{g}(x)|^{\alpha}\frac{d(\mu\circ\phi_{g})}{d\mu}(x)\mathbf{1}\left(|f\circ\phi_{g}(x)|>L\right)d\mu$
$\displaystyle=$
$\displaystyle\frac{2}{\epsilon}\int|f\circ\phi_{g}(x)|^{\alpha}\mathbf{1}\left(|f\circ\phi_{g}(x)|>L\right)d\mu\circ\phi_{g}$
$\displaystyle=$
$\displaystyle\frac{2}{\epsilon}\int|f(x)|^{\alpha}\mathbf{1}\left(|f(x)|>L\right)d\mu.$

Thus,

$\displaystyle\sup_{n}\frac{1}{|F_{n}|}\sum_{g\in F_{n}}\mu\left(x\in
A:|f\mathbf{1}_{|f|>L}\circ\phi_{g}(x)|^{\alpha}\frac{d(\mu\circ\phi_{g})}{d\mu}(x)>\epsilon/2\right)$
$\displaystyle\leq\frac{2}{\epsilon}\int|f(x)|^{\alpha}\mathbf{1}\left(|f(x)|>L\right)d\mu\rightarrow
0$

as $L\to\infty$ by the dominated convergence theorem, since
$f\in{L}^{\alpha}(S,\mu)$. From this, the lemma follows. ∎

Henceforth, we assume that $f$ is bounded, that is, there exists some $K>0$
such that $|f(x)|\leq K$ for all $x\in S$. In the next lemma, we show that we
can assume that the Radon-Nikodym derivatives are bounded away from $0$ and
infinity.

######  Lemma 3.2.

Let $\mu$ be a probability measure and $|f|\leq K$ for some $K>0$. Then

(3.1) $\displaystyle\sup_{n}\frac{1}{|F_{n}|}\sum_{g\in F_{n}}\mu\bigg{(}x\in
A:|f\circ\phi_{g}(x)|^{\alpha}\frac{d(\mu\circ\phi_{g})}{d\mu}(x)>\epsilon,$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{d(\mu\circ\phi_{g})}{d\mu}(x)\not\in[L^{-1},L]\bigg{)}\to
0$

as $L\to\infty$.

###### Proof.

When $\frac{d(\mu\circ\phi_{g})}{d\mu}(x)$ is large, we have

$\displaystyle\mu\left(x\in
A:|f\circ\phi_{g}(x)|^{\alpha}\frac{d(\mu\circ\phi_{g})}{d\mu}(x)>\epsilon,\frac{d(\mu\circ\phi_{g})}{d\mu}(x)>L\right)$
$\displaystyle\leq$
$\displaystyle\mu\left(\frac{d(\mu\circ\phi_{g})}{d\mu}(x)>L\right)$
$\displaystyle\leq$
$\displaystyle\frac{1}{L}\int\frac{d(\mu\circ\phi_{g})}{d\mu}(x)d\mu$
$\displaystyle=$ $\displaystyle\frac{1}{L}\int
d\mu\circ\phi_{g}(x)=\frac{1}{L}\,,$

where we have used the Markov inequality in the third line and the last
equality uses the fact that $\mu$ is a probability measure. On the other hand,
by choosing $L$ large enough so that $L>\epsilon^{-1}K^{\alpha}$, we have

$\displaystyle\mu\left(x\in
A:|f\circ\phi_{g}(x)|^{\alpha}\frac{d(\mu\circ\phi_{g})}{d\mu}(x)>\epsilon,\frac{d(\mu\circ\phi_{g})}{d\mu}(x)<L^{-1}\right)$
$\displaystyle\leq$
$\displaystyle\mu\left(\frac{d(\mu\circ\phi_{g})}{d\mu}(x)>\epsilon
K^{-\alpha},\frac{d(\mu\circ\phi_{g})}{d\mu}(x)<L^{-1}\right)=0\,,$

where we used that $|f|\leq K$. Thus, for $L>\epsilon^{-1}K^{\alpha}$, the
quantity in (3.1) is at most $L^{-1}$, which goes to $0$ as $L\to\infty$. This
proves the lemma. ∎

The next lemma converts the problem of checking condition (2.1) for non-
singular actions to that for measure-preserving actions. First, we need to
recall the Maharam extension (see, e.g., Chapter 3.4 of [1]). For the non-
singular map $\phi_{g}$ of $G$ acting on $(S,\mu)$, let

$w_{g}(x):=\frac{d\mu\circ\phi_{g}}{d\mu}(x),\qquad g\in G,x\in S.$

Then the group action $\phi_{g}^{*}$ of $G$ on
$(S\times(0,\infty),\mu\otimes\mbox{Leb})$ defined, for each $g\in G$, as

(3.2) $\phi_{g}^{*}(x,y):=\left(\phi_{g}(x),\frac{y}{w_{g}(x)}\right),\qquad
x\in S,\,y>0$

preserves the measure $\mu\otimes\mbox{Leb}$; see [24].

######  Lemma 3.3.

Let $\mu$ be a probability measure and $|f|\leq K$ for some $K>0$. For any
$L>1$, define $h^{(L)}\in{L}^{\alpha}(S\times(0,\infty),\mu\otimes\mbox{Leb})$
as

$h^{(L)}(x,y):=\frac{|f(x)|}{y^{1/\alpha}}\mathbf{1}\left(\frac{1}{2L}\leq
y\leq 2L\right)\,,\qquad x\in S,\,y>0\,.$

Then for any $\epsilon>0$,

$\displaystyle\limsup_{n}\frac{1}{|F_{n}|}\sum_{g\in F_{n}}\mu\left(x\in
A:|f\circ\phi_{g}(x)|^{\alpha}\frac{d(\mu\circ\phi_{g})}{d\mu}(x)>\epsilon\right)$
$\displaystyle\leq$
$\displaystyle\frac{2}{3}\limsup_{L\to\infty}\limsup_{n\to\infty}\frac{1}{|F_{n}|}\sum_{g\in
F_{n}}\mu\otimes\mbox{Leb}\left((x,y)\in
A^{\prime}:|h^{(L)}\circ\phi_{g}^{*}(x,y)|^{\alpha}>\epsilon/2\right)\,,$

where

$A^{\prime}:=\\{(x,y)\in S\times(0,\infty):2^{-1}\delta\leq|h^{(L)}(x,y)|\leq
2\delta^{-1}\\}\,.$

