A note on a local ergodic theorem for an infinite tower of coverings

0.18 MB·English

by  Ryokichi Tanaka

A NOTE ON A LOCAL ERGODIC THEOREM FOR AN INFINITE
TOWER OF COVERINGS.
RYOKICHI TANAKA
5
1
0 Abstract. This is a note on a local ergodic theorem for a symmetric exclusion
2
processdefinedonaninfinitetowerofcoverings,whichisassociatedwithafinitely

c generated residually finite amenable group.
e
D
1 1. Introduction
3

The hydrodynamic limit is one of the main frameworks to understand scaling
]
R limit of interacting particle systems in order to capture the relation between micro-
P scopic and macroscopic phenomena in statistical physics. There have been studied
.
h a number of models such as exclusion processes (e.g., [4] and [9]), the stochastic
t
a Ginzburg-Landau model (e.g., [5]), a chain of anharmonic oscillators [13] and sto-
m
chastic energy exchange models [14]. See also [8] and [16] for a background of this
[ problem and its history. While a large number of studies have been devoted to

2 understand stochastic models on the one-dimensional or the d-dimensional discrete
v
torus and their scaling limits, several attempts have been made to generalise the
1
2 underlying graphs to the ones with rich geometric structure. For example, Jara has
5 studied a zero-range process on the Sierpinski gasket [6] and obtained a nonlinear
0
heatequationwhich thelimitofdensityempirical measuresatisfies. Theauthorgave
0
. a unified framework to discuss exclusion processes on a general class of graphs and
1
0 obtained the hydrodynamic limit result for crystal lattices [17]. Recently, Sasada
5 has given a unified and simplified approach to obtain the hydrodynamic limit for
1
an important class of exclusion processes, so-called of non-gradient type, in gen-
:
v
eral settings [15]. Although it is tempting to study a number of stochastic models
i
X on general underlying graphs from viewpoint of physics, the attempts made so far
r depend on the models and also on the structure of graphs. This note proposes a
a
possible approach to unify these generalisations on crystal lattices and self-similar
graphs, and gives a local ergodic theorem for exclusion processes for a step toward
to obtain the scaling limit.
In group theory, there is a remarkable class of groups which have been studied
in the context of dynamical systems. Bartholdi, Grigorchuk and Nekrashevych de-
scribed a number of self-similar (fractal) sets as the scaling limits of finite graphs
associated with group actions [1]. They showed that a class of Julia sets and the
Sierpinski gasket can be obtained as the scaling limits of Schreier graphs of some
groups. See also [7] for another (yet related) construction of such a limit. These
examples appear as groups acting on rooted trees, and a sequence of Schreier graphs
is associated with the action on tree ([12] for a general background on this topic).
Here we discuss the simplest case, an infinite tower of coverings associated with
1
2 RYOKICHITANAKA
those kind of groups (Section 2.1). The group we consider is finitely generated, in-
finite, residually finite amenable (see Section 2.1 for the definition). We define an
exclusion process on each finite covering graph, and take a limit as the size of graphs
goes to infinity.
A limit theorem we are considering is a local ergodic theorem which describes
local equilibrium states of particle systems. The theorem is also referred as the
replacement lemma, e.g., in [8] and [17], and enables us to replace a local average of
number of particles by a global average after a large enough time. There involves
an ergodic theoretic argument to exchange the space average and the time average.
In the hydrodynamic limit, this theorem has a role to describe several different local
equilibrium states, and to verify the derivation of a (possibly non-linear) partial
differential equation by pasting together those states. Here we formulate a local
ergodic theorem by using a notion of local function bundle introduced in [17], and
show it in the form of super exponential estimate (Theorem 2.1). The proof is based
on the entropy method due to [5] and also [10]. In order to obtain the scaling limit
result, one has to describe a limiting space, which we will not discuss in this note.
The problem we discuss here is the first step before taking the scaling limit. It would
be interesting to complete the second step and to obtain the hydrodynamic limit.
