An ergodic description of ground states

Eduardo Garibaldi^{∗}

Departamento de Matemática Universidade Estadual de Campinas 13083-859 Campinas - SP, Brasil

  garibaldi@ime.unicamp.br

Philippe Thieullen^{†}

Institut de Mathématiques Université Bordeaux 1, CNRS, UMR 5251 F-33405 Talence, France

                                   Philippe.Thieullen@math.u-bordeaux1.fr

July 8, 2014

Abstract

Given a translation-invariant Hamiltonian H, a ground state on the lattice ℤ^{d} is a configuration whose energy, calculated with respect to H, cannot be lowered by altering its states on a finite number of sites. The set formed by these configurations is translation-invariant. Given an observable Ψ defined on the space of configurations, a minimizing measure is a translation-invariant probability which minimizes the average of Ψ. If Ψ_{0} is the mean contribution of all interactions to the site 0, we show that any configuration of the support of a minimizing measure is necessarily a ground state.

Keywords: bounded-spin lattice system, ergodic optimization, ground state, minimizing probability

2010 Mathematics Subject Classification: 37A60, 37J50, 82B20

1 Introduction

A dynamical system, in a very broad sense, is given by a compact metric space Ω and a ℤ^{d}-action θ^{j} : Ω → Ω, j ∈ ℤ^{d}, acting continuously on Ω. A probability measure μ is said to be translation invariant if μ(θ^{j}(B)) = μ(B) for any Borel set B of Ω. The central matter of ergodic optimization is to understand the set of the probability measures which minimize the average of a given continuous observable Ψ : Ω → ℝ. We call minimizing ergodic value of Ψ the quantity

Ψ̄ = inf { ∫ Ψ dμ : μ is a translation-invariant probability}.

A minimizing measure is a translation-invariant probability which attains the above infimum. For ℤ-actions, the subject has been extensively studied (for details, see Jenkinson’s notes [5]). Apparently no general result is known for ℤ^{d}-actions.

∗Supported by FAPESP 2009/17075-8 and Brazilian-French Network in Mathematics.

†Supported by FAPESP 2009/17075-8. Funding from the European Union’s Seventh Framework Program (FP7/2007-2013) under grant agreement No. 318999 BREUDS.

A similar problem exists in the theory of Gibbs measures on bounded-spin lattice systems, as described, for instance, in [1, 2, 3]. The configuration space Ω is then given by the lattice ℤ^{d} of sites whose states belong to a fixed compact space Ω_{0}. Formally Ω = Ω_{0}^{ℤ^{d}} is equipped with the product topology. The energy of a configuration ω ∈ Ω is usually understood as an infinite-volume limit, to be properly defined, of energies calculated on finite volumes. A finite volume is simply a finite subset Λ of ℤ^{d} and the energy of a configuration restricted to a finite volume Λ is computed using a single continuous function H_{Λ} : Ω → ℝ. Usually H_{Λ} describes all admissible interaction energies (internal to Λ or representing some coupling with the exterior), and has the form H_{Λ} = ∑_{A∩Λ≠∅} Φ_{A} for some family of continuous functions Φ_{A} : Ω → ℝ indexed by finite subsets A ⊂ ℤ^{d}. A ground-state configuration ω is a configuration whose energy H_{Λ}(ω) on ea ch fixed volume Λ cannot be lowered by changing the states of the sites restricted to Λ. We denote by Ω_{GS}(H) the set of ground-state configurations. Although Ω_{GS}(H) is defined without mentioning the temperature, it may be seen as a set containing the support of limit Gibbs states obtained as the absolute temperature tends to zero. More informations can be found in [2], appendix B.2. We shall not say anything on this issue. Our goal will consist in showing that Ω_{GS}(H) contains the support of any minimizing measure of a particular observable Ψ_{0} = ∑_{A∋0} \frac{1}{#A}Φ_{A} that summarizes the total mean energy contribution to the site 0.

2 Framework and Main Results

The spin values are described here by a compact metrizable space Ω_{0}. We introduce then the configuration space Ω = Ω_{0}^{ℤ^{d}}. A configuration ω ∈ Ω is described by giving the states ω = {ω_{j}}_{j∈ℤ^{d}} at all sites j ∈ ℤ^{d}. Endowed with the product topology, Ω is a compact metrizable space. If d_{0} is a metric compatible with the topology of Ω_{0} and if ∥j∥ := |j_{1}| + … + |j_{d}| denotes the L^{1} norm of an index j = (j_{1}, …, j_{d}) ∈ ℤ^{d}, one may define a metric on Ω compatible with the product topology by d(ω, ω̄) = ∑_{j∈ℤ^{d}} 2^{−∥j∥}d_{0}(ω_{j}, ω̄_{j}).

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+e_{i}}}_{j∈ℤ^{d}}. Given j = (j_{1}, …, j_{d}) ∈ ℤ^{d}, we define θ^{j} := θ_{1}^{j_{1}} ∘ θ_{2}^{j_{2}} ∘ ⋯ ∘ θ_{d}^{j_{d}}.

Let ℱ denote the collection of finite subsets of ℤ^{d}. We call diameter of a set A ∈ ℱ the real number diam(A) = max{∥j − k∥ : j, k ∈ A}. For r > 0, we call inner r-boundary ∂_{r}^{−}(A) and outer r-boundary ∂_{r}^{+}(A) the two sets adjacent to A

∂_{r}^{−}(A) := {j ∈ A : ∥j − k∥ ≤ r for some k ∈ ℤ^{d} \ A},
∂_{r}^{+}(A) := {j ∈ ℤ^{d} \ A : ∥j − k∥ ≤ r for some k ∈ A}.

We call r-boundary ∂_{r}(A) := ∂_{r}^{−}(A) ∪ ∂_{r}^{+}(A). For A ∈ ℱ and ω ∈ Ω, the notation ω|_{A} or simply ω_{A} will denote the restriction of the configuration ω to the set A. The cardinality of a subset Λ ⊂ ℤ^{d} is denoted by #A, and the complement by Λ^{∁}.

