Table Of ContentA remark on a central limit theorem for
non-symmetric random walks on crystal lattices
7
1
Ryuya Namba
0 ∗
2
January 31, 2017
n
a
J
0
3
Abstract
]
R
Recently, Ishiwata, Kawabi and Kotani [4] proved two kinds of central limit
P
. theorems for non-symmetric random walks on crystal lattices from the view point
h
of discrete geometric analysis developed by Kotani and Sunada. In the present
t
a
paper, we obtain yet another kind of the central limit theorem for them. Our
m
argument is based on a measure-change technique due to Alexopoulos [1].
[
2010 AMS Classification Numbers: 60J10, 60F05, 60G50, 60B10.
1
v Keywords: Crystal lattice, non-symmetric random walk, central limit theorem,
8
(modified) harmonic realization.
8
4
8
0 1 Introduction and results
.
1
0
Let X = (V,E) be a locally finite, connected and oriented graph. Here V is the set of all
7
1 vertices and E the set of all oriented edges. For an edge e E, we denote by o(e),t(e)
: ∈
v and e the origin, the terminus and the inverse edge of e, respectively. We denote by
i
X E the collection of all edges whose origin is x V. A path c in X with length n is a
x
∈
r sequence c = (e ,...,e ) of edges e with t(e ) = o(e )(i = 1,...,n 1). We denote by
1 n i i i+1
a −
Ωx,n(X)(x V, n N ) the set of all paths of length n for which origin o(c) = x.
∈ ∈ ∪{∞}
We also denote by o(c),t(c) the origin and the terminus of the path c. For simplicity, we
write Ω (X) := Ω (X).
x x,∞
Let p : E (0,1] be a transition probability, that is, a positive function on E
−→
satisfying
p(e) = 1 (x V). (1.1)
∈
e(cid:88)∈Ex
This induces the probability measure Px on Ωx(X) and the random walk associated with
the transition probability p is the time homogeneous Markov chain (Ωx(X),Px,{wn}∞n=0)
∗GraduateSchoolofNaturalSciences,OkayamaUniversity,3-1-1,Tsushima-Naka,Kita-ku,Okayama
700-8530, Japan ([email protected])
1
with values in X defined by w (c) := o(e ) c = (e ,e ,...) Ω (X) . The graph X
n n+1 1 2 x
∈
is endowed with the graph distance. For a topological space , we denote by C ( ) the
∞
(cid:0) T (cid:1) T
space of all functions f : R vanishing at infinity with the uniform topology f T .
T −→ (cid:107) (cid:107)∞
The (p-)transition operator L acting on C (X) is defined by
∞
Lf(x) := p(e)f t(e) (x V).
∈
e(cid:88)∈Ex (cid:0) (cid:1)
The n-step transition probability p(n,x,y)(n N, x,y V) is given by p(n,x,y) :=
∈ ∈
Lnδ (x), where δ ( ) stands for the Dirac delta function with pole at y. If there is a
y y
·
positive function m : V (0, ) up to constant multiple such that
−→ ∞
p(e)m o(e) = p(e)m t(e) (e E),
∈
the random walk is called (m-(cid:0))symm(cid:1) etric or r(cid:0)ever(cid:1)sible. Otherwise, it is called (m-)non-
symmetric.
As one of the most fundamental examples of infinite graphs, a crystal lattice has been
studied by many authors from both geometric and probabilistic viewpoints. Roughly
speaking, an infinite graph X is called a (Γ-)crystal lattice if X is an infinite-fold cov-
ering graph of a finite graph whose covering transformation group Γ is abelian. Typical
examples we have in mind are the square lattice, the triangular lattice and the hexagonal
lattice, and so on. For basic results, see [5, 6, 7, 8] and literatures therein.
In the theory of random walks on infinite graphs, to investigate the long time asymp-
totics, for instance, the central limit theorem (CLT) is a principal topic for both geometers
andprobabilists. Recently,Ishiwata,KawabiandKotani[4]provedtwokindsoffunctional
CLTs for non-symmetric random walks on crystal lattices using the theory of discrete ge-
ometric analysis developed by Kotani and Sunada. For more details on discrete geometric
analysis, see Section 2. We also refer to [9, 7].