###### Proof.

First observe that for any $L>1$,

$\displaystyle\frac{3}{2}\mu\left(x\in
A:|f\circ\phi_{g}(x)|^{\alpha}w_{g}(x)>\epsilon,w_{g}(x)\in[L^{-1},L]\right)$
$\displaystyle=$
$\displaystyle\mu\otimes\mbox{Leb}\Big{(}(x,y):\delta\leq|f(x)|^{\alpha}\leq\delta^{-1},|f\circ\phi_{g}(x)|^{\alpha}w_{g}(x)>\epsilon,w_{g}(x)\in[L^{-1},L],$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad
y\in[2^{-1},2]\Big{)}$ $\displaystyle\leq$
$\displaystyle\mu\otimes\mbox{Leb}\Big{(}(x,y):2^{-1}\delta\leq\frac{|f(x)|^{\alpha}}{y}\leq
2\delta^{-1},\,\frac{|f\circ\phi_{g}(x)|^{\alpha}}{y}w_{g}(x)>\epsilon/2,$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad
y\in[(2L)^{-1},2L],\,y\in[(2L)^{-1}w_{g}(x),2Lw_{g}(x)]\Big{)}$
$\displaystyle=$
$\displaystyle\mu\otimes\mbox{Leb}\Big{(}(x,y):2^{-1}\delta\leq|h^{(L)}(x,y)|\leq
2\delta^{-1},\,|h^{(L)}\circ\phi_{g}^{*}(x,y)|^{\alpha}>\epsilon/2\Big{)}\,.$

This, together with Lemma 3.2, proves (3.3). ∎

Now, we are ready to prove the main theorem of this section.

######  Theorem 3.4.

Let $\\{\phi_{g}\\}$ be any non-singular group action of $G$ on $(S,\mu)$.
Assume that for the measure-preserving group action $\\{\phi_{g}^{*}\\}$ of
$G$ on $(S\times(0,\infty),\mu\otimes\mbox{Leb})$ defined in (3.2), and any
set $E\subseteq S\times(0,\infty)$ with $\mu\otimes\mbox{Leb}(E)<\infty$, we
have

$\lim_{n\to\infty}\frac{1}{|F_{n}|}\sum_{g\in
F_{n}}\mu\otimes\mbox{Leb}\big{(}E\cap(\phi_{g}^{*})^{-1}(E))=0.$

Then for any $f\in{L}^{\alpha}(S,\mu)$ and all $\epsilon>0,\,\delta>0$,
condition (2.1) holds, that is,

(3.4) $\lim_{n\to\infty}\frac{1}{|F_{n}|}\sum_{g\in F_{n}}\mu\big{(}x\in
S:\delta\leq|f_{e}(x)|\leq\delta^{-1},\,|f_{g}(x)|\geq\epsilon\big{)}=0.$

###### Proof.

Fix $\delta>0,\,\epsilon>0$ and any non-singular $G$-action $\\{\phi_{g}\\}$
on $(S,\mu)$ and any $f\in{L}^{\alpha}(S,\mu)$. By Lemma 3.1 and the
discussion preceding it, we can assume that $\mu$ is a probability measure and
$f$ is bounded, that is, $|f|\leq K$ for some $K>0$. Then by Lemma 3.3, (3.4)
holds if for all $L>1$ and for $\phi_{g}^{*}$ defined from $\phi_{g}$
according to $\eqref{e:Maharram}$, we can show that

(3.5) $\lim_{n\to\infty}\frac{1}{|F_{n}|}\sum_{g\in
F_{n}}\mu\otimes\mbox{Leb}\left((x,y)\in
A^{\prime}:|h^{(L)}\circ\phi_{g}^{*}(x,y)|^{\alpha}>\epsilon/2\right)=0\,,$

where

$h^{(L)}(x,y):=\frac{|f(x)|}{y^{1/\alpha}}\mathbf{1}\left(\frac{1}{2L}\leq
y\leq 2L\right)\qquad\mbox{ and}$ $A^{\prime}:=\\{(x,y)\in
S\times(0,\infty):2^{-1}\delta\leq|h^{(L)}(x,y)|\leq 2\delta^{-1}\\}.$

Fix $L>1$ and note that

$|h^{(L)}(x,y)|=\frac{|f(x)|}{y^{1/\alpha}}\mathbf{1}\left(\frac{1}{2L}\leq
y\leq 2L\right)\leq K(2L)^{1/\alpha}\mathbf{1}\left(\frac{1}{2L}\leq y\leq
2L\right)\,.$

Define the set $E_{L}$ as

$E_{L}:=S\times[(2L)^{-1},2L]\,.$

Then $\mu\otimes\mbox{Leb}(E_{L})=\mbox{Leb}([(2L)^{-1},2L])<\infty$ and

$\displaystyle\mu\otimes\mbox{Leb}\left((x,y)\in
A^{\prime}:|h^{(L)}\circ\phi_{g}^{*}(x,y)|^{\alpha}>\epsilon/2\right)$
$\displaystyle\leq$ $\displaystyle\mu\otimes\mbox{Leb}\left((x,y)\in
S\times(0,\infty):\mathbf{1}_{E_{L}}(x,y)>0,\,\mathbf{1}_{E_{L}}\circ\phi_{g}^{*}(x,y)>0\right)$
$\displaystyle=$
$\displaystyle\mu\otimes\mbox{Leb}\big{(}E_{L}\cap(\phi_{g}^{*})^{-1}(E_{L}))\,.$

As $\mu\otimes\mbox{Leb}(E_{L})<\infty$, by the assumption of the theorem,

$\lim_{n\to\infty}\frac{1}{|F_{n}|}\sum_{g\in
F_{n}}\mu\otimes\mbox{Leb}\big{(}E_{L}\cap(\phi_{g}^{*})^{-1}(E_{L}))=0.$

Hence, (3.5) holds for all $L>1$, thus implying (3.4). This proves the
theorem. ∎

##  4\. Stable Random fields indexed by amenable groups

###  4.1. Criterion for Weak Mixing

In this subsection, we prove one of the main results of this paper that
characterizes weak mixing of a stationary ${S\alpha S}$ random field indexed
by any countable amenable group $G$. In a Rosiński representation of a
stationary ${S\alpha S}$ random field indexed by $G$ as in Theorem 1.4, if the
group action $\\{\phi_{g}\\}_{g\in G}$ has only the null (resp. positive)
part, that is, $\mathcal{P}=\emptyset$ modulo $\mu$ (resp.
$\mathcal{N}=\emptyset$) in the Neveu decomposition, then we say that the
random field is generated by a null (resp. positive) action in that
representation. The following proposition shows that this does not depend on
the particular Rosiński representation chosen as long as the group $G$ is
amenable.