The Hamiltonian of the system will be defined through a family of local interactions Φ_{A} : Ω → ℝ, where each Φ_{A}(ω) takes into account the local interaction energy of the configuration ω_{A}. More precisely, we recall a standard definition.

Definition 2.1. We call translation-invariant interaction family any collection of continuous maps Φ_{A} : Ω → ℝ, indexed by A ∈ ℱ such that

i. Φ_{j+A}(ω) = Φ_{A}(θ^{j}(ω)), for all j ∈ ℤ^{d} and ω ∈ Ω;

ii. ω_{A} = ω̄_{A} implies Φ_{A}(ω) = Φ_{A}(ω̄).

In addition, we say that {Φ_{A}}_{A∈ℱ} is absolutely summable if

∑_{A:0∈A} ∥Φ_{A}∥_{∞} < ∞.

We also recall the following related notions.

Definition 2.2. A translation-invariant interaction family {Φ_{A}}_{A∈ℱ} is said to have finite-range if there exists an integer r > 0 such that Φ_{A} ≡ 0 whenever diam(A) > r. In this case, we also say that the translation-invariant interaction family has range r > 0. In particular for r = 1, the interaction takes into account the nearest neighbors only. (Notice that any translation-invariant interaction family with finiterange is absolutely summable.) We say that {Φ_{A}}_{A∈ℱ} is a long-range interaction family if Φ_{A} ≢ 0 for sets A with arbitrarily large diameter.

Given a translation-invariant absolutely summable interaction family {Φ_{A}}_{A∈ℱ}, the associated Hamiltonian H : ℱ × Ω → ℝ is defined by

H(Λ, ·) = H_{Λ} := ∑_{A: A∩Λ≠∅} Φ_{A}, ∀ Λ ∈ ℱ.  (2.1)

It follows from the absolute summability condition of the interaction family that the associated Hamiltonian is a well defined function. For each Λ ∈ ℱ, H_{Λ} is actually a continuous function obtained as a uniform limit of continuous functions on Ω. Moreover, from its uniform continuity, it follows that H_{Λ} is quasi-local in the sense that

lim_{Λ̄↑ℤ^{d}} sup_{\begin{matrix}ω,ω̄∈Ω \\ ωΛ̄=ω̄Λ̄ \end{matrix}} (H_{Λ}(ω) − H_{Λ}(ω̄)) = 0.

Notice also that the Hamiltonian inherits the invariance of the interaction family

H_{Λ} ∘ θ^{j} = H_{j+Λ} ∀ j ∈ ℤ^{d}, ∀ Λ ∈ ℱ.

The literature is filled with examples of such a formalism. For the Ising model, for instance, the energy of a configuration ω ∈ {−1, 1}^{ℤ^{d}} is formally given by the Hamiltonian H(ω) = −J ∑_{<i j>} ω_{i}ω_{j} − h ∑_{j} ω_{j}, where the first sum is over pairs of adjacent spins, J is a coupling constant which describes (according to its sign) ferromagnetic or antiferromagnetic phenomena, and h represents an external magnetic field. In terms of translation-invariant interactions, the Ising model highlights the nearest-neighbor interaction family given by the functions Φ_{{0}} = −hω_{0}, Φ_{{0,e_{k}}} = −Jω_{0}ω_{e_{k}}, k = 1, …, d, and their translations. The classical Heisenberg model is another example of finite-range interaction model. In this case, the local state space is the unit Euclidean sphere, Ω_{0} = {ω_{0} ∈ ℝ^{3} : ∥ω_{0}∥ = 1}, and the formal Hamiltonian is H(ω) = −J ∑_{<i j>} ω_{i} · ω_{j}. By its turn, Dyson model exhibits long-range interactions with the introduction of a pair-wise coupling which decreases with the distance between the spins. In this model, the lattice is one-dimensional and, for ω ∈ {−1, 1}^{ℤ}, the Hamiltonian takes the form H(ω) = −J ∑_{i>j} ω_{i}ω_{j}/(i − j)^{α} with α ∈ (1, 2). The non-null functions of the absolutely summable interaction family are thus Φ_{{0,k}}(ω) = −Jk^{−α}ω_{0}ω_{k}, k ≥ 1, as well as their translations. All these examples are considered in the results that follow.

In order to be able to apply Birkhoff’s ergodic theorem, we shall consider a unique function Ψ_{0} : Ω → ℝ, which corresponds to the normalized contribution of all interaction energies at the site 0, and is defined by

Ψ_{0} := ∑_{A: 0∈A} \frac{1}{#A}Φ_{A}.  (2.2)

Notice that Ψ_{0} is a continuous real valued function thanks to the absolutely summability condition. Moreover, Ψ_{0} is also quasi-local:

lim_{Λ̄↑ℤ^{d}} sup_{\begin{matrix}ω,ω̄∈Ω \\ ωΛ̄=ω̄Λ̄ \end{matrix}} (Ψ_{0}(ω) − Ψ_{0}(ω̄)) = 0.

Notice also that we could have introduced a notion of energy of a configuration ω restricted to Λ ∈ ℱ by using the Birkhoff’s sum

S_{Λ}Ψ_{0}(ω) := ∑_{k∈Λ} Ψ_{0} ∘ θ^{k}(ω).

For ω, ω̄ ∈ Ω and Λ ∈ ℱ, we denote by ω̄_{Λ}ω_{ℤ^{d}\Λ} the configuration of Ω that coincides with ω̄ on Λ and with ω on ℤ^{d} \ Λ. Let ℳ(Ω) be the set of Borel probability measures equipped with the weak* topology. Let ℳ(Ω, θ) be the subset of translation-invariant probability measures

ℳ(Ω, θ) := {μ ∈ ℳ(Ω) : μ ∘ θ^{j} = μ, ∀ j ∈ ℤ^{d}} .

We shall use Birkhoff’s ergodic theorem for sequences of square boxes

Λ_{n} := [−n, n]^{d} ∩ ℤ^{d}, ∀ n ∈ ℕ.

Our main goal is to describe the set of ground-state configurations. We choose two possible definitions. The first one is more general since no hypothesis of invariance needs to be assumed. The second one is closer to notions that one finds in ergodic optimization.