Before stating our results, we start with a brief review of the setting and results in
[4]. Let us consider a (Γ-)crystal lattice X = (V,E), where the covering transformation
group Γ, acting on X freely, is a torsion free, finitely generated abelian group. Here
we may assume that Γ is isomorphic to Zd, without loss of generality. We denote by
X = (V ,E ) its (finite) quotient graph Γ X. Let p : E (0,1] be a Γ-invariant
0 0 0
\ −→
transition probability. Namely, it satisfies (1.1) and p(σe) = p(e) for every σ Γ, e E.
∈ ∈
Through the covering map π : X X , the transition probability p also induces a
0
−→
Markov chain with values in X . Since the transition probability p on X is positive, the
0
random walk on X is irreducible, that is, for every x,y V, there exists n = n(x,y) N
∈ ∈
such that p(n,x,y) > 0. And so is the random walk on X . Then, thanks to the Perron–
0
Frobenius theorem, we find a unique invariant probability measure on V with
0
m(x) = 1 and m(x) = p(e)m t(e) (x V ). (1.2)
0
∈
x(cid:88)∈V0 e∈(cid:88)(E0)x (cid:0) (cid:1)
It is also called a stationary distribution (see e.g., Durrett [2]). We also write m : V
−→
(0,1] for the Γ-invariant lift of m to X.
2
Let H1(X0,R) and H1(X0,R) be the first homology group and the first cohomology
group of X0, respectively. We take a linear map ρR from H1(X0,R) onto Γ⊗R through
the covering map π : X X . We define the homological direction of the random walk
0
−→
on X by
0
γp := p(e)m o(e) e H1(X0,R),
∈
e(cid:88)∈E0 (cid:0) (cid:1)
and we call ρR(γp)(∈ Γ⊗R) the asymptotic direction. We remark that the random walk
is m-symmetric if and only if γ = 0. Moreover, γ = 0 implies ρ (γ ) = 0. However, the
p p R p
converse does not hold in general. We write g0 for the (p-)Albanese metric on Γ R. (See
⊗
Section 2, for its precise definition.) We call that a periodic realization Φ0 : X Γ R
−→ ⊗
is (p-)modified harmonic if
p(e) Φ t(e) Φ o(e) = ρ (γ ) (x V). (1.3)
0 − 0 R p ∈
e(cid:88)∈Ex (cid:16) (cid:0) (cid:1) (cid:0) (cid:1)(cid:17)
This notion was first proposed in [7] to seek the most canonical periodic realization of a
topological crystal in the geometric context.
Now we are in a position to review two kinds of CLTs formulated in [4]. We first set a
reference point x V with Φ (x ) = 0 and put ξ (c) := Φ w (c) n = 0,1,2,..., c
∗ 0 ∗ n 0 n
∈ ∈
Ωx∗(X) . We define a measurable map X(n) : Ωx∗(X) −→ C(cid:0)0 [0,∞(cid:1))(cid:0),(Γ⊗R,g0) ,µ by
(cid:1) 1 (cid:0) (cid:0) (cid:1) (cid:1)
X(n)(c) := ξ (c)+(nt [nt]) ξ (c) ξ (c) ntρ (γ ) (t 0),
t √n [nt] − [nt]+1 − [nt] − R p ≥
(cid:110) (cid:111)
(cid:0) (cid:1)
whereµistheWienermeasureonC0 [0, ),(Γ R,g0) andthepathspaceC0 [0, ),(Γ
∞ ⊗ ∞ ⊗
R,g0) is equipped with compact un(cid:0)iform topology. W(cid:1)e write P(n)(n = 1,2,(cid:0)...) for the
proba(cid:1)bility measure on C0 [0,∞),(Γ ⊗ R,g0) induced by X(n). Then the CLT of the
first kind is stated as follows:
(cid:0) (cid:1)
Theorem 1.1 ([4, Theorem 2.2]) The sequence P(n) ∞ of probability measures con-
{ }n=1
verges weakly to the Wiener measure µ as n . In other words, the sequence X(n) ∞
→ ∞ { }n=1
converges to a (Γ R,g0)-valued standard Brownian motion (Bt)t≥0 starting from the ori-
⊗
gin in law.
Next we introduce a family p of transition probabilities on X by p (e) :=
ε 0≤ε≤1 ε
{ }
p (e)+εq(e)(e E), where
0
∈
1 m t(e) 1 m t(e)
p (e) := p(e)+ p(e) , q(e) := p(e) p(e) (e E).