######  Proposition 4.1.

If a stationary ${S\alpha S}$ random field indexed by any countable amenable
group $G$ is generated by a positive (resp. null) action in one Rosiński
representation, then in every other Rosiński representation it will be
generated by a positive (resp. null) action.

###### Proof.

This follows by mimicking the proof of [51, Theorem 3.1], which in turn
essentially relies on the work of [49]. The latter work deals with amenable
semigroups of contractions. Hence, it directly applies to our situation. This
yields a characterization for a stationary ${S\alpha S}$ random field indexed
by a discrete amenable group $G$ to be generated by a null (or a positive)
$G$-action. ∎

######  Remark 4.2.

Following closely the proof of [51, Corollary 3.3], we can also establish that
for any stationary ${S\alpha S}$ field $\mathbf{Y}=\\{Y_{g}\\}_{g\in G}$
indexed by an amenable group, the decomposition into null and positive parts
does not depend on the choice of Rosiński representation and hence is unique
in law. We can, therefore, define these to be the positive and null parts,
respectively, of $\mathbf{Y}$.

Following is the main result of this section. It extends significantly the
main theorems of [44, 51] leading to a crucial improvement (see Theorem 4.5
below) of the results of [41].

######  Theorem 4.3.

Any stationary ${S\alpha S}$ random field indexed by a countably infinite
amenable group $G$ is weakly mixing if and only if it is generated by a null
action (in some and hence all Rosiński representations).

For the proof of this theorem, we will need some definitions, which we recall
here from Tempelman [50]. A mapping $T:g\mapsto T_{g}$ on $G$ is said to be a
_right ${L}^{p}$-representation_ of $G$ if, for each $g\in G$, $T_{g}$ is an
operator in ${L}^{p}(\Lambda,\mathcal{L},m)$ (here $(\Lambda,\mathcal{L},m)$
is a $\sigma$-finite standard measure space),
$T_{g_{1}g_{2}}=T_{g_{2}}T_{g_{1}},\,g_{1},g_{2}\in G$ and $T_{e}$ is the
identity operator. A representation $T$ is _bounded_ if $\|T\|_{p}:=\sup_{g\in
G}\|T_{g}\|_{p}<\infty$. For the $T_{g}$-image of $f\in{L}^{p}(\Lambda,m)$, we
use the notation $fT_{g}$. We say that a bounded representation $T$ is a
_Lamperti_ ${L}^{p}$-representation if $T_{g}$ is Lamperti for all $g\in G$,
that is, for all $f,h\in{L}^{p}$ with $fh=0$ $m$-almost everywhere (a.e.), one
has $(fT_{g})(hT_{g})=0$ $m$-a.e. Now, we are ready to recall the following
result from [50] for a countable amenable group $G$.

######  Theorem 4.4.

[50, Theorem 3.9] Let $\\{F_{n}\\}$ be a tempered Følner sequence in $G$. If
$1<p<\infty$ and $T:g\mapsto T_{g}$ is a Lamperti $L^{p}$-representation, then
for each $f\in{L}^{p}(\Lambda,\mathcal{L},m)$,

$\lim_{n\to\infty}\frac{1}{|F_{n}|}\sum_{g\in F_{n}}fT_{g}=h\quad
m\mbox{-a.e.}\,,$

where $h$ is $T_{g}$-invariant.

The $h$ in the above limit is the _mean value_ of the orbit $\\{fT_{g},g\in
G\\}$ and is the projection of $f$ onto the space of all $T$-invariant
functions in ${L}^{p}$.

Now, let us fix any measure-preserving group action $\phi_{g}^{*}$ of $G$
acting on $(\Lambda,\mathcal{L},m)$. We define for each $g\in G$,
$T_{g}:{L}^{2}(\Lambda,\mathcal{L},m)\mapsto{L}^{2}(\Lambda,\mathcal{L},m)$ as
$fT_{g}=f\circ\phi_{g}^{*}$. Note that this is bounded as

$\|fT_{g}\|_{2}^{2}=\int|f\circ\phi_{g}^{*}|^{2}dm=\int|f|^{2}dm=\|f\|_{2}^{2}$

since $\phi_{g}^{*}$ is measure $m$-preserving. Thus, $\|T_{g}\|_{2}=1$ for
all $g\in G$, so that $\|T\|_{2}:=\sup_{g\in G}\|T_{g}\|_{2}=1<\infty$. Also,
this representation is Lamperti because for all $f,h\in{L}^{2}(\Lambda,m)$
with $fh=0$ $m$-a.e. and all $g\in G$,

$(fT_{g})(hT_{g})=(f\circ\phi_{g}^{*})(h\circ\phi_{g}^{*})=0\quad
m\mbox{-a.e.}\,.$

Finally, we are ready to prove Theorem 4.3.

