An Ergodic Description of Ground States

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An Ergodic Description of Ground States

Eduardo Garibaldi, Philippe Thieullen

To cite this version:

Eduardo Garibaldi, Philippe Thieullen. An Ergodic Description of Ground States. Journal of Statis-

tical Physics, Springer Verlag, 2015, 158, pp.359 - 371. �10.1007/s10955-014-1139-z�. �hal-01213721�

Eduardo Garibaldi

^∗

Departamento de Matem´ atica Universidade Estadual de Campinas

13083-859 Campinas - SP, Brasil

garibaldi@ime.unicamp.br

Philippe Thieullen

^†

Institut de Math´ ematiques Universit´ e Bordeaux 1, CNRS, UMR 5251

F-33405 Talence, France

Philippe.Thieullen@math.u-bordeaux1.fr

October 5, 2010

Abstract

1 Introduction

2 Framework and Main Results

Given a finite spin set F, we introduce the configuration space Ω = F

^Zd

. Its elements ω ∈ Ω are written as ω = {ω

_j

}

_j∈

Z^d

. Endowed with the product topology of discrete topologies, Ω is a compact metrizable space. Indeed, for j = (j

₁

, . . . , j

_d

) ∈ Z

^d

one may consider the L

¹

norm kjk = |j

₁

| + . . . + |j

_d

| and define then a metric on Ω by

d(ω, ω) = 2 ¯

^{−inf{kjk:ωj6=¯ωj}}

, which is compatible with the product topology.

Notice that Z

^d

acts on Ω by translation. Let {e

₁

, . . . , e

_d

} be the canonical basis for the lattice Z

^d

. For each i ∈ {1, . . . , d}, we consider the shift transformation θ

i

: Ω → Ω given by θ

i

(ω) = {ω

_j+ei

}

_j∈Zd

. Given j = (j

1

, . . . , j

d

) ∈ Z

^d

, we define θ

^j

:= θ

₁^j1

◦ θ

^j₂²

◦ · · · ◦ θ

_d^jd

.

Let F denote the collection of finite subsets of Z

^d

. The diameter and the boundary of A ∈ F are respectively diam(A) = max{kj − kk : j, k ∈ A} and

∂A := {j ∈ Z

^d

\ A : kj − kk = 1 for some k ∈ A}. For A ∈ F and ω ∈ Ω, we denote the restriction ω|

_A

simply by ω

A

.

Definition 2.1. By an invariant interaction family we mean any collection of con- tinuous maps Φ

A

: Ω → R , indexed by A ∈ F , such that

1. Φ

_j+A

(ω) = Φ

_A

(θ

^j

(ω)), for all j ∈ Z

^d

and ω ∈ Ω;

∗

supported by FAPESP 2009/17075-8

†

supported by ANR BLANC07-3 187245, Hamilton-Jacobi and Weak KAM Theory

1

2 Eduardo Garibaldi and Philippe Thieullen

2. ω

_A

= ¯ ω

_A

implies Φ

_A

(ω) = Φ

_A

( ¯ ω).

We say that {Φ

_A

}

_A∈F

is an invariant short range interaction family if in addi- tion there exists an integer r > 0 such that Φ

A

≡ 0 whenever diam(A) > r. In this case, we say also that the invariant interaction family has range r > 0. For r = 1, one in particular says to have an interaction of nearest neighbors.

Given an invariant interaction family {Φ

_A

}

_A∈F

, we introduce the associated hamiltonian H : F × Ω → R defined by

H(Λ, ω) = H

_Λ

(ω) := X

A:A∩Λ6=∅

Φ

_A

(ω), ∀ Λ ∈ F , ω ∈ Ω. (2.1)

Notice also that the hamiltonian inherits the invariance of the interaction family, namely, H

Λ

◦ θ

^j

= H

j+Λ

for any j ∈ Z

^d

.