The organisation of this note is the following. In Section 2, we introduce the
setting and examples of infinite tower of coverings, define exclusion processes and
formulate a local ergodic theorem (Theorem 2.1). In Section 3, we show Theorem
2.1 and also prove auxiliary results: the one-block estimate (Theorem 3.2) and the
two-blocks estimate (Theorem 3.3).
2. Notation and results
2.1. Groups and associated infinite towers of coverings. Let Γ = S be
h i
a finitely generated group, where S is a finite symmetric set of generators, i.e.,
S = S 1. Throughout this note Γ is an infinite group. We assume that Γ is
−
residually finite, i.e., there is a descending sequence of finite index normal subgroups
{Γi}∞i=0 such that Γi ⊳Γ with [Γ : Γi] < ∞ for each i, and ∞i=1Γi = {id}. We also
assume that Γ is amenable, i.e., there exists a sequence of finite subsets F of
T { i}∞i=0
Γ such that lim ∂ F / F = 0 where ∂ F := F S F . We call such a sequence
i S i i S i i i
F a Følner→s∞eq|uenc|e |and| each F a Følner set. H\enceforth we fix some finite
{ i}∞i=0 i
generating set S in Γ and denote by x the corresponding word norm of x in Γ,
Γ
| |
i.e., the minimum number of elements in S to obtain x.
Example 2.1.
(i) The group of integers Z. We can take S = 1,1 as a finite symmetric set
of generators. Considering the subgroups Γ{−:= 2}iZ for positive integers i,
i
one can check that the group is residually finite. The sets F := [ i,i] Z
i
gives a Følner sequence. In the same way, Zd(d 1) is also residu−ally fi⊂nite
≥
and amenable.
(ii) Grigorchuck group. Grigorchuck group is residually finite and a subsequence
of balls forms a Følner sequence since it has subexponential volume growth.
(See, e.g., Chapter 6 in [3].)
A NOTE ON A LOCAL ERGODIC THEOREM FOR AN INFINITE TOWER OF COVERINGS.3
(iii) Basilica group. Basilica group is realized as a finitely generated subgroup in
the automorphism group of the binary tree, and this implies that the group
is residually finite. The group is also amenable as proved in [2].
LetX = (V,E)beaCayley graphofΓassociatedwithS, i.e., theset ofvertices V
is Γ, and the set of oriented edges E := (x,xs) Γ Γ : s S . Here we consider
{ ∈ × ∈ }
the Cayley graph as an oriented graph where both possibilities of orientation are
included, i.e., if e E, then its reversed edge e E. Denote the origin of e by
∈ ∈
oe and the terminus by te. The group Γ acts on X from the left hand side freely
and vertex transitively. The quotient graph X := Γ X consists of one vertex and
0
\
(oriented) loop edges. We use the graph distance d in X, and denote by B(x,r) the
ball in V centred at x with radius r. Abusing the notation, we denote the graph
distance by d in other graphs as well.
Remark 2.1. We can extend our results to quasi-transitive graphs, i.e., Γ acts on
X with a finite number of orbits. If Γ = Zd, the quasi-transitive graph X is called
a crystal lattice ([11] and [17]).
Let Γ be a descending sequence of normal subgroups such that Γ ⊳Γ and
{ i}∞i=0 i
[Γ : Γi] < ∞ for each i, and ∞i=1Γi = {id}. For each i, define the quotient finite
graph X := Γ X. Each X is a finite graph since Γ is a finite index subgroup
i i i i
\ T
of Γ. Here Γ/Γ acts on X freely and vertex transitively, where Γ/Γ is a finite
i i i
group. Then we have an infinite tower of coverings of finite graphs: X X for
i 0
→
i = 1,2,....
Let us define a particle system on X. Define the configuration space by Z :=
0,1 V, and denote each configuration by η := η . The action of Γ lifts on Z
x x V
{ } { } ∈
naturally, by setting (ση) := η for σ Γ, η Z and z V. In the same way,
z σ 1z
− ∈ ∈ ∈
for each quotient finite graph X = (V ,E ) = Γ X, we define a configuration
m m m m
\
space Z := 0,1 Vm. The action of Γ/Γ on X lifts on Z as above. Here we
m m m m
{ }
define a local function bundle, which is defined on the product space of the state
space V and the configuration space Z. This is used to formulate our local ergodic
theorem.