Definition 2.3. We say that ω ∈ Ω is a ground-state configuration with respect to the Hamiltonian H if

H_{Λ}(ω) ≤ H_{Λ}(ω̄_{Λ}ω_{ℤ^{d}\Λ}), ∀ ω̄ ∈ Ω, ∀ Λ ∈ ℱ.

Let Ω_{GS}(H) be the set of ground-state configurations.

We say that μ ∈ ℳ(Ω, θ) is a minimizing measure for the function Ψ_{0} if

∫_{Ω} Ψ_{0} dμ ≤ ∫_{Ω} Ψ_{0} dν, ∀ ν ∈ ℳ(Ω, θ).

Let ℳ_{min}(Ω, θ, H) be the set of minimizing measures.

We call ground-state energy of the Hamiltonian H the constant

H̄ := inf_{ω∈Ω} lim inf_{n→∞} \frac{1}{#Λ_{n}} H_{Λ_{n}}(ω).

It is easy to show that ground-state configurations do exist for a translationinvariant absolutely summable interaction family. We shall give a short proof for completeness (see proposition 3.1). The existence of minimizing measures is clearly guaranteed by the weak* compactness of ℳ(Ω, θ). Moreover, by the ergodic decomposition theorem (see, for instance, [3], section 7.3), there always exist ergodic minimizing measures.

Our theorem states that a translation-invariant probability measure is minimizing if, and only if, its support lies on the set of ground-state configurations. The necessity part is more difficult to prove and uses a kind of maximal lemma for ground-state configurations.

Theorem 2.4. Let H be an Hamiltonian defined by a translation-invariant absolutely summable interaction family. Then, one has

i. H̄ = lim_{n→∞} \frac{1}{#Λ_{n}} inf_{ω∈Ω} H_{Λ_{n}}(ω);

ii. H̄ = ∫_{Ω} Ψ_{0} dμ, ∀ μ ∈ ℳ_{min}(Ω, θ, H);

iii. H̄ = lim_{n→∞} \frac{1}{#Λ_{n}} H_{Λ_{n}}(ω), ∀ ω ∈ Ω_{GS}(H);

iv. (Subordination Principle) if μ is a translation-invariant probability, then

supp(μ) ⊆ Ω_{GS}(H) ⟹ μ ∈ ℳ_{min}(Ω, θ, H);

v. (Main Result) if μ is a translation-invariant probability, then

μ ∈ ℳ_{min}(Ω, θ, H) ⟹ supp(μ) ⊆ Ω_{GS}(H).

The previous theorem extends several results in Schrader’s article [7]. In his work, Schrader first considers a configuration space of the form {0, 1}^{ℤ^{d}}, which enables him to identify configurations to subsets of ℤ^{d}. We do not restrict our analysis to a finite state space Ω_{0}. In order to define the ground-state energy H̄, he only considers Hamiltonians with free boundary conditions H_{Λ,F ree} := ∑_{A⊆Λ} Φ_{A}. We use the more natural Hamiltonian H_{Λ} = ∑_{A∩Λ≠∅} Φ_{A}, which takes into account the interactions across the boundary of Λ. Moreover, Schrader does not introduce the normalized contribution of all interaction energies Ψ_{0} and makes no connection between the present notion of minimizing measures (as found in ergodic optimization theory) and his notion of ground-state translation-invariant measures. We have changed a little bit the terminology: we use the expression ground-state configuration for configurations ω ∈ Ω_{GS}(H), no name is given in [7]; we use the expression minimizing measure for translation-invariant probability measures μ which minimize ∫ Ψ_{0} dμ as in ergodic optimization theory, a similar notion is used in [7] and is called ground state without asking the shift invariance. Part iv of theorem 2.4 is similar to theorem 4.6 in [7], part v is similar to theorem 4.8 there. Our motivation to extend Schrader’s article from the simple state space Ω_{0} = {0, 1} to any general compact state space is to make clear the connection between two notions: a notion of configuration with the lowest possible energy where no average is computed, and a notion of ground-state energy which uses mean values of a unique energy function Ψ_{0}.

3 Proof of Theorem 2.4

From now on, without being restated each time, we assume that {Φ_{A}}_{A∈ℱ} is a translation-invariant absolutely summable interaction family. We begin by showing that ground-state configurations do exist. We give a short proof for completeness.

Proposition 3.1. The set of ground-state configurations with respect to H is a non-empty, closed and translation-invariant set.

Proof. For Λ ∈ ℱ, let Ω_{GS,Λ} be the set of the configurations ω ∈ Ω such that

∀ ω′ ∈ Ω, ω′_{ℤ^{d}\Λ} = ω_{ℤ^{d}\Λ} ⟹ H_{Λ}(ω) ≤ H_{Λ}(ω′).

We choose a reference configuration ω̄ ∈ Ω. Notice that Ω_{GS,Λ} clearly contains the minimum points of the continuous map ω_{Λ} ∈ Ω_{0}^{Λ} ↦ H_{Λ}(ω_{Λ}ω̄_{ℤ^{d}\Λ}) ∈ ℝ. Moreover, Ω_{GS,Λ} is closed and monotone with respect to Λ in the following sense: if Λ ⊂ Λ′, then Ω_{GS,Λ′} ⊂ Ω_{GS,Λ}. (We use the fact that H_{Λ′} = H_{Λ} + ∑_{A∩Λ=∅, A∩Λ′≠∅} Φ_{A}.) In particular, the family {Ω_{GS,Λ}}_{Λ∈ℱ} has the finite intersection property, and by compactness, Ω_{GS}(H) = ⋂_{Λ∈ℱ} Ω_{GS,Λ} is non-empty and translation invariant.

The proof of the main result of theorem 2.4 will be given at the end of this section. We shall first need the following two lemmas which can be found in the literature. We have nevertheless included short proofs for convenience of the reader and for clarification of the notations. We show in the first lemma how one can neglect the long-range interaction.

Lemma 3.2. lim_{n→∞} \frac{1}{#Λ_{n}} ∑_{j∈Λ_{n}} ∑_{\begin{matrix}A∋j \\ A⊄Λ_{n} \end{matrix}} ∥Φ_{A}∥_{∞} = 0.