Remark 2.2. We say that {Φ

_A

}

_A∈F

is an invariant long range interaction family if Φ

A

6≡ 0 for sets A with arbitrarily large diameter. In this case, one shall assume a summability condition:

X

0∈A

kΦ

_A

k

_∞

< ∞,

which is obviously trivial for the short range situation. Notice that the associated hamiltonian is then well defined, since

kH

_Λ

k

_∞

≤ X

j∈Λ

X

j∈A

kΦ

_A

k

_∞

= X

j∈Λ

X

0∈A

kΦ

_A

◦ θ

^j

k

_∞

= #Λ X

0∈A

kΦ

_A

k

_∞

< ∞, ∀ Λ ∈ F .

For ω, ω ¯ ∈ Ω and Λ ∈ F , we denote by ¯ ω

Λ

ω

_Z^d_\Λ

the configuration of Ω that coincides with ¯ ω on Λ and with ω on Z

^d

\ Λ. We also introduce the following notation

Λ

_n+1

:= {−n, −n + 1, . . . , 0, . . . , n − 1, n}

^d

, ∀ n ∈ N .

Definition 2.3. We say that ω ∈ Ω is a minimizing configuration with respect to the hamiltonian H if

H

_Λ

(ω) ≤ H

_Λ

( ¯ ω

_Λ

ω

_Zd\Λ

), ∀ ω ¯ ∈ Ω, Λ ∈ F .

We define the minimizing ergodic value of the hamiltonian H as the constant H ¯ := inf

ω∈Ω

lim inf

n→∞

1

#Λ

n

H

Λn

(ω). (2.2)

Our main result concerning these minimizing objects is the following one.

Theorem 2.4. Let H be a hamiltonian defined by an invariant short range inter- action family. The set of minimizing configurations with respect to the hamiltonian H is a non-empty invariant closed set. For a minimizing configuration ω ∈ Ω, the limit lim

n→∞ 1

#Λn

H

_Λn

(ω) always exists. Moreover, there are minimizing configu- rations ω ∈ Ω for which

n→∞

lim 1

#Λ

_n

H

_Λn

(ω) = ¯ H.

3 Proof of Theorem 2.4

The proof of Theorem 2.4 follows from a series of results regrouped in this section.

Proposition 3.1. Let H be a hamiltonian defined by an invariant short range interaction family. There exist minimizing configurations for the hamiltonian H.

The set of minimizing configurations with respect to H is invariant and closed.

Proof. For each Λ ∈ F, denote Ω

_Λ

:= n

ω ∈ Ω : H

_Λ

(ω) ≤ H

_Λ

( ¯ ω), ∀ ω ¯ ∈ Ω with ¯ ω

_Zd\Λ

= ω

_Zd\Λ

o .

Notice that Ω

Λ

is non-empty: for any fixed configuration ω ∈ Ω, it clearly contains the minimum points of the map ¯ ω

_Λ

∈ F

^Λ

7→ H

_Λ

( ¯ ω

_Λ

ω

_Zd\Λ

) ∈ R . Besides, each Ω

_Λ

is a closed set, since the map ω ∈ Ω 7→ H

_Λ

( ¯ ω

_Λ

ω

_Zd\Λ

) − H

_Λ

(ω) ∈ R is continuous (actually, locally constant) for all ¯ ω

_Λ

∈ F

^Λ

.

The family {Ω

_Λ

}

_Λ∈F

is monotone: Λ ⊂ Λ

⁰

⇒ Ω

_Λ⁰

⊂ Ω

_Λ

. Indeed, suppose that ω

⁰

∈ Ω

_Λ⁰

. Let ¯ ω ∈ Ω be such that ¯ ω

_Z^d_\Λ

= ω

⁰

Z^d\Λ

. As Λ ⊂ Λ

⁰

, we get

¯

ω

_Zd\Λ⁰

= ω

⁰

Z^d\Λ⁰

. Hence, H

_Λ⁰

(ω

⁰

) ≤ H

_Λ⁰

( ¯ ω). Since H

_Λ⁰

= H

_Λ

+ X

A∩Λ⁰6=∅

A∩Λ=∅

Φ

_A

and Φ

A

(ω

⁰

) = Φ

A

( ¯ ω) whenever ∅ 6= A ∩ Λ

⁰

⊂ Λ

⁰

\ Λ, we clearly obtain that H

Λ

(ω

⁰

) ≤ H

Λ

( ¯ ω), which means that ω

⁰

∈ Ω

Λ

.