Definition 2.1. A Γ-invariant local function bundle for vertices is a function f :
V Z R such that:
× →
There exists r 0 such that for every x V, the function f : Z R
x
• ≥ ∈ →
depends only on η .
z z B(x,r)
For every σ Γ,{x }V∈,η Z, it holds that f(σx,ση) = f(x,η).
• ∈ ∈ ∈
In a similar way, a Γ-invariant local function bundle for edges is a function f :
E Z R such that:
× →
There exists r 0 such that for every e E, the function f : Z R
e
• ≥ ∈ →
depends only on η .
z z B(oe,r) B(te,r)
For all σ Γ,e {E,}η∈ Z, i∪t holds that f(σe,ση) = f(e,η).
• ∈ ∈ ∈
Example 2.2.
4 RYOKICHITANAKA
(i) For x V, if we define f (η) := η , then f is a Γ-invariant local function
x x
∈
bundle for vertices. In this case, for each x V, the function f depends
x
∈
only on a configuration on x. The f is Γ-invariant by definition.
(ii) For x V, if we define f (η) := η , where E := e E : oe = x ,
∈ x e Ex te x { ∈ }
then f is a Γ-invariant local function∈ bundle for vertices.
Q
(iii) Fore E, if we define f (η) := η +η , then f is aΓ-invariant local function
e oe te
∈
bundle for edges.
(iv) For e E, if we define f (η) := η + η + c, where c > 0,
thenf∈isaΓ-invariantlocealfunctione′b∈uEonedltee′foredeg′e∈sEtaendte′satisfiesf(e,η) c
Q Q ≥
for all e E,η Z.
∈ ∈
Let F be a Følner sequence for Γ. For a Γ-invariant local function bundle
for vert{iceis}∞if=1: V Z R, we define a local average associated with F . For
× → { i}∞i=1
x V and F , let
i
∈
1
f := f .
x,i F σx
i
| | σX∈Fi
Note that f is again a Γ-invariant local function bundle.
,i
·
Let us introduce a local average on a Følner set controlled by its size. For a non
negative real number K, we define
b(K) := max i : F B(o,K) ,
i
{ ⊆ }
i.e., b(K) is the largest number L such that all F ,i = 1,...,L are included in the
i
ball centred at o with radius K. Note that b(K) is non-decreasing and goes to
∞
as K goes to . We also consider a local average of the following type:
∞
1
f := f .
x,b(K) F σx
b(K)
| | σ∈XFb(K)
Fix a distinguished vertex o V. Now Γ acts on V transitively, so every vertex
∈
x can be written in the form x = γo for some γ Γ. For every m 1, denote by
∈ ≥
the same character o V , the image of o via the covering map X X . For
m m
∈ →
σ Γ, denote by σ Γ/Γ the image of σ via the canonical surjection. Since f is
m
∈ ∈
Γ-invariant, the Γ-invariant local function bundles f for vertices and edges induce
functions on V Z , and E Z respectively, they are Γ/Γ -invariant under
m m m m m
× ×
the diagonal action for each m. We also use the same character for these induced
ones.
2.2. Particle systems. Assume that Z and Z are equipped with the prodiscrete
m
topology, i.e., the product of discrete topology. Denote by (Z ) and by (Z) the
m
P P
spaces of Borel probability measures on Z and on Z, respectively. We also define
m
the (1/2)-Bernoulli measures ν on Z and ν on Z, as the direct product of the
m m
(1/2)-Bernoulli measures on 0,1 .