An Ergodic Description of Ground States 7

Proof. For Λ_{m} ⊂ Λ_{n}, we have that

∑_{j∈Λ_{n}} ∑_{\begin{matrix}A∋j \\ A⊄Λ_{n} \end{matrix}} ∥Φ_{A}∥_{∞} ≤ ∑_{\begin{matrix}j∈Λ_{n} \\ j+Λ_{m}⊂Λ_{n} \end{matrix}} ∑_{\begin{matrix}A∋j \\ A⊄j+Λ_{m} \end{matrix}} ∥Φ_{A}∥_{∞} + ∑_{\begin{matrix}j∈Λ_{n} \\ j+Λ_{m}⊄Λ_{n} \end{matrix}} ∑_{A∋j} ∥Φ_{A}∥_{∞}

≤ ∑_{j∈Λ_{n}} ∑_{\begin{matrix}A∋j \\ A⊄j+Λ_{m} \end{matrix}} ∥Φ_{A}∥_{∞} + #\{j ∈ Λ_{n} : j + Λ_{m} ⊄ Λ_{n}\} ∑_{A∋0} ∥Φ_{A}∥_{∞}

= #Λ_{n} ∑_{\begin{matrix}A∋0 \\ A⊄Λ_{m} \end{matrix}} ∥Φ_{A}∥_{∞} + #(Λ_{n} \ Λ_{n−m}) ∑_{A∋0} ∥Φ_{A}∥_{∞}.

By the absolute summability, given ϵ > 0, there is m ∈ ℕ with ∑\begin{matrix}A∋0 \\ A⊄Λ_{m} \end{matrix} ∥Φ_{A}∥_{∞} < \frac{ϵ}{2}. One may find n_{0} > m such that \frac{#(Λ_{n}\Λ_{n−m})}{#Λ_{n}} ∑_{A∋0} ∥Φ_{A}∥_{∞} < \frac{ϵ}{2} for all n ≥ n_{0}. Therefore, whenever n ≥ n_{0}, we obtain

∑_{j∈Λ_{n}} ∑_{\begin{matrix}A∋j \\ A⊄Λ_{n} \end{matrix}} ∥Φ_{A}∥_{∞} ≤ ϵ#Λ_{n},

which finishes the proof.

We show in the second lemma that the two average energies \frac{1}{#Λ_{n}} H_{Λ_{n}}(ω) and \frac{1}{#Λ_{n}} S_{Λ_{n}}Ψ_{0}(ω) are comparable and have uniformly controlled oscillations. Lemma 3.3.

i. lim_{n→∞} \frac{1}{#Λ_{n}} sup_{ω_{Λn}=ω̄_{Λn}} (S_{Λ_{n}}Ψ_{0}(ω) − S_{Λ_{n}}Ψ_{0}(ω̄)) = 0;

ii. lim_{n→∞} \frac{1}{#Λ_{n}} sup_{ω_{Λn}=ω̄_{Λn}} (H_{Λ_{n}}(ω) − H_{Λ_{n}}(ω̄)) = 0;

iii. lim_{n→∞} \frac{1}{#Λ_{n}} ∥H_{Λ_{n}} − S_{Λ_{n}}Ψ_{0}∥_{∞} = 0.

Proof. Notice first that

S_{Λ}Ψ_{0} = ∑_{j∈Λ} ∑_{\begin{matrix}A∋j \\ A⊂Λ \end{matrix}} \frac{1}{#A}Φ_{A} + ∑_{j∈Λ} ∑_{\begin{matrix}A∋j \\ A⊄Λ \end{matrix}} \frac{1}{#A}Φ_{A} = ∑_{A⊂Λ} Φ_{A} + ∑_{\begin{matrix}A∩Λ≠∅ \\ A∩Λ^{∁}≠∅ \end{matrix}} \frac{#(A ∩ Λ)}{#A}Φ_{A}. (3.1)

So for ω_{Λ} = ω̄_{Λ}, we have

S_{Λ}Ψ_{0}(ω_{Λ}) − S_{Λ}Ψ_{0}(ω̄_{Λ}) = ∑_{j∈Λ} ∑_{\begin{matrix}A∋j \\ A⊄Λ \end{matrix}} \frac{1}{#A}(Φ_{A}(ω_{Λ}) − Φ_{A}(ω̄_{Λ}))
≤ 2 ∑_{j∈Λ} ∑_{\begin{matrix}A∋j \\ A⊄Λ \end{matrix}} \frac{1}{#A}∥Φ_{A}∥_{∞} ≤ ∑_{j∈Λ} ∑_{\begin{matrix}A∋j \\ A⊄Λ \end{matrix}} ∥Φ_{A}∥_{∞},

and item i follows then from lemma 3.2.

In order to prove item iii, we decompose H_{Λ} similarly

H_{Λ} = ∑_{A⊂Λ} Φ_{A} + ∑_{\begin{matrix}A∩Λ≠∅ \\ A∩Λ^{∁}≠∅ \end{matrix}} Φ_{A} = ∑_{j∈Λ} ∑_{\begin{matrix}A∋j \\ A⊂Λ \end{matrix}} \frac{1}{#A}Φ_{A} + ∑_{\begin{matrix}A∩Λ≠∅ \\ A∩Λ^{∁}≠∅ \end{matrix}} Φ_{A}.  (3.2)

From the above equality and from (3.1), we have

S_{Λ}Ψ_{0} − H_{Λ} = ∑_{\begin{matrix}A∩Λ≠∅ \\ A∩Λ^{∁}≠∅ \end{matrix}} (\frac{#(A ∩ Λ)}{#A} − 1)Φ_{A},

∥S_{Λ}Ψ_{0} − H_{Λ}∥_{∞} ≤ ∑_{\begin{matrix}A∩Λ≠∅ \\ A∩Λ^{∁}≠∅ \end{matrix}} ∥Φ_{A}∥_{∞} = ∑_{\begin{matrix}A∩Λ≠∅ \\ A∩Λ^{∁}≠∅ \end{matrix}} ∑_{j∈A∩Λ} \frac{1}{#(A ∩ Λ)}∥Φ_{A}∥_{∞}

= ∑_{j∈Λ} ∑_{\begin{matrix}A∋j \\ A⊄Λ \end{matrix}} \frac{1}{#(A ∩ Λ)}∥Φ_{A}∥_{∞} ≤ ∑_{j∈Λ} ∑_{\begin{matrix}A∋j \\ A⊄Λ \end{matrix}} ∥Φ_{A}∥_{∞}.  (3.3)

Then item iii follows from lemma 3.2 and item ii follows from i and iii.