Therefore, the family of closed sets {Ω

_Λ

}

_Λ∈F

has the finite intersection property:

T

N

i=1

Ω

Λi

⊃ Ω

^SN

i=1Λi

6= ∅. By compactness, there exists ω ∈ Ω

Λ

for all Λ ∈ F.

The set of minimizing configurations with respect to H is exactly T

Λ∈F

Ω

Λ

, which is clearly closed. Besides, since θ

^−j

(Ω

_Λ

) = Ω

_j+Λ

for all j ∈ Z

^d

, this set is also invariant.

Denote the Birkhoff sum by S

_Λ

Ψ := P

j∈Λ

Ψ ◦ θ

^j

.

Definition 3.2. Given an invariant short range interaction family {Φ

_A

}

_A∈F

, we introduce

Ψ

₀

:= X

A: 0∈A

1

#A Φ

_A

.

Notice that Ψ

0

is a real valued function since the above sum is actually finite.

Moreover, we have the following property.

Lemma 3.3. The map Ψ

0

satisfies sup

Λ∈F

sup

ωΛ= ¯ωΛ

1

#∂Λ (S

_Λ

Ψ

₀

(ω) − S

_Λ

Ψ

₀

( ¯ ω)) < ∞.

In order to prove this lemma, given Λ ∈ F and r > 0, we define

Int

_r

Λ := {j ∈ Λ : d(j, ∂Λ) ≥ r} .

4 Eduardo Garibaldi and Philippe Thieullen

Proof. Notice that

S

_Λ

Ψ

0

= X

j∈Λ

X

A: 0∈A

1

#A Φ

_A

◦ θ

^j

= X

j∈Λ

X

A:j∈A

1

#A Φ

_A

= X

A:A∩Λ6=∅

X

j∈A∩Λ

1

#A Φ

_A

= X

A:A∩Λ6=∅

#(A ∩ Λ)

#A Φ

_A

. (3.1) Assume the invariant interaction family {Φ

_A

}

_A∈F

has range r > 0. Whenever ω

Λ

= ¯ ω

Λ

, it is easy to see that

S

_Λ

Ψ

₀

(ω) − S

_Λ

Ψ

₀

( ¯ ω) = X

A∩(Λ\Int_rΛ)6=∅

#(A ∩ Λ)

#A (Φ

_A

(ω) − Φ

_A

( ¯ ω))

≤ 2 X

A∩(Λ\IntrΛ)6=∅

kΦ

_A

k

_∞

≤ 2 #(Λ \ Int

r

Λ) X

0∈A

kΦ

_A

k

_∞

,

from which the statement follows immediately.

The relation of Ψ

₀

with the associated hamiltonian is given below.

Proposition 3.4. Suppose {Φ

_A

}

_A∈F

is an invariant interaction family with range r > 0. If H is the associated hamiltonian, then

kH

_Λ

− S

_Λ

Ψ

₀

k

_∞

≤ #(Λ \ Int

_r

Λ) X

0∈A

kΦ

_A

k

_∞

.

Proof. Using (3.1), we obtain H

Λ

− S

Λ

Ψ

0

= X

A∩IntrΛ6=∅

1 − #(A ∩ Λ)

#A

Φ

A

+ X

A∩(Λ\IntrΛ)6=∅

1 − #(A ∩ Λ)

#A

Φ

A

.

Notice that if A ∈ F has diameter less than r and intersects Int

r

Λ, then A ⊂ Λ.

Thus, the first term in the right side of the above equation is equal to zero. Hence, we get

kH

_Λ

− S

Λ

Ψ

0

k

_∞

≤ X

A∩(Λ\IntrΛ)6=∅

kΦ

_A

k

_∞

≤ #(Λ \ Int

r

Λ) X

0∈A

kΦ

_A

k

_∞

.

We remark that, for Λ

_n

= (−n, n)

^d

∩ Z

^d

, one has #(Λ

_n

\ Int

_r

Λ

_n

) ≤ C

_d

r

^d

n

^d−1

, for some constant C

_d

> 0 which depends just on the dimension d. In particular, we immediately obtain the following corollary.