0 2 m o(e) 2 − m o(e) ∈
(cid:0) (cid:1) (cid:0) (cid:1)
(cid:16) (cid:17) (cid:16) (cid:17)
This is nothing but the in(cid:0)terpo(cid:1)lation of the original transitio(cid:0)n pro(cid:1)bability p = p and the
1
m-symmetric transition probability p . We should note that, in this setting, the relation
0
ρ (γ ) = ερ (γ ) plays a crucial role to obtain the CLT of the second kind. We write
R pε R p
3
g0(ε) for the (pε-)Albanese metric on Γ⊗R. Moreover, let Φ(0ε) : X −→ (Γ⊗R,g0(0)) be the
p -modified harmonic realization of X.
ε
Now set a reference point x V satisfying Φ(ε)(x ) = 0 for all 0 ε 1 and
∗ ∈ 0 ∗ ≤ ≤
put ξ(ε)(c) := Φ(ε) w (c) n = 0,1,2,...,c Ω (X) . We define a measurable map
n 0 n ∈ x∗
Y(ε,n) : Ωx∗(X) −→(cid:0)C0 [0(cid:1),∞(cid:0) ),(Γ⊗R,g0(0)) by (cid:1)
1(cid:0) (cid:1)
Y(ε,n)(c) := ξ(ε)(c)+(nt [nt]) ξ(ε) (c) ξ(ε)(c) (t 0).
t √n [nt] − [nt]+1 − [nt] ≥
(cid:110) (cid:111)
(cid:0) (cid:1)
Let ν be the probability measure on C0 [0,∞),(Γ ⊗ R,g0(0)) induced by the stochastic
process Bt+ρR(γp)t t≥0, where(Bt)t≥0 is(cid:0)a(Γ⊗R,g0(0))-valued(cid:1)standardBrownianmotion
with B0(cid:0)= 0. Moreov(cid:1)er let Q(ε,n) be the probability measure on C0 [0,∞),(Γ⊗R,g0(0))
induced by Y(ε,n). Then the CLT of the second kind is the following.
(cid:0) (cid:1)
Theorem 1.2 ([4, Theorem 2.4]) The sequence Q(n−1/2,n) ∞ of probability measures
{ }n=1
converges weakly to the probability measure ν as n . In other words, the sequence
→ ∞
Y(n−1/2,n) ∞ converges to a (Γ R,g(0))-valued standard Brownian motion with drift
{ }n=1 ⊗ 0
ρ (γ ) starting from the origin in law.
R p
The main purpose of the present paper is to establish yet another CLT for non-
symmetricrandomwalksoncrystallattices. Ourapproachisinspiredbyameasure-change
techinique due to Alexopoulos [1] in which several limit theorems for random walks on
discrete groups of polynomial volume growth are obtained.
Now consider the (m-)non-symmetric transition probability p : E (0,1]. In partic-
−→
ular, we assume that ρR(γp) (cid:54)= 0. Let Φ0 : X −→ (Γ⊗R,g0) be a (p-)modified harmonic
realization of X. We define a function F = Fx(λ) : V0 Hom(Γ,R) (0, ) by
× −→ ∞
F (λ) := p(e)exp λ,Φ t(e) Φ o(e) , (1.4)
x Hom(Γ,R) 0 − 0 Γ⊗R
e∈(cid:88)(E0)x (cid:16) (cid:10) (cid:0) (cid:1) (cid:0) (cid:1)(cid:11) (cid:17)
(cid:101) (cid:101)
for x V0, λ Hom(Γ,R), where e stands for the lift of e E0 to X. Then we can verify
∈ ∈ ∈
that, for every x V0, the function Fx( ) : Hom(Γ,R) (0, ) has a unique minimizer
∈ · −→ ∞
λ∗ = λ∗(x) Hom(Γ,R). (See Lem(cid:101)ma 3.1.) We define a positive function p : E0 (0,1]
∈ −→
by
p(e)exp λ o(e) ,Φ t(e) Φ o(e)
p(e) := Hom(Γ,R) ∗ 0 − 0 Γ⊗R (e E ). (1.5)
(cid:16) (cid:10) F(cid:0)o(e) λ(cid:1)∗(o(e(cid:0))) (cid:1) (cid:0) (cid:1)(cid:11) (cid:17) ∈ 0
(cid:101) (cid:101)
Thenitisstraightforwardtocheckth(cid:0)atthefun(cid:1)ctionpalsogivesatransitionprobabilityon
X . Noting the random walk w(p) ∞ associated with p is also irreducible, the Perron–
0 { n }n=0
Frobenius theorem yields a unique normalized invariant measure m : V (0,1] in the
0
−→
sense of (1.2). We write p : E (0,1] and m : V (0,1] for the Γ-invariant lifts of
−→ −→
4
p : E (0,1] and m : V (0,1], respectively. We write g(p) for the (p-)Albanese
0 −→ 0 −→ 0
metric associated with the transition probability p.