Remark 3.4. For a translation-invariant interaction family with finite range r, the properties pointed out in the above lemma may be more precisely stated as follows:

i. sup_{Λ∈ℱ} sup_{ω_{Λ}=ω̄_{Λ}} \frac{1}{#∂_{1}^{+}Λ} (S_{Λ}Ψ_{0}(ω) − S_{Λ}Ψ_{0}(ω̄)) < ∞;

ii. sup_{Λ∈ℱ} sup_{ω_{Λ}=ω̄_{Λ}} \frac{1}{#∂_{1}^{+}Λ} (H_{Λ}(ω) − H_{Λ}(ω̄)) < ∞;

iii. ∥H_{Λ} − S_{Λ}Ψ_{0}∥_{∞} ≤ #∂_{r}^{−}(Λ) ∑_{A∋0} ∥Φ_{A}∥_{∞}.

From lemma 3.3, item iii, we immediately obtain the following corollary which shows that H̄ is a minimizing ergodic value of a unique observable in the context of ergodic optimization (see [5]). We shall show soon in proposition 3.7 that we can actually permute inf_{ω∈Ω} and lim inf_{n→+∞} and that H̄ is obtained by minimizing ∫ Ψ_{0} dμ over all translation-invariant probabilities μ.

Corollary 3.5. The minimizing ergodic value of H is given by

H̄ = inf_{ω∈Ω} lim inf_{n→∞} \frac{1}{#Λ_{n}} S_{Λ_{n}}Ψ_{0}(ω).

In the following proposition, we first show that the limit in theorem 2.4, item iii, does exist. The proof is similar to Birkhoff’s ergodic proof for uniquely ergodic systems. The identification of the limit to H̄ will be done in proposition 3.8.

Proposition 3.6. If ω is a ground-state configuration with respect to H, then both

limits lim_{n→∞} \frac{1}{#Λ_{n}} H_{Λ_{n}}(ω) = lim_{n→∞} \frac{1}{#Λ_{n}} S_{Λ_{n}}Ψ_{0}(ω) do exist.

Proof. Let ω be a ground-state configuration for H. By lemma 3.3, item iii, it is enough to show that the second limit exists. Set

L = lim inf_{n→∞} \frac{1}{#Λ_{n}} S_{Λ_{n}}Ψ_{0}(ω).

Given ϵ > 0, consider a positive integer N large enough in such a way that

\frac{1}{#Λ_{N}} S_{Λ_{N}} Ψ_{0}(ω) < L + ϵ,

\frac{1}{#Λ_{N}} ∥S_{Λ_{N}} Ψ_{0} ∘ θ^{j} − H_{j+Λ_{N}} ∥_{∞} = \frac{1}{#Λ_{N}} ∥S_{Λ_{N}} Ψ_{0} − H_{Λ_{N}} ∥_{∞} ≤ ϵ, (3.4)
and \frac{1}{#Λ_{N}} sup_{ω′_{Λ_{N}} =ω′′_{Λ_{N}}} (S_{Λ_{N}} Ψ_{0}(ω′) − S_{Λ_{N}} Ψ_{0}(ω′′)) ≤ ϵ.  (3.5)

Suppose now that the integers m, n ≥ 1 and ℓ ∈ {0, 1, …, 2N} are such that 2n + 1 = m(2N + 1) + ℓ. We choose a subset of indices J of Λ_{n} so that the translates of Λ_{N}, {j + Λ_{N}}_{j∈J}, are pairwise disjoint, are contained inside Λ_{n}, and essentially cover Λ_{n} in the sense that

#(Λ_{n} \ ∪_{j∈J}(j + Λ_{N})) ≤ C_{d,N}m^{d−1},

where C_{d,N} is a constant that depends only on d and N. We then decompose the Birkhoff’s sum

S_{Λ_{n}}Ψ_{0} = ∑_{j∈J} S_{Λ_{N}} Ψ_{0} ∘ θ^{j} + ∑_{k∈Λ_{n}\∪_{j∈J} (j+Λ_{N})} Ψ_{0} ∘ θ^{k}.

We define for each index j ∈ J a configuration ω^{j} which coincides with θ^{−j}(ω) on j + Λ_{N} and with ω on ℤ^{d} \ (j + Λ_{N}). Since ω is minimizing, we have

S_{Λ_{N}} Ψ_{0} ∘ θ^{j}(ω) = S_{j+Λ_{N}} Ψ_{0}(ω) ≤ H_{j+Λ_{N}} (ω) + ϵ#Λ_{N}
≤ H_{j+Λ_{N}} (ω^{j}) + ϵ#Λ_{N} = H_{Λ_{N}} (θ^{j}(ω^{j})) + ϵ#Λ_{N}
≤ S_{Λ_{N}} Ψ_{0}(θ^{j}(ω^{j})) + 2ϵ#Λ_{N} ≤ S_{Λ_{N}} Ψ_{0}(ω) + 3ϵ#Λ_{N}.

By adding these inequalities over j ∈ J, one obtains

S_{Λ_{n}}Ψ_{0}(ω) ≤ #J(S_{Λ_{N}} Ψ_{0}(ω) + 3ϵ#Λ_{N}) + C_{d,N}m^{d−1}∥Ψ_{0}∥_{∞}.

Since #J ≤ #Λ_{n}/#Λ_{N}, it follows that

lim sup_{n→∞} \frac{1}{#Λ_{n}} S_{Λ_{n}}Ψ_{0}(ω) ≤ L + 4ϵ.

Since ϵ > 0 can be chosen as close as one wants to zero, the proof is complete.