Corollary 3.5. Let H be a hamiltonian defined by an invariant short range inter- action family. Then

H ¯ = inf

ω∈Ω

lim inf

n→∞

1

#Λ

n

S

_Λn

Ψ

₀

(ω).

We may now prove the second statement of Theorem 2.4.

Proposition 3.6. Let H be a hamiltonian defined by an invariant short range interaction family. Then, for any minimizing configuration ω with respect to H, the limit

n→∞

lim 1

#Λ

_n

H

Λn

(ω) does exist.

Proof. Let ω be a minimizing configuration for H. By Proposition 3.4, it is equiv- alent to show that the limit lim

n→∞ 1

#Λn

S

Λn

Ψ

0

(ω) exists. Notice also that we may assume without loss of generality that Ψ

0

≥ 0. So denote

L = lim inf

n→∞

1

#Λ

n

S

_Λn

Ψ

₀

(ω) and, given > 0, consider a positive integer N such that

_#Λ¹

N

S

ΛN

Ψ

0

(ω) < L + . Consider integers m and ` with m ≥ 1 and ` = 0, 1, . . . , N − 1. Notice that, for suitable integers j

₁

, . . . , j

_md

∈ Λ

_mN

, we may write

S

ΛmN+`

Ψ

0

=

m^d

X

k=1

S

ΛN

Ψ

0

◦ θ

^jk

+ X

j∈Λ_mN+`\ΛmN

Ψ

0

◦ θ

^j

. Define then ω

_N,k

:= θ

^jk

(ω)

j_k+ΛN

ω

_Zd\(j_k+ΛN)

and ¯ ω

_N,k

:= θ

^−jk

(ω

_N,k

). As ω is a minimizing configuration, one thus gets

H

j_k+ΛN

(ω) ≤ H

j_k+ΛN

(ω

N,k

) = H

ΛN

( ¯ ω

N,k

).

Recall that kS

_ΛN

Ψ

₀

◦ θ

^j

− H

_j+ΛN

k

_∞

≤ ΓN

^d−1

for every j ∈ Z

^d

, where Γ = Γ(d, r, {Φ

_A

}) = C

d

r

^d

P

0∈A

kΦ

_A

k

∞

. Therefore, we have that S

_ΛmN+`

Ψ

₀

(ω) ≤

m^d

X

k=1

H

_ΛN

( ¯ ω

_N,k

) + m

^d

ΓN

^d−1

+ X

j∈ΛmN+`\ΛmN

Ψ

₀

◦ θ

^j

(ω)

≤

m^d

X

k=1

S

ΛN

Ψ

0

( ¯ ω

N,k

) + 2m

^d

ΓN

^d−1

+ 2

^d

h

(m + 1)

^d

− m

^d

i

N

^d

kΨ

₀

k

_∞

. Notice that ( ¯ ω

_N,k

)

_ΛN

= ω

_ΛN

. We use now Lemma 3.3 to guarantee that there exists a constant κ > 0 such that S

ΛN

Ψ

0

( ¯ ω

N,k

) − S

ΛN

Ψ

0

(ω) ≤ κ #∂Λ

N

.

We obtain that S

_ΛmN+`

Ψ

₀

(ω) ≤

≤ m

^d

"

S

ΛN

Ψ

0

(ω) + κ #∂Λ

N

+ 2ΓN

^d−1

+ 2

^d

1 + 1 m

d

− 1

!

N

^d

kΨ

₀

k

_∞

# ,

which yields 1

#Λ

_mN+`

S

_ΛmN+`

Ψ

₀

(ω) ≤

≤ 1

#Λ

_N

S

_ΛN

Ψ

₀

(ω) + κ #∂Λ

_N

#Λ

_N

+ 2 Γ N +

1 + 1

m

d

− 1

!

kΨ

₀

k

_∞

.