###### Proof of Theorem 4.3.

We first note the easy direction: if the random field $\mathbf{Y}$ is not
generated by a null action, then it is not weakly mixing. To see this, we can
first without loss of generality assume that the action of $G$ is positive
(because of the positive-null decomposition of $\mathbf{Y}$ as in Remark 4.2
and the criterion in Lemma 2.7). Thus, $\mathbf{Y}$ has an integral
representation on $(S,\nu)$ where $\nu$ is a probability measure, which is
preserved by the action of the amenable group $G$. Consider a sequence of
Folner sets $F_{n}$ for $G$. Then for any set $A$ with $\nu(A)>0$,
$\frac{1}{|F_{n}|}\sum_{g\in F_{n}}\mathbf{1}_{g.A}$ converges by the ergodic
theorem of Lindenstrauss for amenable groups [23] to some invariant $h$ such
that $\int hd\nu>0$ (since by the dominated convergence theorem, this equals
$\nu(A)$). Hence, again by the dominated convergence theorem,
$1/|F_{n}|\int_{S}\mathbf{1}_{A}\sum_{g\in F_{n}}\mathbf{1}_{g.A}d\nu$
converges to $\int_{A}hd\nu$, which we claim is a positive number; this in
turn violates the weak mixing criterion in Lemma 2.7 for
$f_{e}=\mathbf{1}_{A}$. To see the claim note that, if $\int_{A}hd\nu=0$, then
$\int_{g.A}hd\nu=0$ for all $g\in G$, and hence $\int hd\nu=0$. From here, it
is easy to generalize to any $f_{e}\in{L}^{\alpha}(S,\nu)$ using standard
arguments.

Now we assume that the random field is generated by a null non-singular action
$\phi_{g}$ acting on $(S,\mu)$. Define the measure-preserving action
$\phi_{g}^{*}$ of $G$ on
$(\Lambda:=S\times(0,\infty),\,\mathcal{L}:=\mathcal{S}\otimes\mathcal{B}_{(0,\infty)},\,m:=\mu\otimes\mbox{Leb})$
as defined in (3.2). We first claim that $\phi_{g}^{*}$ is also null. This is
because if $B\subseteq S$ is a weakly wandering set (see Section 2) for
$\phi_{g}$, then $B\times(0,\infty)$ is a weakly wandering set for
$\phi_{g}^{*}$. Therefore, it follows that $\Lambda:=S\times(0,\infty)$ is the
measurable union of its weakly wandering subsets since $S$ is so.

For $f\in{L}^{2}(\Lambda,m)$, the operator $fT_{g}=f\circ\phi_{g}^{*}$ is a
bounded Lamperti ${L}^{2}$-representation by the discussion following Theorem
4.4. Fix any set $E\subseteq S\times(0,\infty)$ with
$m(E)=\mu\otimes\mbox{Leb}(E)<\infty$, and let $f=\mathbf{1}_{E}$. Clearly
$f\in{L}^{2}(\Lambda,m)$ and by Theorem 4.4 we have

$\lim_{n\to\infty}\frac{1}{|F_{n}|}\sum_{g\in
F_{n}}\mathbf{1}_{(\phi_{g}^{*})^{-1}(E)}=h\quad m\mbox{-a.e.}$

for some $h$ which is $\phi_{g}^{*}$ invariant. Clearly $h\geq 0$ and by the
Fatou’s Lemma,

$\displaystyle\int_{\Lambda}hdm\leq\liminf_{n\to\infty}\int_{\Lambda}\frac{1}{|F_{n}|}\sum_{g\in
F_{n}}\mathbf{1}_{(\phi_{g}^{*})^{-1}(E)}dm=\liminf_{n\to\infty}\frac{1}{|F_{n}|}\sum_{g\in
F_{n}}\int_{\Lambda}\mathbf{1}_{(\phi_{g}^{*})^{-1}(E)}dm$
$\displaystyle=\int_{\Lambda}\mathbf{1}_{E}dm=m(E)<\infty\,,$

using the fact that $\phi_{g}^{*}$ preserves the measure $m$. Thus
$h\in{L}^{1}(\Lambda,m)$. It follows that $d\nu=hdm$ is a finite
$\phi_{g}^{*}$-invariant measure equivalent to $m$ on $\\{h>0\\}$. Since the
action $\phi_{g}^{*}$ is null, it follows that $h=0$ $m$-a.e. Hence

$\lim_{n\to\infty}\frac{1}{|F_{n}|}\sum_{g\in
F_{n}}\mathbf{1}_{(\phi_{g}^{*})^{-1}(E)}=0\quad m\mbox{-a.e.}\,.$

Multiplying both sides by $\mathbf{1}_{E}$, we have

$\lim_{n\to\infty}\frac{1}{|F_{n}|}\sum_{g\in
F_{n}}\mathbf{1}_{E\cap(\phi_{g}^{*})^{-1}(E)}=0\quad m\mbox{-a.e.}\,.$

Moreover, for all $n$,

$0\leq\frac{1}{|F_{n}|}\sum_{g\in
F_{n}}\mathbf{1}_{E\cap(\phi_{g}^{*})^{-1}(E)}\leq\mathbf{1}_{E}$

and $\int_{\Lambda}\mathbf{1}_{E}dm=m(E)<\infty$. Hence, by the dominated
convergence theorem,

$\lim_{n\to\infty}\frac{1}{|F_{n}|}\sum_{g\in
F_{n}}m\left(E\cap(\phi_{g}^{*})^{-1}(E)\right)=0\,.$

Thus by Theorem 3.4, condition (2.1) holds for any $f\in{L}^{\alpha}(S,\mu)$
and all $\epsilon>0,\,\delta>0$. Hence by Lemma 2.7, the ${S\alpha S}$ random
field is weakly mixing since $G$ is amenable. ∎

###  4.2. Connections to von Neumann Algebras

In this subsection, we give a characterization of weak mixing of a stationary
${S\alpha S}$ random field indexed by a countable amenable group $G$ in terms
of von Neumann algebras. We refer to [48] for basics of von Neumann algebras
and factors. Recall that a von Neumann algebra
$M\subseteq\mathcal{B}(\mathcal{H})$ (for some separable Hilbert space
$\mathcal{H}$) is called a factor if its center $Z(M):=M\cap M^{\prime}$ is
trivial, i.e., $Z(M)=\mathbb{C}1$. We shall, in general, need von Neumann’s
_central decomposition_

(4.1) $M=\int_{Y}M_{y}\,\nu(dy)$

of $M$ as a direct integral of factors $\\{M_{y}:y\in Y\\}$ over a
$\sigma$-finite measure space $(Y,\mathcal{Y},\nu)$. Recall (see [48]) that a
factor is of type $II_{1}$ if it is infinite-dimensional and it admits a
normalized trace. We say [41, Definition 3.6] that a von Neumann algebra $M$
does not admit a $II_{1}$ factor in its central decomposition (4.1) if for
$\nu$-almost all $y\in Y$, $M_{y}$ is a not a $II_{1}$ factor.