We have then the following characterization of the minimizing ergodic value H̄.

Proposition 3.7 (Items i and ii of Theorem 2.4).

H̄ = min_{μ∈ℳ(Ω,θ)} ∫_{Ω} Ψ_{0} dμ = lim_{n→∞} \frac{1}{#Λ_{n}} inf_{ω∈Ω} S_{Λ_{n}}Ψ_{0}(ω) = lim_{n→∞} \frac{1}{#Λ_{n}} inf_{ω∈Ω} H_{Λ_{n}}(ω).

Proof. The last equality will follow from lemma 3.3, item iii. Besides, by standard superaditivity argument (see, for instance, Proposition 4.10 in [4]), one can ensure that the limit lim_{n→∞} \frac{1}{#Λ_{n}} inf_{ω∈Ω} S_{Λ_{n}}Ψ_{0}(ω) exists. Therefore, we notice that

min_{μ} ∫ Ψ_{0} dμ ≥ H̄ ≥ lim_{n→+∞} \frac{1}{#Λ_{n}} inf_{ω∈Ω} S_{Λ_{n}}Ψ_{0}(ω)  (3.6)

The second inequality of (3.6) comes from corollary 3.5. To prove the first one, we use Birkhoff’s ergodic theorem (see, for example, Theorem 2.1.5 in [6]). Suppose that μ ∈ ℳ(Ω, θ) is ergodic. By Birkhoff’s ergodic theorem, μ-almost every configuration ω satisfies

∫ Ψ_{0} dμ = lim_{n→∞} \frac{1}{#Λ_{n}} S_{Λ_{n}}Ψ_{0}(ω) ≥ H̄.

Hence, the first inequality of (3.6) follows from the existence of a minimizing ergodic probability.

To conclude the proof, it thus suffices to show that

lim_{n→∞} \frac{1}{#Λ_{n}} inf_{ω} S_{Λ_{n}}Ψ_{0}(ω) ≥ min_{μ} ∫ Ψ_{0} dμ.

For each n, consider a configuration ω^{n} ∈ Ω such that S_{Λ_{n}}Ψ_{0}(ω^{n}) = inf_{ω} S_{Λ_{n}}Ψ_{0}(ω) and define a Borel probability measure

μ_{n} := \frac{1}{#Λ_{n}} ∑_{j∈Λ_{n}} δ_{θ^{j}(ω^{n})} ∈ ℳ(Ω).

Let μ ∈ ℳ(Ω) be any weak* limit for a subsequence {μ_{n_{k}}}. Clearly by construction,

lim_{n→∞} inf_{ω∈Ω} \frac{S_{Λ_{n}}Ψ_{0}(ω)}{#Λ_{n}} = lim_{k→∞} \frac{S_{Λ_{n_{k}}} Ψ_{0}(ω^{n_{k}})}{#Λ_{n_{k}}} = lim_{k→∞} ∫ Ψ_{0} dμ_{n_{k}} = ∫ Ψ_{0} dμ.

Moreover, μ is translation-invariant: for any continuous function f, one has

∀ i = 1, …, d,  |∫ (f ∘ θ_{i} − f) dμ_{n_{k}} | ≤ 2\frac{#∂_{1}^{+}Λ_{n_{k}}}{#Λ_{n_{k}}} ∥f∥_{∞} ⟶^{k→∞} 0,

which indeed shows the invariance of μ.

The next proposition also contains items of our theorem.

Proposition 3.8 (Items iii and iv of Theorem 2.4).

∀ ω ∈ Ω_{GS}(H), lim_{n→∞} \frac{1}{#Λ_{n}} H_{Λ_{n}}(ω) = H̄,

∀ μ ∈ ℳ(Ω, θ), supp(μ) ⊂ Ω_{GS}(H) ⟹ H̄ = ∫ Ψ_{0} dμ.

Proof. Let ω ∈ Ω be a ground-state configuration. From proposition 3.6, we know that the limit lim_{n→∞} \frac{1}{#Λ_{n}} H_{Λ_{n}}(ω) exists. Besides, from the very definition of H̄, we get H̄ ≤ lim_{n→∞} \frac{1}{#Λ_{n}} H_{Λ_{n}}(ω). Let ω′ be a configuration of Ω and let Λ be a finite subset of ℤ^{d}. Define ω̄ equal to ω on ℤ^{d} \ Λ and equal to ω′ on Λ. Using equation (3.2) and the upper bound given by (3.3), we have

H_{Λ}(ω) ≤ H_{Λ}(ω̄) = ∑_{A⊂Λ} Φ_{A}(ω̄) + ∑_{\begin{matrix}A∩Λ≠∅ \\ A∩Λ^{∁}≠∅ \end{matrix}} Φ_{A}(ω̄)

≤ H_{Λ}(ω′) + 2 ∑_{\begin{matrix}A∩Λ≠∅ \\ A∩Λ^{∁}≠∅ \end{matrix}} ∥Φ_{A}∥_{∞} ≤ H_{Λ}(ω′) + 2 ∑_{j∈Λ} ∑_{\begin{matrix}A∋j \\ A⊄Λ \end{matrix}} ∥Φ_{A}∥_{∞}.

We thus obtain, thanks to lemma 3.2 and for any configuration ω′,

lim_{n→∞} \frac{1}{#Λ_{n}} H_{Λ_{n}}(ω) ≤ lim inf_{n→∞} \frac{1}{#Λ_{n}} H_{Λ_{n}}(ω′),

which yields lim_{n→∞} \frac{1}{#Λ_{n}} H_{Λ_{n}}(ω) ≤ H̄.

If μ is translation invariant and has support included in Ω_{GS}(H), by Birkhoff’s ergodic theorem, the support of μ contains a generic configuration ω, which implies that μ is minimizing thanks to lemma 3.3, item iii, and the first part of the proof.

Before proving item v of Theorem 2.4, we will need the following estimate.

Lemma 3.9. For any Λ ∈ ℱ and M ∈ ℕ,

∥H_{Λ} − ∑_{\begin{matrix}A∩Λ≠∅ \\ diam(A)≤2dM \end{matrix}} Φ_{A}∥_{∞} ≤ #Λ ∑_{\begin{matrix}A∋0 \\ A⊄Λ_{M} \end{matrix}} ∥Φ_{A}∥_{∞}.