Let L be the transition operator, acting on C (X), associated with the transition
(p) ∞
probability p. Namely,
L f(x) = p(e)f t(e) (x V).
(p)
∈
e(cid:88)∈Ex (cid:0) (cid:1)
Recalling that the function F = F (λ) has the (unique) minimizer λ = λ (x) for every
x ∗ ∗
x V , it follows that
0
∈
L Φ (x) Φ (x) = p(e) Φ t(e) Φ o(e) = 0 (x V). (1.6)
(p) 0 0 0 0
− − ∈
e(cid:88)∈Ex (cid:16) (cid:0) (cid:1) (cid:0) (cid:1)(cid:17)
This equation means that the p-modified harmonic realization Φ0 : X Γ R is a p-
−→ ⊗
harmonic realization. In particular, we obtain ρ (γ ) = 0. Here we should emphasize that
R p
the transition probability p : E (0,1] coincides with the original one p : E (0,1]
0 0
−→ −→
provided that ρ (γ ) = 0.
R p
We fix a reference point x V such that Φ (x ) = 0 and put
∗ 0 ∗
∈
ξ(p)(c) := Φ w(p)(c) n = 0,1,2,..., c Ω (X) .
n 0 n ∈ x∗
We define a measurable map(cid:0)X(n) : Ω(cid:1)x∗(X)(cid:0)−→ C0 [0,∞),(Γ⊗R,g0(p(cid:1))) ,µ by
1 (cid:0) (cid:0) (cid:1) (cid:1)
X(n)(c) := ξ(p)(c)+(nt [nt]) ξ(p) (c) ξ(p)(c) (t 0), (1.7)
t √n [nt] − [nt]+1 − [nt] ≥
(cid:110) (cid:111)
(cid:0) (cid:1)
where µ = µ(p) is the Wiener measure on C0 [0,∞),(Γ ⊗ R,g0(p)) . We write P(n)(n =
1,2,...) for the probability measure on C0 [0(cid:0),∞),(Γ⊗R,g0(p)) i(cid:1)nduced by X(n). Then
our main theorem is stated as follows:
(cid:0) (cid:1)
Theorem 1.3 The sequence P(n) ∞ of probability measures converges weakly to the
{ }n=1
Wiener measure µ as n . Namely, the sequence X(n) ∞ converges to a (Γ R,g(p))-
→ ∞ { }n=1 ⊗ 0
valued standard Brownian motion (B(p)) starting from the origin in law.
t t≥0
Finally, we give a relationship between the n-step transition probabilities p(n,x,y)
and p(n,x,y) as follows:
Theorem 1.4 There exists some positive constant C such that
p(n,x,y)
lim = C
n→∞ p(n,x,y)exp nM
p
uniformly for x,y V, where (cid:0) (cid:1)
∈
M := m(x) λ (x),ρ (γ ) logF λ (x) .
p Hom(Γ,R) ∗ R p Γ⊗R − x ∗
x(cid:88)∈V0 (cid:16) (cid:10) (cid:11) (cid:0) (cid:1)(cid:17)
5
For the precise long time asymptotic behavior of p(n,x,y), we refer to [4, Theorem 2.5]
and Trojan [10].
The rest of the present paper is organized as follows: In Section 2, we provide a brief
review on the theory of discrete geometric analysis. In Section 3, we state our measure-
change technique in details and give proofs of Theorems 1.3 and 1.4. We also discuss a
relationship between our measure-change technique and a discrete analogue of Girsanov’s
theorem due to Fujita [3]. Finally, in Section 4, we give some concrete examples of non-
symmetric random walks on crystal lattices.
2 A quick review on discrete geometric analysis
In this section, we give basic materials of the theory of discrete geometric analysis on
graphs quickly. For more details, we refer to Kotani–Sunada [7] and Sunada [9].