{ }
A NOTE ON A LOCAL ERGODIC THEOREM FOR AN INFINITE TOWER OF COVERINGS.5
Let us define the symmetric exclusion process on X. For a configuration η Z
∈
and an edge e E, define by ηe a configuration exchanging states on oe and te, i.e.,
∈
η z = oe,
te
ηe := η z = te,
z  oe
η otherwise,
z
for z V. Note that ηe = ηe. Furthermore, for a function F on Z and e E, we
∈ ∈
define π F(η) := F(ηe) F(η). We also define the corresponding ones for η Z
e m
− ∈
and e E in the same way. The symmetric exclusion process on X is defined in
m
∈
terms of a Γ-invariant local function bundle for edges. We call a Γ-invariant local
function bundle c : E Z R is a jump rate if it satisfies the following properties:
× →
(symmetric) c(e,η) = c(e,η) and c(e,η) = c(e,ηe) for all e E,η Z.
• ∈ ∈
(non-degenerate) There exists a positive constant c such that c c(e,η)
0 0
• ≤
for all e E,η Z.
∈ ∈
DefinethegeneratorofasymmetricexclusionprocessbytheoperatorL : L2(Z ,ν )
m m m
→
L2(Z ,ν ),
m m
L F(η) := c(e,η)π F(η),
m e
eX∈Em
for F L2(Z ,ν ).
m m
∈
We consider a family of exclusion processes on an infinite tower of coverings.
Each process on a covering graph X is speeded up by some time scaling factor
m
t . Suppose that t is an increasing sequence of positive numbers: t < t <
m { m}∞m=1 1 2
< t with the condition √t 2diamX , where diamX denotes the
m m m m
··· → ∞ ≤
diameter of X .
m
Consider the continuous time Markov chain generated by L speeded up by t .
m m
Fix an arbitrary time T > 0 and denote by D([0,T],Z ) the space of paths being
m
right continuous and having left limits. For a probability measure µ on Z , we
m m
define by P the distribution on D([0,T],Z ) of the continuous time Markov chain
m m
η (t) generated by t L with the initial measure µ .
m m m m
For 0 ρ 1, denote by ν the ρ-Bernoulli measure on Z, that is the direct
ρ
≤ ≤
product of the ρ-Bernoulli measures on 0,1 . We define a global average of a Γ-
invariant local function bundle f : V Z{ R}by f (ρ) := E [f ] the expectation
of the function f : Z R with respe×ct to→ν . h oi νρ o
o ρ
→
The following estimate enables us to approximate a global average of a local
function bundle by a local average of one under a suitable time-space average. This
estimate is referred as a local ergodic theorem, which we show in the super expo-
nential estimate.
Theorem 2.1. Fix T > 0. For every Γ-invariant local function bundles f : V Z
R and every δ > 0, × →
1 1 T
lim limsuplimsup logP V (η(t))dt δ = ,
m o,m,ε,i
i→∞ ε→0 m→∞ [Γ : Γm] (cid:18)[Γ : Γm] Z0 ≥ (cid:19) −∞
6 RYOKICHITANAKA
where
V (η) = f (η) f η .
o,m,ε,i σo,i −h oi σo,b(ε√tm)
σ∈XΓ/Γm(cid:12) (cid:0) (cid:1)(cid:12)
(cid:12) (cid:12)
Hereweremarkthatthistheoremgeneralises[17][Theorem4.1]forcrystallattices.
3. Proof of Theorem 2.1
The super exponential estimate for P is reduced to Peq which is the distribution
m m
of continuous time Markov chain generated by t L with the initial measure the
m m
(1/2)-Bernoulli measure ν , i.e., an equilibrium measure. Indeed, for every mea-
m
surable set A D([0,T],Z ), P (A) 2[Γ:Γm]Peq(A) and the factor 2[Γ:Γm] does
⊂ m m ≤ m
not contribute to the super exponential estimate. (Note that [Γ : Γ ] = V .)
m m
| |
Moreover, the super exponential estimate Theorem 2.1 is reduced to the following
theorem.