Proof. Indeed

H_{Λ} − ∑_{\begin{matrix}A∩Λ≠∅ \\ diam(A)≤2dM \end{matrix}} Φ_{A} = ∑_{\begin{matrix}A∩Λ≠∅ \\ diam(A)>2dM \end{matrix}} Φ_{A} = ∑_{k∈Λ} ∑_{\begin{matrix}A∋k \\ diam(A)>2dM \end{matrix}} \frac{1}{#(A ∩ Λ)}Φ_{A}.

The result follows then from the translation invariance of the family {Φ_{A}}_{A∈ℱ}.

We conclude by the proving the main result of this paper. The proof uses an estimate which is similar to the one that we find in the proof of the ergodic maximal lemma.

Proposition 3.10 (Item v of Theorem 2.4). The support of any minimizing probability is included in the set of ground-state configurations Ω_{GS}(H).

Proof. The proof is done by contradiction. Let μ be an ergodic minimizing probability whose support is not included in Ω_{GS}(H). The open set U = Ω \ Ω_{GS}(H) intersects supp(μ) and satisfies μ(U) > 0. We choose a generic configuration ω ∈ U in the following sense. Let {V_{ℓ}}_{ℓ≥0} be a countable basis of open sets for the product topology of Ω. For each ℓ ≥ 0, consider the characteristic function χ_{ℓ} : Ω → {0, 1} of V_{ℓ}. By Birkhoff’s ergodic theorem, there exists a Borel set B ⊂ U such that μ(B) = μ(U) and, for any ω ∈ B and ℓ ≥ 0,

lim_{n→∞} \frac{S_{Λ_{n}}Ψ_{0}(ω)}{#Λ_{n}} = ∫ Ψ_{0} dμ and lim_{n→∞} \frac{S_{Λ_{n}}χ_{ℓ}(ω)}{#Λ_{n}} = μ(V_{ℓ}).  (3.7)

We choose once for all ω ∈ B ∩ supp(μ).

Since ω is not a ground-state configuration, there exist ω̃ ∈ Ω, Ñ ∈ ℕ and η̃ > 0 such that

ω̃_{ℤ^{d}\ΛÑ} = ω_{ℤ^{d}\ΛÑ} and H_{ΛÑ} (ω̃) < H_{ΛÑ} (ω) − η̃.  (3.8)

Notice that ω̃ is not any more generic. Thanks to lemma 3.9 and the summability condition of {Φ_{A}}_{A∈ℱ}, we may choose a positive integer M such that

∥H_{ΛÑ} − ∑_{\begin{matrix}A∩ΛÑ ≠∅ \\ diam(A)≤2dM \end{matrix}} Φ_{A}∥_{∞} = ∥ ∑_{\begin{matrix}A∩ΛÑ ≠∅ \\ diam(A)>2dM \end{matrix}} Φ_{A}∥_{∞} ≤ \frac{η̃}{8}.  (3.9)

Set M̃ = Ñ + 2dM. Note that by the triangular inequality

diam(A) ≤ 2dM and A ∩ ΛÑ ≠ ∅ ⟹ A ⊂ ΛM̃.

We now choose an open neighborhood V_{ℓ} of ω in the following way. The map

ω′ ∈ Ω ↦ P (ω′) := ω′_{ΛM̃ \ΛÑ} ω̃_{ℤ^{d}\(ΛM̃ \ΛÑ)} ∈ Ω

is continuous. Then, there is an open set V_{ℓ} containing ω such that, for all ω′ ∈ V_{ℓ},

| ∑_{\begin{matrix}A∩ΛÑ ≠∅ \\ diam(A)≤2dM \end{matrix}} Φ_{A} ∘ P (ω′) − ∑_{\begin{matrix}A∩ΛÑ ≠∅ \\ diam(A)≤2dM \end{matrix}} Φ_{A} ∘ P (ω)| ≤ \frac{η̃}{8},  (3.10)

|H_{ΛÑ} (ω′) − H_{ΛÑ} (ω)| ≤ \frac{η̃}{8}.  (3.11)

We choose once for all such a neighborhood V_{ℓ}.

Notice that V_{ℓ} ∩ supp(μ) ≠ ∅ and in particular μ(V_{ℓ}) > 0. Since ω is generic in the sense of the two equalities in (3.7), we can choose n large enough so that

\frac{S_{Λ_{n}}χ_{ℓ}(ω)}{#Λ_{n+Ñ}} ≥ \frac{μ(V_{ℓ})}{2} and \frac{H_{Λ_{n+Ñ}} (ω)}{#Λ_{n+Ñ}} < H̄ + \frac{η̃}{8} \frac{μ(V_{ℓ})}{#Λ_{2M̃}} .  (3.12)

Denote A_{n} := {j ∈ Λ_{n} : θ^{j}(ω) ∈ V_{ℓ}}. Let then B_{n} ⊂ A_{n} be a maximal subcollection of indices such that (j + ΛM̃) ∩ (k + ΛM̃) = ∅ whenever j, k ∈ B_{n} are distinct. Since for all j ∈ A_{n} there must exist k ∈ B_{n} such that (j + ΛM̃) ∩ (k + ΛM̃) ≠ ∅, we have

#A_{n} ≤ #{(j, k) ∈ A_{n} × B_{n} : (j + ΛM̃) ∩ (k + ΛM̃) ≠ ∅}

≤ ∑_{k∈B_{n}} #(A_{n} ∩ (k + ΛM̃ − ΛM̃)) ≤ #B_{n} #Λ_{2M̃},

which together with (3.12) yields

\frac{#B_{n}}{#Λ_{n+Ñ}} > \frac{μ(V_{ℓ})}{2#Λ_{2M̃}} .  (3.13)

Let ω^{n} ∈ Ω be the configuration which coincides with θ^{−j}(ω̃) on j + ΛÑ for every j ∈ B_{n} and with ω on the complement ℤ^{d} \ ⊔_{j∈B_{n}}(j + ΛÑ):

∀ j ∈ B_{n}, ω_{j+ΛÑ}^{n} = (θ^{−j}(ω̃))_{j+ΛÑ} and ω_{ℤ^{d}\⊔_{j∈Bn}(ΛÑ +j)}^{n} = ω_{ℤ^{d}\⊔_{j∈Bn}(ΛÑ +j)}.