6 Eduardo Garibaldi and Philippe Thieullen

Hence, it follows that lim sup

n→∞

1

#Λ

n

S

Λn

Ψ

0

(ω) ≤ L + + κ #∂Λ

N

#Λ

N

+ 2 Γ N .

Since the integer N can be taken arbitrarily large and > 0 can be chosen as close as one wants to zero, the proof is complete.

We will adopt an ergodic point of view. To that end, denote by M the set of Borel probability measures, equipped with the weak* topology. We consider the compact convex subset of invariant probabilities

M

θ

:= n

µ ∈ M : µ ◦ θ

^j

= µ, ∀ j ∈ Z

^d

o

. (3.2)

We have then the following characterization of the minimizing ergodic value ¯ H.

Proposition 3.7. Let H be a hamiltonian defined by an invariant short range interaction family. Then

H ¯ = min

µ∈Mθ

Z

Ω

Ψ

0

(ω) dµ(ω).

Proof. By the ergodic decomposition theorem (see, for example, Theorem 2.3.3 in [3]), one may suppose that µ ∈ M

θ

is ergodic. Therefore, by Birkhoff’s ergodic theorem (see, for instance, Theorem 2.1.5 in [3]), any configuration ω belonging to the support of µ satisfies

n→∞

lim 1

#Λ

n

S

_Λn

Ψ

₀

(ω) = Z

Ω

Ψ

₀

(ω) dµ(ω).

So thanks to Corollary 3.5, we have that ¯ H ≤ inf

µ∈Mθ

R Ψ

0

dµ.

For > 0, consider a configuration ω

∈ Ω and an arbitrarily large integer n

> 0 such that

_#Λ¹

n

S

_Λn

Ψ

₀

(ω

) < H ¯ + and define a Borel probability measure µ := 1

#Λ

n

X

j∈Λ_n

δ

_θ^j_(ω)

∈ M.

Let µ be any weak* accumulation probability for the family {µ

}

_>0

when goes to zero. Clearly by construction, R

Ω

Ψ

0

(ω) dµ(ω) ≤ H. So in order to obtain the ¯ opposite inequality, it is enough to argue that µ is invariant. However, notice that, for every i = 1, 2, . . . , n and for all f ∈ C

⁰

(Ω), one has

Z

(f ◦ θ

_i

− f ) dµ

≤ 1

#Λ

_n

2 #∂Λ

_n

kf k

_∞

→ 0 as n

→ ∞, which indeed shows the invariance of µ.

By a minimizing probability we mean an invariant probability µ that minimizes the average value R

Ψ

0

dµ, namely, such that ¯ H = R

Ψ

0

dµ. Their existence is guaranteed by the previous proposition. Moreoveor, by the ergodic decomposition theorem, there always exist ergodic minimizing probabilities.

Recall now that a point at the support of an invariant probability is said to be

generic if it belongs to a subset of full measure. We may complete the proof of

Theorem 2.4 with the following result.

Theorem 3.8. Suppose {Φ

_A

}

_A∈F

is an invariant interaction family with range r > 0. Then there are generic points ω at the support of any ergodic minimizing probability which are minimizing configurations for the associated hamiltonian H and satisfy

n→∞

lim 1

#Λ

n

H

Λn

(ω) = ¯ H.

Proof. For W ∈ F

^ΛM

, M ∈ N, consider the characteristic function χ

W

: Ω → {0, 1}

which has value 1 at a point ¯ ω ∈ Ω if, and only if, ¯ ω

_ΛM

= W .

Let µ ∈ M

θ

be an ergodic minimizing probability. Let us denote by b (φ) the subset of Ω of full measure for which Birkhoff’s ergodic theorem holds with respect to the integrable map φ : Ω → R . Obviously, any point

ω ∈ supp(µ) ∩ b (Ψ

0

) ∩ \

W∈F^ΛM, M∈N

b (χ

W

)

is generic and verify

n→∞

lim 1

#Λ

n

H

_Λn

(ω) = lim

n→∞

1

#Λ

n

S

_Λn

Ψ

₀

(ω) = Z

Ψ

₀

dµ = ¯ H.