WeconsiderarandomwalkonafinitegraphX = (V ,E )associatedwithatransition
0 0 0
probability p : E (0,1]. Thanks to the Perron–Frobenius theorem, there is a unique
0
−→
(normalized) invariant measure m : V (0,1] in the sense of (1.2). The random walk
0
−→
is called (m-)symmetric if p(e)m o(e) = p(e)m t(e) (e E ).
0
∈
First we define the 0-chain group and the 1-chain group of X by
0
(cid:0) (cid:1) (cid:0) (cid:1)
C0(X0,R) := axx ax R , C1(X0,R) := aee ae R, e = e ,
∈ ∈ −
(cid:110)x(cid:88)∈V0 (cid:12) (cid:111) (cid:110)e(cid:88)∈E0 (cid:12) (cid:111)
(cid:12) (cid:12)
respectively. Let ∂ : C1(X0,R(cid:12)) C0(X0,R) be the boundary m(cid:12)ap, given by the homo-
−→
morphism satisfying ∂(e) := t(e) o(e)(e E0). The first homology group H1(X0,R) of
− ∈
X0 is defined by Ker(∂) C1(X0,R) .
⊂
On the other hand, we define the 0-cochain group and the 1-cochain group of X by
0
(cid:0) (cid:1)
C0(X0,R) := f : V0 R , C1(X0,R) := ω : E0 R ω(e) = ω(e) ,
{ −→ } { −→ | − }
respectively. The difference operator d : C0(X0,R) C1(X0,R) is defined by the
−→
homomorphism with df(e) := f t(e) f o(e) (e E0). We also define H1(X0,R) :=
− ∈
C1(X0,R)/Im(d), called the first cohomology group of X0.
(cid:0) (cid:1) (cid:0) (cid:1)
Next we define the transition operator L : C0(X0,R) C0(X0,R) by
−→
Lf(x) := (I δpd)f(x) = p(e)f t(e) x V0, f C1(X0,R) ,
− ∈ ∈
e∈(cid:88)(E0)x (cid:0) (cid:1) (cid:0) (cid:1)
where the operator δp : C1(X0,R) C0(X0,R) is defined by
−→
δpω(x) := p(e)ω(e) x V0, ω C1(X0,R) .
− ∈ ∈
e∈(cid:88)(E0)x (cid:0) (cid:1)
We introduce the quantity γ , called the homological direction of the given random walk
p
on X , by
0
γp := m(e)e C1(X0,R),
∈
e(cid:88)∈E0
(cid:101)
6
where m(e) := p(e)m o(e) (e E ). It is easy to show that ∂(γ ) = 0, that is, γ
0 p p
∈ ∈
H1(X0,R). It should be noted that the transition probability p gives an m-symmetric
(cid:0) (cid:1)
random(cid:101)walk on X0 if and only if γp = 0. A 1-form ω C1(X0,R) is said to be modified
∈
harmonic if
δ ω(x)+ γ ,ω = 0 (x V ).
p C1(X0,R)(cid:104) p (cid:105)C1(X0,R) ∈ 0
We denote by 1(X ) the space of modified harmonic 1-forms on X , and equip it with
0 0
H
the inner product
ω,η := m(e)ω(e)η(e) γ ,ω γ ,η ω,η 1(X ) .
p p p 0
(cid:104)(cid:104) (cid:105)(cid:105) −(cid:104) (cid:105)(cid:104) (cid:105) ∈ H
e(cid:88)∈E0 (cid:0) (cid:1)
Due to the discrete analogu(cid:101)e of Hodge–Kodaira theorem (cf. [7, Lemma 5.2]), we may
identify 1(X0), , p with H1(X0,R).
H (cid:104)(cid:104)· ·(cid:105)(cid:105)
Now let X = (V,E) be a Γ-crystal lattice. Namely, X is a covering graph of a finite
(cid:0) (cid:1)
graph X with an abelian covering transformation group Γ. We write p : E (0,1] and
0
−→
m : V (0,1] for the Γ-invariant lifts of p : E (0,1] and m : V (0,1], respec-
0 0
−→ −→ −→
tively. Through the covering map π : X X , we take the surjective linear map ρ :
−→ 0 R
H1(X0,R) −→ Γ⊗R(∼= Rd). We consider the transpose tρR : Hom(Γ,R) −→ H1(X0,R),
which is a injective linear map. Here Hom(Γ,R) denotes the space of homomorphisms
from Γ into R. We induce a flat metric g0 on the Euclidean space Γ R through the
⊗
following diagram:
(Γ R(cid:79)(cid:79) ,g0)(cid:111)(cid:111)(cid:111)(cid:111) ρR H1(X(cid:79)(cid:79)0,R)
⊗
dual dual
(cid:15)(cid:15) (cid:15)(cid:15)
Hom(Γ,R)(cid:31)(cid:127) tρR (cid:47)(cid:47)H1(X0,R) ∼= H1(X0),(cid:104)(cid:104)·,·(cid:105)(cid:105)p .