Notice that θ^{j}(ω^{n}) coincides with ω̃ on ΛÑ and with θ^{j}(ω) on ΛM̃ \ ΛÑ. From the definition of the map P, we obtain that

(θ^{j}(ω^{n}))_{ΛM̃} = (P (θ^{j}(ω^{n})))_{ΛM̃} and P (θ^{j}(ω^{n})) = P (θ^{j}(ω)), ∀ j ∈ B_{n}.

Hence, since θ^{j}(ω) ∈ V_{ℓ} for any j ∈ B_{n}, thanks to (3.10) we get

∑_{\begin{matrix}A∩ΛÑ ≠∅ \\ diam(A)≤2dM \end{matrix}} Φ_{A}(θ^{j}(ω^{n})) ≤ \frac{η̃}{8} + ∑_{\begin{matrix}A∩ΛÑ ≠∅ \\ diam(A)≤2dM \end{matrix}} Φ_{A}(ω̃).  (3.14)

We decompose H_{Λ_{n+Ñ}} (ω^{n}) = H^{I}(ω^{n})+H^{II}(ω^{n})+H^{III}(ω^{n}) in three terms, where for a configuration ω̄ of Ω

H^{I}(ω̄) := ∑_{\begin{matrix}A∩(⨆_{j∈Bn}(j+ΛÑ))≠∅ \\ diam(A)≤2dM \end{matrix}} Φ_{A}(ω̄),

H^{II}(ω̄) := ∑_{\begin{matrix}A∩Λ_{n+Ñ} ≠∅ \\ A∩(⨆_{j∈Bn}(j+ΛÑ))=∅ \end{matrix}} Φ_{A}(ω̄),

H^{III}(ω̄) := ∑_{\begin{matrix}A∩(⨆_{j∈Bn}(j+ΛÑ))≠∅ \\ diam(A)>2dM \end{matrix}} Φ_{A}(ω̄).

For the first term, we obtain

H^{I}(ω^{n}) = ∑_{j∈B_{n}} ∑_{\begin{matrix}A∩ΛÑ ≠∅ \\ diam(A)≤2dM \end{matrix}} Φ_{A}(θ^{j}(ω^{n}))  (\{j + ΛM̃\}_{j∈B_{n}} are disjoint)

≤ #B_{n}( ∑_{\begin{matrix}A∩ΛÑ ≠∅ \\ diam(A)≤2dM \end{matrix}} Φ_{A}(ω̃) + \frac{η̃}{8})  (inequality (3.14))

≤ #B_{n}(H_{ΛÑ} (ω̃) + \frac{η̃}{4})  (inequality (3.9))
≤ #B_{n}(H_{ΛÑ} (ω) − \frac{3η̃}{4})  (inequality (3.8))
≤ ∑_{j∈B_{n}} (H_{ΛÑ} (θ^{j}(ω)) − \frac{5η̃}{8})  (inequality (3.11))

≤ ∑_{j∈B_{n}} ∑_{\begin{matrix}A∩ΛÑ ≠∅ \\ diam(A)≤2dM \end{matrix}} (Φ_{A}(θ^{j}(ω)) − \frac{η̃}{2})  (inequality (3.9))

= ∑_{\begin{matrix}A∩(⨆_{j∈Bn}(j+ΛÑ))≠∅ \\ diam(A)≤2dM \end{matrix}} Φ_{A}(ω) − \frac{η̃}{2}#B_{n}  (\{j + ΛM̃\}_{j∈B_{n}} are disjoint)

= H^{I}(ω) − \frac{η̃}{2}#B_{n}

For the second term, we use the fact that ω^{n} and ω coincide on ℤ^{d} \ ⊔_{j∈B_{n}}(j + ΛÑ) so that

H^{II}(ω^{n}) = ∑_{\begin{matrix}A∩Λ_{n+Ñ} ≠∅ \\ A∩(⨆_{j∈Bn}(j+ΛÑ))=∅ \end{matrix}} Φ_{A}(ω) = H^{II}(ω).

For the third term, we use again twice inequality (3.9). Here A may intersect several j + ΛÑ, j ∈ B_{n}, but

H^{III}(ω^{n}) − H^{III}(ω) = ∑_{\begin{matrix}A∩(⨆_{j∈Bn}(j+ΛÑ))≠∅ \\ diam(A)>2dM \end{matrix}} Φ_{A}(ω^{n}) − Φ_{A}(ω)

≤ ∑_{j∈B_{n}} ∑_{\begin{matrix}A∩ΛÑ ≠∅ \\ diam(A)>2dM \end{matrix}} |Φ_{A}(θ^{j}(ω^{n})) − Φ_{A}(θ^{j}(ω))| ≤ \frac{η̃}{4}#B_{n}.

We finally collect the three terms and obtain for n sufficiently large

inf_{ω̄∈Ω} \frac{H_{Λ_{n+Ñ}} (ω̄)}{#Λ_{n+Ñ}} ≤ \frac{H_{Λ_{n+Ñ}} (ω^{n})}{#Λ_{n+Ñ}} ≤ \frac{H_{Λ_{n+Ñ}} (ω)}{#Λ_{n+Ñ}} − \frac{η̃}{4} \frac{#B_{n}}{#Λ_{n+Ñ}} ≤ H̄ − \frac{η̃}{8} \frac{μ(V_{ℓ})}{#Λ_{2M̃}} ,

which contradicts the characterization of the constant H̄ in proposition 3.7.

Since any periodic configuration gives rise to a translation-invariant probability measure supported on its orbit, we point out an immediate corollary.

Corollary 3.11. A periodic configuration ω ∈ Ω that verifies

lim_{n→∞} \frac{1}{#Λ_{n}} H_{Λ_{n}}(ω) = H̄

is a ground-state configuration with respect to H.

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