We now describe a crossed product von Neumann algebra called the _group
measure space construction_ [26]. This encodes ergodic properties of a non-
singular $G$-action $\\{\phi_{g}\\}$. Any such action induces, for each $g\in
G$, the isometry $\pi_{g}:L^{2}(S,\mu)\to L^{2}(S,\mu)$ given by

$(\pi_{g}h)(s)=h\circ\phi_{g}(s)\left(\frac{d\mu\circ\phi_{g}}{d\mu}(s)\right)^{1/2},\;s\in
S.$

The unitary representation $\\{\pi_{g}\\}_{g\in G}$ of $G$ on $L^{2}(S,\mu)$
is called the _Koopman representation._ For all $a\in L^{\infty}(S,\mu)$
(thought of as acting on $L^{2}(S,\mu)$ by multiplication), for all $g\in G$
and for all $h\in L^{2}(S,\mu)$,

(4.2) $(\pi_{g}\,a\,\pi_{g^{-1}}h)(s)=((\sigma_{g}a)h)(s),\;\;s\in S,$

where $\sigma_{g}a:=a\circ\phi_{g}$. Thus, the Koopman representation
“normalizes” $L^{\infty}(S,\mu)$ inside $\mathcal{B}(L^{2}(S,\mu))$.

Consider the von Neumann algebra $\mathcal{B}(l^{2}(G)\otimes
L^{2}(S,\mu))=\overline{\mathcal{B}(l^{2}(G))\otimes\mathcal{B}(L^{2}(S,\mu))}$
(where the closure is with respect to the weak/strong operator topology).
Define a representation of $G$ by $g\mapsto u_{g}:=\lambda_{g}\otimes\pi_{g}$,
where $\\{\lambda_{g}\\}$ is the left regular representation and
$\\{\pi_{g}\\}$ is the Koopman representation. We also represent
$L^{\infty}(S,\mu)$ by $a\mapsto 1\otimes\mathcal{M}_{a}$, where
$\mathcal{M}_{a}$ is the multiplication (by $a$) operator on $L^{2}(S,\mu)$.
Then,

(4.3)
$u_{t}(1\otimes\mathcal{M}_{a})u_{t^{-1}}=1\otimes\mathcal{M}_{\sigma_{t}a}\,.$

Define the _group measure space construction_ as the double commutant

$L^{\infty}(S,\mu)\rtimes G:=\\{u_{t},1\otimes\mathcal{M}_{a}:t\in G,\,a\in
L^{\infty}(S,\mu)\\}^{\prime\prime}.$

It was shown in [41, Theorem 5.4] that for any countable group $G$, non-
singular $G$-actions coming from two minimal (and hence Rosiński)
representations of a stationary ${S\alpha S}$ random field
$\mathbf{Y}=\\{Y_{g}\\}_{g\in G}$ are $W^{\ast}$-equivalent, i.e. the
corresponding group measure space constructions are isomorphic as von Neumann
algebras. This algebra was defined as the _minimal group measure space
construction_ of $\mathbf{Y}$.

Two stationary ${S\alpha S}$ fields $\big{\\{}X^{(1)}_{g}\big{\\}}_{g\in
G_{1}}$ and $\big{\\{}X^{(2)}_{g}\big{\\}}_{g\in G_{2}}$ indexed by two
(possibly non-isomorphic) countable groups are called
$W^{\ast}_{R}$-equivalent (resp., $W^{\ast}_{m}$-equivalent) if the group
measure space construction corresponding to a Rosiński (resp., minimal)
representation of $\big{\\{}X^{(1)}_{g}\big{\\}}_{g\in G_{1}}$ is isomorphic
(as a von Neumann algebra) to the group measure space construction
corresponding to a Rosiński (resp., minimal) representation of
$\big{\\{}X^{(2)}_{g}\big{\\}}_{g\in G_{2}}$. A property $P$ of stationary
${S\alpha S}$ random fields (indexed by a class $\mathcal{G}$ of countable
groups) is called $W^{\ast}_{R}$-rigid (resp., $W^{\ast}_{m}$-rigid) for
$\mathcal{G}$ if whenever two such fields (not necessarily indexed by the same
group) are $W^{\ast}_{R}$-equivalent (resp., $W^{\ast}_{m}$-equivalent), one
enjoys property $P$ if and only if the other one does. By [35, Theorem 3.1],
any minimal representation is a Rosiński representation (but the converse does
not hold). Hence $W^{\ast}_{m}\mbox{-equivalence}$ implies
$W^{\ast}_{R}\mbox{-equivalence}$ and $W^{\ast}_{R}$-rigidity implies
$W^{\ast}_{m}$-rigidity.

In [41], it was shown that weak mixing is a $W^{\ast}_{R}$-rigid (and hence
$W^{\ast}_{m}$-rigid) property for stable random fields indexed by
$\mathcal{G}:=\\{\mathbb{Z}^{d}:d\in\mathbb{N}\\}$ by showing that such a
random field is weakly mixing if and only if the group measure space
construction corresponding to some (equivalently, any) Rosiński representation
does not admit a $II_{1}$ factor in its central decomposition. The same proof
now combines with Theorem 4.3 to give the following, resolving completely
Problem 3 (and hence Problem 1 and Conjecture 2) of [41].

######  Theorem 4.5.

Suppose that $G$ is a countable amenable group and
$\mathbf{Y}=\\{Y_{g}\\}_{g\in G}$ is a stationary ${S\alpha S}$ random field.
Suppose that $\mathbf{Y}$ admits a Rosiński representation such that the
underlying non-singular $G$-action $\\{\phi_{g}\\}$ is free. Then the
following are equivalent.

  1. (1)

$\\{Y_{g}\\}_{g\in G}$ is weakly mixing;

  2. (2)

the group measure space construction corresponding to $\\{\phi_{g}\\}_{g\in
G}$ does not admit a $II_{1}$ factor in its central decomposition.

In particular, weak mixing is a $W^{\ast}_{R}$-rigid (and hence
$W^{\ast}_{m}$-rigid) property for the class $\mathcal{G}$ of countably
infinite amenable groups.