This metric g0 is called the Albanese metric on Γ (cid:0)R. (cid:1)
⊗
From now on, we realize the crystal lattice X into the continuous model (Γ R,g0)
⊗
in the following manner. A periodic realization of X into Γ R is defined by a piecewise
⊗
linear map Φ : X Γ R with Φ(σx) = Φ(x)+σ 1(σ Γ, x V). We introduce a
−→ ⊗ ⊗ ∈ ∈
special periodic realization Φ0 : X Γ R by
−→ ⊗
x
Hom(Γ,R) ω,Φ0(x) Γ⊗R = ω x ∈ V, λ ∈ Hom(Γ,R) , (2.1)
(cid:90)x∗
(cid:10) (cid:11) (cid:0) (cid:1)
where x∗ is a fixed reference point satisfying(cid:101)Φ0(x∗) = 0 and ω is the lift of ω to X. Here
x n
ω = ω := ω(e) (cid:101)
(cid:90)x∗ (cid:90)c i=1
(cid:88)
for a path c = (e ,...,e ) with o(e(cid:101)) = x (cid:101)and t(e (cid:101)) = x. It should be noted that this
1 n 1 ∗ n
line integral does not depend on the choice of a path c. The periodic realization Φ given
0
by above enjoys the so-called modified harmonicity in the sense that
LΦ (x) Φ (x) = ρ (γ ) (x V).
0 − 0 R p ∈
7
We note that this equation is also written as (1.3). Further, such a realization is uniquely
determined up to translation. We call the quantity ρ (γ ) the asymptotic direction of
R p
the given random walk. We should emphasize that γ = 0 implies ρ (γ ) = 0. However,
p R p
the converse does not always hold, that is, there is a case γ = 0 and ρ (γ ) = 0. (See
p (cid:54) R p
Subsection 4.2, for an example.) If we equip Γ R with the Albanese metric, then the
⊗
modified harmonic realization Φ0 : X (Γ R,g0) is said to be a modified standard
−→ ⊗
realization.
3 Proofs of the main results
3.1 A mesure–change technique
In what follows, we write λ[x]Γ⊗R := Hom(Γ,R)(cid:104)λ,x(cid:105)Γ⊗R (λ ∈ Hom(Γ,R), x ∈ Γ⊗R) and
dΦ (e) := Φ t(e) Φ o(e) (e E),
0 0 0
− ∈
for brevity. Take an orthonormal ba(cid:0)sis (cid:1)ω1,...(cid:0),ωd (cid:1)in Hom(Γ,R) ( 1(X0), , p) ,
{ } ⊂ H (cid:104)(cid:104)· ·(cid:105)(cid:105)
and denote by {v1,...,vd} its dual basis in Γ⊗R. Namely, ωi[vj]Γ(cid:0)⊗R = δij(1 ≤ i,j ≤ d(cid:1)).
Then, v1,...,vd isanorthonormalbasisinΓ RwithrespecttotheAlbanesemetricg0.
{ } ⊗
We may identify λ = λ1ω1+ +λdωd Hom(Γ,R) with (λ1,...,λd) Rd. Furthermore,
··· ∈ ∈
we write x := ω [x] , Φ (x) := ω [Φ (x)] and ∂ := ∂/∂λ (i = 1,...,d, x V).
i i Γ⊗R 0 i i 0 Γ⊗R i i ∈
We denote by O( ) the Landau symbol.
·
At the beginning, consider the function F = Fx(λ) : X0 Hom(Γ,R) R defined
× −→
by (1.4). We easily see that F = Fx(λ) is a positive function on X0 Hom(Γ,R) with
×
F (0) = 1(x V ). In our setting, the following lemma plays a siginificant role so as to
x 0
∈
obtain Theorem 1.3.