For µ (Z ), define the Dirichlet form for ϕ := dµ/dν by
m m
∈ P
I (µ) := √ϕL √ϕdν ,
m m m
−
ZZm
2
which is also equal to (1/2) c(e,η) π √ϕ dν . For every C > 0, we
Zm e∈Em e m
define the subset of (Z ) by
P m R P (cid:0) (cid:1)
[Γ : Γ ]
m
:= µ (Z ) : µ is Γ/Γ -invariant and I (µ) C .
m,C m m m
P ∈ P ≤ t
(cid:26) m (cid:27)
Theorem 3.1. For every C > 0, it holds that
lim limsuplimsup sup E f f (η ) = 0.
i→∞ ε→0 m→∞ µ∈Pm,C µ o,i −h oi o,b(ε√tm)
(cid:12) (cid:12)
(cid:12) (cid:12)
First, we see how to deduce Theorem 2.1 from Theorem 3.1.
Proof of Theorem 2.1. As we observe, it suffices to prove the required estimate for
Peq. Recall that V (η) = f (η) f η .
m o,m,ε,i σ Γ/Γm σo,i −h oi σo,b(ε√tm)
∈
By the Chebychev inequaliPty, for all(cid:12)a > 0 and for (cid:0)all δ > 0, (cid:1)(cid:12)
(cid:12) (cid:12)
1 T T
Peq V (η(t))dt δ Eeqexp a V (η(t))dt aδ[Γ : Γ ] .
m [Γ : Γ ] o,m,ε,i ≥ ≤ m o,m,ε,i − m
(cid:18) m Z0 (cid:19) (cid:18) Z0 (cid:19)
For all a > 0, we consider the operator
t L +aV : L2(Z ,ν ) L2(Z ,ν ),
m m o,m,ε,i m m m m
→
which is self-adjoint for all a > 0 by the definition of L . Denote by λ (a)
m o,m,ε,i
the largest eigenvalue of t L + aV . By the Feynmann-Kac formula (e.g.,
m m o,m,ε,i
[8][Lemma 7.2, Appendix 1]),
T
Eeqexp a V dt expTλ (a).
m o,m,ε,i ≤ o,m,ε,i
(cid:18) Z0 (cid:19)
A NOTE ON A LOCAL ERGODIC THEOREM FOR AN INFINITE TOWER OF COVERINGS.7
Therefore it suffices to show that for all a > 0,
1
lim limsuplimsup λ (a) = 0. (3.1)
o,m,ε,i
i→∞ ε→0 m→∞ [Γ : Γm]
Indeed, by using (3.1), we have that
1 1 T
lim limsuplimsup logPeq V (η(t))dt δ aδ.
i→∞ ε→0 m→∞ [Γ : Γm] m(cid:18)[Γ : Γm] Z0 o,m,ε,i ≥ (cid:19) ≤ −
Taking a , we obtain Theorem 2.1.
→ ∞
It remains to prove (3.1). By the variational principle, the largest eigenvalue
λ (a) can be expressed in the following form:
o,m,ε,i
λ (a) := sup a V dµ t I (µ) .
o,m,ε,i o,m,ε,i m m
−
µ∈P(Zm)(cid:26) ZZm (cid:27)
It is enough to consider only the case when a V dµ t I (µ) for some
Zm o,m,ε,i ≥ m m
µ (Z ). For µ (Z ), we deduce by µ the average of µ by the Γ/Γ -action,
∈ P m ∈ P m R m
that is,
1
µ := µ σ.
[Γ : Γ ] ◦
m
σ∈XΓ/Γm
Here µ is a Γ/Γ -invariant probability measure. Now we have that
m
1
V dµ = E f f (η ) .
[Γ : Γ ] o,m,ε,i µ o,i −h oi o,b(ε√tm)
m ZZm
(cid:12) (cid:12)
There exists a constant C(f) > 0 depending(cid:12)only on f such that(cid:12)
V C(f)[Γ : Γ ].
o,m,ε,i m
≤
By the convexity of I ,
m
1
I (µ) I (µ σ),
m m
≤ [Γ : Γ ] ◦
m
σ∈XΓ/Γm
and since I is Γ/Γ -invariant, we have that
m m
[Γ : Γ ]
m
I (µ) aC(f) .
m
≤ t
m
When we denote by the set of probability measures µ which is Γ/Γ -
m,aC(f) m
P
invariant with I (µ) aC(f)[Γ : Γ ]/t , then
m m m
≤
1
λ (a) a sup E f f (η ) .