###### Proof.

Because of Theorem 4.3 above, it is enough to establish that
$\\{\phi_{g}\\}_{g\in G}$ is a null action if and only if the associated group
measure space construction does not admit a $II_{1}$ factor in its central
decomposition. This follows verbatim the proof of equivalence of (3) and (4)
of [41, Theorem 5.10]. On the other hand, $W^{\ast}_{R}$-rigidity (and hence
$W^{\ast}_{m}$-rigidity) of weak mixing follows directly from the
characterization (2) above. This completes the proof. ∎

Following the argument used in the proof of Theorem 5.20 in [41], one can
establish that having a nontrivial _weakly mixing part_ (i.e., _null part_ as
in Remark 4.2) is also a $W^{\ast}_{R}$-rigid (and hence $W^{\ast}_{m}$-rigid)
property for the class $\mathcal{G}$ of countably infinite amenable groups.

The canonical action of any finitely generated group $G$ on its Furstenberg-
Poisson boundary $(\partial G,\mu^{FP})$ is free and ergodic, and the
associated group measure space construction is a type $III$ factor. This
shows, thanks to Theorem 4.5, that the corresponding stationary ${S\alpha S}$
random fields are nontrivial and weakly mixing as long as $G$ is amenable and
$\partial G$ is nontrivial. Examples arise from amenable lamplighter groups
[21].

##  5\. Boundary actions of hyperbolic groups and Stable Fields

We extract from Theorem 3.4, two sufficient conditions to prove ergodicity of
a $G$-indexed ${S\alpha S}$ random field. Suppose that we are given a non-
singular action $g\to\phi_{g}$ of a discrete group $G$ on a sigma-finite
measure space $(Y,\mu)$. Let
$(X,\nu)=(Y\times(0,\infty),\mu\otimes\mbox{Leb})$ denote the Maharam
extension (Equation 3.2) equipped with a _$\nu$ \- preserving action_
$\phi_{g}^{*}$ as described before Lemma 3.3. There are two ways in which we
shall apply Theorem 3.4 in this section.

Case 1: $\mu(Y)<\infty$ and the $G$-action on $Y$ is non-singular with no
restriction on Radon-Nikodym derivatives:
Without loss of generality, we may assume that $\mu(Y)=1$, i.e. $Y$ is a
probability measure space. In this case, any $E\subset X$ satisfying the
condition $\nu(E)<\infty$ in Theorem 3.4 is contained (up to measure zero) in
a countable _disjoint_ union of sets $B$ of the form
$B(a_{i},b_{i})=Y\times(a_{i},b_{i})$ with $0<a_{i}<b_{i}<\infty$. We shall
show in Section 5.1 that for a number of actions with hyperbolic properties,
the following condition holds for a non-singular action of a group $G$ on $X$.

######  Condition 5.1.

$\frac{1}{|F_{n}|}\sum_{g\in
F_{n}}\nu\bigg{(}\left(Y\times(\frac{1}{K},K)\right)\,\cap\,(\phi_{g}^{*})^{-1}\left(Y\times(\frac{1}{K},K)\right)\bigg{)}\to
0\,\ {\rm as}\,\ n\to\infty.$

By Theorem 3.4 this will suffice to prove ergodicity of the corresponding
$G$-indexed ${S\alpha S}$ random field. Note further that if

$(y,t)\in
A_{K}:=\left(Y\times\left(\frac{1}{K},K\right)\right)\,\cap\,(\phi_{g}^{*})^{-1}\left(Y\times\left(\frac{1}{K},K\right)\right),$

by the definition of $\phi_{g}^{*}$, the Radon-Nikodym derivative satisfies

$\frac{1}{K^{2}}<\frac{d\mu\circ\phi_{g}^{-1}}{d\mu}(y)<K^{2}.$

Hence, $\nu(A_{K})$ in Condition 5.1 is at most $(K-K^{-1})\mu(B_{K})$, where
$B_{K}$ is given by

$B_{K}:=\bigg{\\{}y\in
Y:\frac{1}{K^{2}}<\frac{d\mu\circ\phi_{g}^{-1}}{d\mu}(y)<K^{2}\bigg{\\}}.$

Case 2: $\mu(Y)=\infty$ and the $G$-action on $Y$ is non-singular with Radon-
Nikodym derivatives uniformly bounded away from both $0$ and $\infty$:
In this case, it follows immediately from Theorem 3.4 that to prove ergodicity
of the corresponding $G$-indexed ${S\alpha S}$ random field, it suffices to
check the following for $\mu(B)<\infty$:

######  Condition 5.2.

$\frac{1}{|F_{n}|}\sum_{g\in
F_{n}}\bigg{(}\mu\big{(}B\cap\phi_{g}(B)\big{)}\bigg{)}\to 0\,\ {\rm as}\,\
n\to\infty.$

We shall check Condition 5.2 in Section 5.2 below.

###  5.1. Boundary action of hyperbolic groups

We refer the reader to [2, Section 3.1] for the relevant material on
Patterson-Sullivan measures that we need in this subsection, and to the
original sources [30, 47, 31, 10] for details. We summarize what we need here
from Patterson-Sullivan theory. Let $X$ be a Gromov hyperbolic metric space.
Let $\partial_{G}X$ denote the Gromov boundary [16]. For $x,y\in X$ and
$\xi\in\partial_{G}X$, $\beta_{\xi}(y,x)$ will denote the _Busemann function_
(see [10] for details). In all applications below, $x$ will be chosen as an
origin of the hyperbolic space $X$.

######  Definition 5.3.

[9, p. 721] For $X$ Gromov-hyperbolic, let $M(\partial_{G}X)$ denote the
collection of positive finite Borel measures on $\partial_{G}X$. A
$G-$equivariant map $X\to M(\partial_{G}X)$ sending $x\to\mu_{x}$ is said to
be a $C-$quasiconformal density of dimension $v$ ($v\geq 0$), for some $C\geq
1$, if

(5.1)
$\frac{1}{C}exp(-v\beta_{\xi}(y,x))\leq\frac{d\mu_{x}}{d\mu_{y}}(\xi)\leq
Cexp(-v\beta_{\xi}(y,x))$

for all $x,y\in X$, $\xi\in\partial X$; in particular,

$\frac{1}{C}exp(-v\beta_{\xi}(o,g.o))\leq\frac{d\mu_{g.o}}{d\mu_{o}}(\xi)\leq
Cexp(-v\beta_{\xi}(o,g.o)),$

for all $g\in G$.