Lemma 3.1 For every x V0, the function Fx( ) : Hom(Γ,R) (0, ) has a unique
∈ · −→ ∞
minimizer λ = λ (x).
∗ ∗
Proof. Fix a fixed x V , we have
0
∈
∂ F (λ) = ∂ p(e)exp λ dΦ (e)
i x i(cid:32) 0 Γ⊗R (cid:33)
e∈(cid:88)(E0)x (cid:16) (cid:2) (cid:3) (cid:17)
d (cid:101)
= ∂ p(e)exp λ ω dΦ (e)
i(cid:32) i · i 0 Γ⊗R (cid:33)
e∈(cid:88)(E0)x (cid:16)(cid:88)i=1 (cid:2) (cid:3) (cid:17)
= p(e)exp λ dΦ0(e) Γ⊗R dΦ0(e(cid:101))i i = 1,...,d, λ ∈ Hom(Γ,R) .
e∈(cid:88)(E0)x (cid:16) (cid:2) (cid:3) (cid:17) (cid:0) (cid:1)
In other words, (cid:101) (cid:101)
∂ F (λ),...,∂ F (λ)
1 x d x
(cid:16) (cid:17)
= p(e)exp λ dΦ0(e) Γ⊗R dΦ0(e)(∈ Γ⊗R) λ ∈ Hom(Γ,R) . (3.1)
e∈(cid:88)(E0)x (cid:16) (cid:2) (cid:3) (cid:17) (cid:0) (cid:1)
(cid:101) (cid:101)
8
Repeating the above calculation, we have
∂ ∂ F (λ) = p(e)exp λ dΦ (e) dΦ (e) dΦ (e)
i j x 0 Γ⊗R 0 i 0 j
e∈(cid:88)(E0)x (cid:16) (cid:2) (cid:3) (cid:17)
(cid:101) (cid:101) (cid:101)
for λ ∈ Hom(Γ,R) and 1 ≤ i,j ≤ d. Then, we know that ∂i∂jFx(·) 1≤i,j≤d, the Hessian
matrix of the function F ( ), is positive definite. Indeed, consider the quadratic form
x · (cid:0) (cid:1)
corresponding to the Hessian matrix. Since
p(e)exp λ dΦ (e) dΦ (e) dΦ (e) ξ ξ
0 Γ⊗R 0 i 0 j i j
1≤(cid:88)i,j≤de∈(cid:88)(E0)x (cid:16) (cid:2) (cid:3) (cid:17)
(cid:101) d (cid:101) (cid:101)
2
= p(e)exp λ dΦ0(e) Γ⊗R dΦ0(e)iξi ≥ 0 ξ = (ξ1,...,ξd) ∈ Rd (3.2)
e∈(cid:88)(E0)x (cid:16) (cid:2) (cid:3) (cid:17)(cid:110)(cid:88)i=1 (cid:111) (cid:0) (cid:1)
(cid:101) (cid:101)
and the transition probability p is positive, we easily see that the Hessian matrix is non-
negative definite. By multiplying both sides of (3.2) by m(x) and taking the sum which
runs over all vertices of X , it readily follows that
0
d
2
m(e)exp λ dΦ0(e) Γ⊗R dΦ0(e)iξi ≥ 0, ξ = (ξ1,...,ξd) ∈ Rd .
e(cid:88)∈E0 (cid:16) (cid:2) (cid:3) (cid:17)(cid:110)(cid:88)i=1 (cid:111) (cid:0) (cid:1)
Next sup(cid:101)pose that the left(cid:101)-hand side of (3.2)(cid:101)is zero. Then we have
d
dΦ (e) ξ = 0
0 i i
i=1
(cid:88)
for all e E . This equation implies Φ (x)(cid:101),ξ = Φ (y),ξ for all x,y V, where
∈ 0 (cid:104) 0 (cid:105)Rd (cid:104) 0 (cid:105)Rd ∈
(cid:104)·,·(cid:105)Rd stands for the standard inner product on Rd. Let σ1,...,σd be generators of
Γ ∼= Zd. It follows from the periodicity of Φ0 that (cid:104)σi,ξ(cid:105)Rd = 0(1 ≤ i ≤ d). Hence we
conclude ξ = 0. Namely, we have proved the positive definiteness of the Hessian matrix.