[Γ : Γ ] o,m,ε,i ≤ µ o,i −h oi o,b(ε√tm)
m µ∈Pm,aC(f)
(cid:12) (cid:12)
Hence (3.1) follows from Theorem 3.1. (cid:12) (cid:12) (cid:3)
Henceforth we often identify a probability measure on Z with the one on Z by
m
the periodic extension: Let π : V V be the covering map induced by the Γ-
m m
→
action. Define a periodic inclusion ι : Z Z by (ι η) := η , η Z ,z V.
m m → m z πm(z) ∈ m ∈
We identify µ on Z with its push forward by ι , which is a periodic extension of
m m
µ on Z. Conversely, we identify a Γ -invariant probability measure on Z with the
m
one on Z in a natural way.
m
8 RYOKICHITANAKA
Theorem 3.1 follows the one-block estimate (Theorem 3.2) and the two-blocks
estimate (Theorem 3.3). First, we prove the one-block estimate.
Theorem 3.2 (Theone-blockestimate). Forevery Γ-invariantlocal function bundle
f : V Z R and for every C > 0, it holds that
× →
lim limsup sup E f f (η ) = 0.
i→∞ m→∞ µ∈Pm,C µ o,i −h oi o,i
(cid:12) (cid:12)
(cid:12) (cid:12)
We discuss a restricted region in X and define the corresponding Dirichlet form.
Let Λ = (V ,E ) be a subgraph of X. Define the restricted configuration space by
Λ Λ
Z := 0,1 VΛ, and the (1/2)-Bernoulli measure ν on Z by the direct product of
Λ Λ Λ
{ }
the (1/2)-Bernoulli measures on 0,1 .
{ }
In our setting, Γ acts on X vertex transitively, thus o V is a fundamental
∈
domain in V. We can choose a fundamental domain E0 in E as the set of those
edges; oe or te = o. We use the same notation E0 for the image of E0 on X via
m
the covering map. Define the operator on L2(Z ,ν ) by
Λ Λ
1
L := π .
◦Λ 2 e
eX∈EΛ
For µ (Z), we denote by µ the restriction of µ on Z , that is, defined by taking
Λ Λ
∈ P |
the average outside of Λ. Set ϕ := dµ /dν the density of µ . The corresponding
Λ Λ Λ Λ
| |
Dirichlet form of √ϕ is
Λ
I (µ) := √ϕ L √ϕ dν .
Λ◦ − Λ ◦Λ Λ Λ
ZZΛ
Proof of Theorem 3.2. Define a subgraph Λ of X by setting V := B(o,K) and
Λ
E := e E : oe,te V . For every µ , by the convexity of the Dirichlet
Λ Λ m,C
{ ∈ ∈ } ∈ P
form and by the (Γ/Γ )-invariance of µ and ν , we have
m m
1
I (µ) = (π √ϕ )2dν
Λ◦ 2 e Λ Λ
eX∈EΛZZΛ
1
(π √ϕ)2dν
e m
≤ 2
eX∈EΛZZm
1
(π √ϕ)2dν
e m
≤ 2
σ∈Γ/ΓXm,|σ|Γ≤Ke∈XσE0ZZm
1
= B (K) (π √ϕ)2dν ,
Γ e m
2| |
e E0ZZm
X∈
where in the last term B (K) denotes the ball of radius K in Γ about id in Γ by
Γ
the word norm . On the other hand in the same way, we have that
Γ
|·|
I (µ) = [Γ : Γ ](1/2) c(e,η)(π √ϕ)2dν . (3.2)
m m e m
e E0ZZm
X∈
A NOTE ON A LOCAL ERGODIC THEOREM FOR AN INFINITE TOWER OF COVERINGS.9
By the uniformly boundedness of c(e,η) c > 0 (the non-degeneracy of c( , )), it
0
≥ · ·
holds that
B (K)
Γ
I (µ) | | I (µ).