######  Example 5.4.

Examples of quasiconformal densities arise in a hyperbolic setting from the
following:

  1. (1)

Patterson-Sullivan measures ${\mu^{PS}}$ on the boundary of non-elementary
hyperbolic groups, see [10].

  2. (2)

More generally, Patterson-Sullivan measures ${\mu^{PS}}$ on limit sets
$\Lambda_{G}$ of non-elementary groups $G$ acting properly by isometries on a
hyperbolic space $X$. In this case, without loss of generality, we may choose
$X$ to be the weak convex hull of $\Lambda_{G}\subset\partial_{G}X$, i.e. $X$
may be chosen to be the union of all bi-infinite geodesics joining pairs of
points in $\Lambda_{G}$. Thus, $\Lambda_{G}=\partial_{G}X$ in this case. See
[10, 47].

  3. (3)

A non-elementary hyperbolic group $G$ acting on its Poisson boundary
$(\partial_{G}G,\mu_{p})$, where $\mu_{p}$ denotes the hitting measure of
random walks on the Poisson boundary. By a result of Kaimanovich [20], the
underlying space of the Poisson boundary in this case may be taken to be the
Gromov boundary $\partial_{G}G$. Further, the natural metric in this context
is the Green metric [8]–a metric quasi-isometric to the word metric on $G$.
See [20, 8].

  4. (4)

Anosov representations of hyperbolic groups $G$ in semi-simple Lie groups $L$
(of possibly higher rank) furnish Patterson-Sullivan measures on their limit
sets $\Lambda_{G}$ contained in $L/P$, where $P$ is a parabolic subgroup.
Fundamental work of Labourie [22] and Guichard-Wienhard [18] shows that the
limit set $\Lambda_{G}$ is homeomorphic to the Gromov boundary $\partial_{G}G$
in this case. Dey and Kapovich [12] prove that the Patterson-Sullivan measure
turns out to be a conformal density with certain uniqueness properties in this
case. See [12].

Given a quasiconformal $\mu$ on $\partial_{G}X$, a basepoint $o\in X$, and
$g\in G$, we shall be interested in the $\mu-$measure of the set

$A(K,o,g):=\\{\xi\in\partial_{G}X:|\beta_{\xi}(o,g.o)|\leq K\\}.$

Equivalently, $A(K,o,g)$ is the set of points $\xi$ in $\partial_{G}X$ where

$\frac{1}{C}\leq\frac{d\mu_{o}\circ\phi_{g}^{-1}}{d\mu_{o}}(\xi)\leq C,$

and the relationship between $K,C$ is obtained from Definition 5.3
(essentially $C$ is an exponential of $K$).

######  Lemma 5.5.

Let $G,\partial_{G}X$ be any of the examples in Example 5.4, and $\mu$ be a
conformal density as in Example 5.4. Then $\mu(A(K,o,g))\to 0$ as
$g\to\infty$. (Here, we interpret $g\to\infty$ as the sequence $g.o$ exiting
any bounded set.)

###### Proof.

For any geodesic segment $\lambda=[a,b]$ in a hyperbolic space $X$, let
$\pi_{\lambda}$ denote a nearest point projection from $X$ onto $\lambda$.
Then $\pi_{\lambda}$ extends to a (coarsely well-defined) map
$\Pi_{\lambda}:\partial_{G}X\to\lambda$. For $o\in X$ as above, and $g\in G$,
let $\lambda_{g}$ denote a geodesic joining $o,g.o$. Let
$\Pi_{g}:\partial_{G}X\to\lambda_{g}$ denote the resulting nearest point
projection $\partial_{G}X$ to $\lambda_{g}$.

Let $m_{g}$ denote the mid-point of $\lambda_{g}$, and $[m_{g}-a,m_{g}+a]$
denote the geodesic subsegment of $\lambda_{g}$ of length $2a$ centered at
$m_{g}$. We next observe that in each of the examples in Example 5.4, there
exists $D$ depending on $K$ such that $A(K,o,g)$ is contained in
$\Pi_{g}^{-1}([m_{g}-D,m_{g}+D])$.

Hence $\mu(A(K,o,g))\leq e^{-{v}(d(o,m_{g})-D)}$. In particular, if
$g\to\infty$, $\mu(A(K,o,g))\to 0$. ∎

######  Theorem 5.6.

In all the examples in Example 5.4, Condition 5.1 holds. Hence, the associated
$G-$indexed ${S\alpha S}$ random field is ergodic.

###### Proof.

For any $\epsilon>0$, Lemma 5.5 implies that there exists $N>0$ such that
$\mu(A(K,o,g))<\epsilon$, whenever $d(o,g.o)\geq N$. In Condition 5.1, define
$F_{m}$ to be

$F_{m}=\\{g\in G:g(o,g.o)\leq m\\},$

though any other compact exhaustion would work as well. Next, given $r\geq 1$
choose $m>N$ such that $|F_{M}|>r|F_{N}|$ for all $M\geq m$. Finally, as
pointed out in the discussion in the beginning of this section, it suffices to
consider sets in Condition 5.1 to be equal to
$(\partial_{G}X)\times[\frac{1}{K},K]$ for some $K>1$, so that the measure of
$B_{K}$ equals $\mu(A(K^{\prime},o,g))$ (for some $K^{\prime}$ depending on
$K$). Then

$\sum_{g\in F_{m}}\big{(}\mu(B_{K})\big{)}=\sum_{g\in
F_{m}}\big{(}\mu(A(K^{\prime},o,g))\big{)}<|F_{m}|\epsilon+|F_{N}|.$

Hence,

$\frac{1}{|F_{m}|}\sum_{g\in
F_{m}}\big{(}\mu(B_{K})\big{)}<\epsilon+\frac{1}{r}.$

Since $\epsilon$ can be chosen arbitrarily small, and $r$ arbitrarily large,
Condition 5.1 follows. ∎

###  5.2. Double boundary action and Bowen-Margulis-Sullivan measures

We recall the Bader-Furman construction of Bowen-Margulis-Sullivan measures
[15, 5] for $X$ Gromov-hyperbolic and $G$ acting properly discontinuously and
cocompactly by isometries on it (e.g. $X$ may be taken to be a Cayley graph of
$G$ with respect to a finite generating set). Let $[\mu]$ be the Patterson-
Sullivan conformal density in Example 5.4 (1) on $\partial_{G}X$. The square
class $[\mu\times\mu]$ is supported on $\partial_{G}X\times\partial_{G}X$. The
Gromov inner product on $X$ with respect to the basepoint $o$ will be denoted
by $\langle{x},{y}\rangle_{o}$.