This implies that the function Fx( ) : Hom(Γ,R) (0, ) is strictly convex for
· −→ ∞
every x V . Consequently, we know that there exists a unique minimizer λ = λ (x)
0 ∗ ∗
∈ ∈
Hom(Γ,R) of Fx(λ) for each x X0, thereby completing the proof of Lemma 3.1.
∈
Now consider the positive function p : E (0,1] given by (1.5). By defini-
0
−→
tion, we easily see that the function p also gives a positive transition probability on
X . Thus the transition probability p : E (0,1] yields an irreducible random
0 0
−→
walk (Ωx∗(X),Px∗,{wn(p)}∞n=0) with values in X. Applying the Perron-Frobenius theo-
rem again, we find a unique positive function m : V (0,1] satisfying (1.2). Put
0
−→
m(e) := p(e)m((cid:98)o(e))(e E ). We also denote by p : E (0,1] and m : V (0,1]
0
∈ −→ −→
the Γ-invariant lifts of p : E (0,1] and m : V (0,1], respectively. As in the
0 0
−→ −→
p(cid:101)revious section, we construct the (p-)Albanese metric g(p) on Γ R associated with the
0 ⊗
transition probability p. We take an orthonormal basis ω(p),...,ω(p) in Hom(Γ,R)
{ 1 d } ⊂
( 1(X ), , ) .
0 p
H (cid:104)(cid:104)· ·(cid:105)(cid:105) (cid:0)
(cid:1)
9
We introduce the transition operator L : C (X) C (X) associated with the
(p) ∞ ∞
−→
transition probability p by
L f(x) := p(e)f(t(e)) (x V).
(p)
∈
e(cid:88)∈Ex
Recalling (3.1) and the definition of λ = λ (x), we see that
∗ ∗
∂ F λ (x) ,...,∂ F λ (x) = p(e)exp λ (x) dΦ (e) dΦ (e) = 0
1 x ∗ d x ∗ ∗ 0 Γ⊗R 0
(cid:16) (cid:0) (cid:1) (cid:0) (cid:1)(cid:17) e∈(cid:88)(E0)x (cid:16) (cid:2) (cid:3) (cid:17)
holds for every x V . This immediately leads to (cid:101) (cid:101)
0
∈
L Φ (x) Φ (x) = p(e)dΦ (e) = 0 (x V). (3.3)
(p) 0 0 0
− ∈
e(cid:88)∈Ex
From this equation, one concludes that the given p-modified standard realization Φ :
0
X (Γ R,g0) in the sense of (1.3) is the harmonic realization associated with the
−→ ⊗
changed transition probability p.
Remark 3.1 Equation (3.3) implies ρ (γ ) = 0. We also emphasize that the transition
R p
probability p : E (0,1] coincides with the original one p : E (0,1] provided that
0 0
−→ −→
ρ (γ ) = 0.
R p
Remark 3.2 In our setting, it is essential to assume that the given transition probability
p is positive. Because, if it were not for the positivity of p, the assertion of Lemma 3.1
would not hold in general. (There is a case where the function F ( ) has no minimizers.)
x
·
On the other hand, to obtain Theorems 1.1 and 1.2, it is sufficient to impose that the
given transition probability p is non-negative with p(e)+p(e) > 0(e E).
∈
3.2 Proofs of Theorems 1.3 and 1.4
This subsection is devoted to proofs of Theorems 1.3 and 1.4. Following the argument as
in[4, Theorem2.2]fortherandomwalkassociatedwiththechangedtransitionprobability
p, we can carry out the proof of Theorem 1.3. Though a minor change of the proof is
required, the argument is a little bit easier due to ρ (γ ) = 0.
R p
As the first step, we prove the following lemma.
Lemma 3.2 For any f C∞(Γ R), as N , ε 0 and N2ε 0, we have
∈ 0 ⊗ → ∞ (cid:38) (cid:38)
1 ∆ X
(I LN )P f P (p)f 0.
Nε2 − (p) ε − ε 2 ∞ −→
(cid:13) (cid:16) (cid:17)(cid:13)
Here Pε : C∞(Γ⊗R) −(cid:13)(cid:13)→ C∞(X)(0 ≤ ε ≤ 1) is a scaling(cid:13)(cid:13)operator defined by
P f(x) := f εΦ (x) (x X)
ε 0
∈
and ∆(p) stands for the positive Laplacia(cid:0)n − (cid:1)di=1(∂2/∂x2i) on Γ⊗R associated with the
p-Albanese metric g(p).
0 (cid:80)
10