Λ◦ ≤ c [Γ : Γ ] m
0 m
Since µ satisfies I (µ) C[Γ : Γ ]/t , we have that I (µ) C /t 0 as
m ≤ m m Λ◦ ≤ K m →
m , where C is a constant depending only on K. The space of probability
K
→ ∞
measures (Z) is compact with the weak topology. Thus, every sequence µ in
P { i}∞i=1
(Z) has a convergence subsequence. Let (Z) be the set of all limit points of
P A ⊂ P
µ in (Z). By the above argument, we have that I (µ) = 0 for every µ .
{ i}∞i=1 P Λ◦ ∈ A
We obtain that µ (ηe) = µ (η) for all e E and all η Z since
Λ Λ Λ Λ
| | ∈ ∈
I (µ) = (1/2) (π µ (η))2 = 0.
Λ◦ e |Λ
eX∈EΛηX∈ZΛ p
This holds for every Λ with radius K and implies that random variables η
x x V
{ } ∈
are exchangeable under µ. By the de Finetti theorem, there exists a probability
1
measure λ on [0,1] such that µ = ν λ(dρ), where ν is the ρ-Bernoulli measure
0 ρ ρ
on Z. Then we have
R
limsup sup E f f (η ) supE f f (η )
µ o,i −h oi o,i ≤ µ o,i −h oi o,i
m→∞ µ∈Pm,C µ∈A
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) sup E(cid:12) f f (η (cid:12)) .
≤ νρ o,i −h oi o,i
ρ [0,1]
∈
(cid:12) (cid:12)
Therefore, itisenoughtoshowthatlim sup E (cid:12)f f (η )(cid:12) = 0. Since
f is a Γ-invariant local function bundlie→,∞thereρ∈e[x0i,1s]tsνLρ o,0i −suhchoithao,ti f : Z R
o
(cid:12)≥ (cid:12) →
depends only on η : d(o,z) L and there exists a co(cid:12)nstant C(f) > 0(cid:12) depending
z
{ ≤ }
only on f such that
E f E [f ] 2 C(f)C / F 0 (3.3)
νρ o,i − νρ o,i ≤ L | i| →
as i , where C is a c(cid:12)onstant depend(cid:12)ing only on L. Here we have that f (ρ) =
L o
→ ∞ (cid:12) (cid:12) h i
E [f ] since f is Γ-invariant. Since f (ρ) is a polynomial with respect to ρ, in
νρ o,i h oi
particular, uniformly continuous on [0,1], it holds that
sup E f (η ) f (ρ) 0,i .
νρ h oi o,i −h oi → → ∞
ρ [0,1]
∈
(cid:12) (cid:12)
By the triangular inequality, (cid:12) (cid:12)
sup E f f (η )
νρ o,i −h oi o,i
ρ [0,1]
∈
(cid:12) (cid:12)
sup E(cid:12) f f (ρ)(cid:12) + sup E f (ρ) f (η ) 0,i .
≤ νρ o,i −h oi νρ h oi −h oi o,i → → ∞
ρ [0,1] ρ [0,1]
∈ ∈
(cid:12) (cid:12) (cid:12) (cid:12)
This concludes tha(cid:12)t lim lims(cid:12)up sup(cid:12) E f f (cid:12)(η ) = 0. (cid:3)
i→∞ m→∞ µ∈Pm,C µ o,i −h oi o,i
(cid:12) (cid:12)
Next, we prove the two-blocks estimate. (cid:12) (cid:12)
Theorem 3.3 (The two-blocks estimate). For every C > 0, it holds that
lim limsuplimsuplimsup sup sup E η η = 0.