Suppose on the contrary that ω is not a minimizing configuration. Hence, there shall exist ˜ ω ∈ Ω, ˜ N ∈ N and ˜ η > 0 such that

˜

ω

_Zd\ΛN˜

= ω

_Zd\ΛN˜

and H

_Λ˜

N

( ˜ ω) < H

_Λ˜

N

(ω) − η. ˜ Since in particular ω ∈ T

M∈N

b (χ

ωΛM

), we have that

n→∞

lim 1

#Λ

_n

S

_Λn

χ

_{ωΛ ˜}

N+r

(ω) = µ [

¯ ω∈Ω

ω

_Λ˜

N+r

ω ¯

_Zd\ΛN+r˜

!

=: λ

_N˜+r

> 0.

Therefore, for n large enough, one guarantees that S

Λn

χ

ω_{Λ ˜}

N+r

(ω) > #Λ

n λN+r˜

2

. Denote A

n

:= {j ∈ Λ

n

: ω

ΛN+r˜ +j

= ω

ΛN+r˜

}. Let then B

n

⊂ A

n

be a maximal subcollection of indices such that (Λ

N˜+r

+ j) ∩ (Λ

N+r˜

+ k) = ∅ whenever j, k ∈ B

n

. Since for all j ∈ A

_n

there must exist k ∈ B

_n

such that (Λ

_N˜+r

+ j) ∩ (Λ

_N+r˜

) 6= ∅, it follows that S

Λn

χ

ω_{Λ ˜}

N+r

(ω) = #A

n

≤ #B

n

· #Λ

N+r˜

, which yields for n large enough

1

#Λ

_n

#B

n

> 1

#Λ

N+r˜

λ

N˜+r

2 .

Thus, for n sufficiently large, let us introduce the configuration ω

ⁿ

∈ Ω as ω

ⁿ_Λ

N˜+j

= ˜ ω

_Λ˜

N

for all j ∈ B

_n

, and ω

ⁿ

Z^d\tj∈Bn(ΛN˜+j)

= ω

_Zd\t_j∈Bn(ΛN˜+j)

. From the construction, one gets that H

_Λ˜

N+j

(ω

ⁿ

) = H

_Λ˜

N

( ˜ ω) for each j ∈ B

_n

. Notice then

8 Eduardo Garibaldi and Philippe Thieullen

that

H

Λn

(ω

ⁿ

) = X

j∈Bn

H

ΛN˜+j

(ω

ⁿ

) + X

A∩Λn6=∅, A∩tj∈Bn

(

^ΛN˜+j

)

^=∅

Φ

A

(ω

ⁿ

)

< #B

n

H

ΛN˜

(ω) − η ˜

+ X

A∩Λn6=∅, A∩tj∈Bn

(

^ΛN˜+j

)

^=∅

Φ

A

(ω)

= H

_Λn

(ω) − η ˜ #B

_n

+ X

j∈Bn

H

_Λ˜

N

(ω) − H

_Λ˜

N

(θ

^j

(ω)) .

By the very definition of A

_n

, one obtains that H

_Λ˜

N

(ω) = H

_Λ˜

N

(θ

^j

(ω)) for any j ∈ B

_n

. In this way, let us also assume that n is large enough in order that

1

#Λ

_n

H

Λn

(ω

ⁿ

) < H ¯ + 1

#Λ

N+r˜

λ

N+r˜

4 η. ˜ Hence, for n sufficiently large, it is not difficult to see that

1

#Λ

n

H

Λn

(ω

ⁿ

) < H ¯ − 1

#Λ

N+r˜

λ

_N+r˜

4 η, ˜ which contradicts the definition (2.2) of the constant ¯ H.

References

[1] A. Bovier, Statistical mechanics of disordered systems: A mathematical perspective, Cam- bridge Series in Statistical and Probabilistic Mathematics

18, Cambridge University Press,

Cambridge, 2006.

[2] H. O. Georgii, Gibbs measures and phase transitions, De Gruyter Studies in Mathematics

9,

Walter de Gruyter, Berlin, 1988.

[3] G. Keller, Equilibrium states in ergodic theory, London Mathematical Society Students Texts

42, Cambridge University Press, Cambridge, 1998.