######  Proposition 5.7.

[15, Proposition 1][5, Proposition 3.3] There exists a $G-$invariant Radon
measure, denoted ${\mu^{BMS}}$, in the measure class $[\mu\times\mu]$ on
$(\partial_{G}X\times\partial_{G}X)\setminus\Delta$, where $\Delta$ denotes
the diagonal. Moreover, ${\mu^{BMS}}$ has the form

$d{\mu^{BMS}}(x,y)=e^{F(x,y)}\,d\mu(x)\,d\mu(y)$

where $F$ is a measurable function on $\partial_{G}X\times\partial_{G}X$ of
the form $F(x,y)=2v\,\langle{x},{y}\rangle_{o}+O(1),$ and $v$ is the Hausdorff
dimension of $\mu$.

The measure ${\mu^{BMS}}$ is called the _Bowen-Margulis-Sullivan measure._

######  Proposition 5.8.

Let $G$ be a Gromov hyperbolic group acting properly discontinuously and
cocompactly by isometries on $X$. Let ${\mu^{BMS}}$ be the Bowen-Margulis-
Sullivan measure on $(\partial_{G}X\times\partial_{G}X)\setminus\Delta$. Then
Condition 5.2 is satisfied for the diagonal $G-$action on
$(\partial_{G}X\times\partial_{G}X,{\mu^{BMS}})$. Hence, the associated
$G-$indexed ${S\alpha S}$ random field is ergodic.

###### Proof.

We need to check condition 5.2 for finite measure subsets of
$\partial_{G}X\times\partial_{G}X$. Note that
$\partial_{G}X\times\partial_{G}X\setminus\Delta$ may be identified with bi-
infinite geodesics in $X$ joining pairs of distinct points on $\partial_{G}X$.
We further note that finite measure subsets of
$(\partial_{G}X\times\partial_{G}X,{\mu^{BMS}})$ can equivalently be generated
by the collection of bi-infinite geodesics in $X$ passing through a compact
subset of $X$. Let ${\Omega}_{A}\subset\partial_{G}X\times\partial_{G}X$ then
denote the collection of bi-infinite geodesics in $X$ passing through a
compact $A\subset X$.

It thus suffices to show that for any compact $A\subset X$, Condition 5.2 is
satisfied for ${\Omega}_{A}$. We shall prove the stronger assertion, as in
Lemma 5.5 that ${\mu^{BMS}}({\Omega}_{A}\cap g.{\Omega}_{A})\to 0$ as
$g\to\infty$. The same argument as in the proof of Theorem 5.6 will then
furnish Condition 5.2.

To prove that ${\mu^{BMS}}({\Omega}_{A}\cap g.{\Omega}_{A})\to 0$ as
$g\to\infty$, observe that ${\mu^{BMS}}({\Omega}_{A}\cap g.{\Omega}_{A})$ is
the same as the measure of geodesics passing through both $A$ and $g.A$. Let
$d_{g}=d(A,g.A)$ denote the distance (in $X$) between $A$ and $g.A$. Let
$\lambda(g,A)$ denote a shortest geodesic joining $A$ and $g.A$, and $l_{g}$
denote its length. Let $m_{g}$ denote the mid-point of $\lambda(g,A)$. Since
$G$ acts cocompactly on $X$, there is a uniformly bounded constant $D_{0}$
(the diameter of a fundamental domain) and an element $h\in G$ such that
$h.m_{g}$ lies within $D_{0}$ of a fixed basepoint $o\in X$. Further, since
${\mu^{BMS}}$ is $G-$invariant by Proposition 5.7,
${\mu^{BMS}}({\Omega}_{A}\cap g.{\Omega}_{A})={\mu^{BMS}}(h.{\Omega}_{A}\cap
hg.{\Omega}_{A})$, and ${\mu^{BMS}}(h.{\Omega}_{A}\cap hg.{\Omega}_{A})$ is
the measure of geodesics passing through both $h.A$ and $hg.A$. The distance
of both $h.A$ and $hg.A$ from $o$ is at least $(\frac{1}{2}d(o,g.o)-D_{0})$ as
$h.\lambda_{g}$ is (coarsely) centered at $h.m_{g}$. Let $D_{g}=d(o,h.A)$.
Clearly, $D_{g}\to\infty$ as $g\to\infty$. Finally, the measure
${\mu^{BMS}}(h.{\Omega}_{A}\cap hg.{\Omega}_{A})$ is at most
$C_{0}e^{-vD_{g}}$, where

  1. (1)

$C_{0}$ depends only on $A$,

  2. (2)

$v$ is the Hausdorff dimension of $\partial_{G}X$ with respect to the
Patterson-Sullivan measure $\mu$.

In particular, ${\mu^{BMS}}(h.{\Omega}_{A}\cap hg.{\Omega}_{A})\to 0$ as
$g\to\infty$, proving our claim. ∎

Open Problems: We conclude this paper with some open questions for stationary
${S\alpha S}$ random fields indexed by amenable and hyperbolic groups.
1) Does Theorem 4.3 hold for left-stationary max-stable random fields indexed
by amenable groups?
2) Does Theorem 4.3 hold for amenable actions (in the sense of Zimmer)?
3) In the case of hyperbolic groups, does null action characterize ergodicity
of the associated ${S\alpha S}$ random field? (Note that the actions in
Example 5.4 are null.)
4) Is it possible to characterize strong mixing for stationary ${S\alpha S}$
random fields indexed by amenable groups in terms of the underlying non-
singular action?
5) What is the role of type $III$ factors on mixing properties?

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