i→∞ ε→0 L→∞ m→∞ σ∈Γs.t.L<|σ|≤ε√tmµ∈Pm,C µ(cid:12) o,i − σo,i(cid:12)
(cid:12) (cid:12)
10 RYOKICHITANAKA
We introduce the following notions: Denote by (Z Z) the space of probability
P ×
measures on Z Z. For σ Γ, define σˆ : Z Z Z by σˆ(η) := (η,σ 1η). For
−
× ∈ → ×
µ (Z), denote by σˆµ (Z Z) the push forward by σˆ, i.e., σˆµ := µ σˆ 1. We
−
∈ P ∈ P × ◦
define the subset in (Z Z) as the set of all limit points of σˆµ : L < σ
ε,L
A P × { | | ≤
ε√t ,µ as m , and the subset in (Z Z) as the set of all limit
m m,C ε
∈ P } → ∞ A P ×
points of as L .
ε,L
A → ∞
Then we have that:
limsuplimsup sup sup η η (σˆµ)(dηdη)
o,i − ′o,i ′
L→∞ m→∞ σ∈Γs.t.L<|σ|≤ε√tmµ∈Pm,C Z(η,η′)∈Z×Z (cid:12) (cid:12)
(cid:12) (cid:12)
sup η η µ(dηdη ).
≤ o,i − ′o,i ′
µ∈AεZ(η,η′)∈Z×Z (cid:12) (cid:12)
As in the proof of the one block estim(cid:12)ate, we i(cid:12)ntroduce a subgraph Λ in X, by
setting V := B(o,K), E := e E : oe,te V and consider the generator L
Λ Λ { ∈ ∈ Λ} ◦Λ
on L2(Z,ν) by
1
L = π .
◦Λ 2 e
eX∈EΛ
Then let us define the two generators on L2(Z Z,ν ν) and the corresponding
× ⊗
Dirichlet forms. First, we define the generator on L2(Z Z,ν ν) by L 1 +
× ⊗ ◦Λ ⊗
1 L . For µ (Z Z), denote by µ the restriction of µ on Z Z ,
⊗ ◦Λ ∈ P × |Λ×Λ Λ × Λ
i.e., µ (η,η ) := µ( (η,η ) : η = η,η = η ) for (η,η ) Z Z . The
Λ Λ ′ ′ Λ ′ Λ ′ ′ Λ Λ
| × { | | } ∈ ×
corresponding Dirichlet form of √ϕ , where ϕ := dµ /dν ν is defined
Λ Λ Λ Λ Λ Λ Λ Λ
× × | × ⊗
by
e e e e
IΛ◦×Λ(µ) := −ZZΛ×ZΛ√ϕΛ×Λ(L◦Λ ⊗1+1⊗L◦Λ)√ϕΛ×ΛdνΛ ⊗νΛ.
Second, we define thegenerator onL2(Z Z,ν ν) inthe following way: For(x,y)
× ⊗ ∈
V V and (η,η ) Z Z, we construct a new configuration (η,η )(x,y) Z Z
′ ′
× ∈ × ∈ ×
by setting (η ,η ) on (x,y), (η ,η ) on (y,x) and keeping unchanged otherwise. For
y′ x x′ y
F L2(Z Z,ν ν), (x,y) V V, defineπ F((η,η)) = F((η,η )(x,y)) F((η,η )).
x,y ′ ′ ′
∈ × ⊗ ∈ × −
Define the generator L on L2(Z Z,ν ν) by
◦o,o × ⊗
L :=eπ ,
◦o,o o,o
and the corresponding Dirichlet form of √ϕ by
Λ Λ
e×
(o,o)
I (µ) := √ϕ L √ϕ dν ν .
Λ×Λ −ZZΛ×ZΛ Λ×Λ ◦o,o Λ×Λ Λ ⊗ Λ
Then we use the following lemma. The proof appears in Lemma 4.2 in [17], so
we omit it.
Lemma 3.1. For all F L2(Z ,ν ) and all σ Γ/Γ , it holds that
m m m
∈ ∈
(π F)2dν 4d(o,σo)2 (π F)2dν .
o,σo m e m
≤
ZZm e E0ZZm
X∈