A Quantum Mermin-Wagner Theorem for a Generalized Hubbard Model

Abstract

This paper is the second in a series of papers considering symmetry properties of bosonic quantum systems over 2D graphs, with continuous spins, in the spirit of the Mermin-Wagner theorem. In the model considered here the phase space of a single spin is ℋ₁=L₂(𝑀), where 𝑀 is a 𝑑-dimensional unit torus 𝑀=ℝ^𝑑/ℤ^𝑑 with a flat metric. The phase space of 𝑘 spins is ℋ_𝑘=L^sym₂(𝑀^𝑘), the subspace of L₂(𝑀^𝑘) formed by functions symmetric under the permutations of the arguments. The Fock space 𝓗=⊕_𝑘=0,1,…ℋ_𝑘 yields the phase space of a system of a varying (but finite) number of particles. We associate a space 𝓗≃𝓗(𝑖) with each vertex 𝑖∈Γ of a graph (Γ,ℰ) satisfying a special bidimensionality property. (Physically, vertex 𝑖 represents a heavy “atom” or “ion” that does not move but attracts a number of “light” particles.) The kinetic energy part of the Hamiltonian includes (i) −Δ/2, the minus a half of the Laplace operator on 𝑀, responsible for the motion of a particle while “trapped” by a given atom, and (ii) an integral term describing possible “jumps” where a particle may join another atom. The potential part is an operator of multiplication by a function (the potential energy of a classical configuration) which is a sum of (a) one-body potentials 𝑈⁽¹⁾(𝑥), 𝑥∈𝑀, describing a field generated by a heavy atom, (b) two-body potentials 𝑈⁽²⁾(𝑥,𝑦), 𝑥,𝑦∈𝑀, showing the interaction between pairs of particles belonging to the same atom, and (c) two-body potentials 𝑉(𝑥,𝑦), 𝑥,𝑦∈𝑀, scaled along the graph distance 𝚍(𝑖,𝑗) between vertices 𝑖,𝑗∈Γ, which gives the interaction between particles belonging to different atoms. The system under consideration can be considered as a generalized (bosonic) Hubbard model. We assume that a connected Lie group G acts on 𝑀, represented by a Euclidean space or torus of dimension 𝑑^′≤𝑑, preserving the metric and the volume in 𝑀. Furthermore, we suppose that the potentials 𝑈⁽¹⁾, 𝑈⁽²⁾, and 𝑉 are G-invariant. The result of the paper is that any (appropriately defined) Gibbs states generated by the above Hamiltonian is G-invariant, provided that the thermodynamic variables (the fugacity 𝑧 and the inverse temperature 𝛽) satisfy a certain restriction. The definition of a Gibbs state (and its analysis) is based on the Feynman-Kac representation for the density matrices.

1. Introduction

1.1. Basic Facts on Bi-Dimensional Graphs

As in [1], we suppose that the graph (Γ,ℰ) has been given, with the set of vertices Γ and the set of edges ℰ. The graph has the property that whenever edge (𝑗^′,𝑗^′′)∈ℰ, the reversed edge (𝑗^′′,𝑗^′) belongs to ℰ as well. Furthermore, graph (Γ,ℰ) is without multiple edges and has a bounded degree; that is, the number of edges (𝑗,𝑗^′) with a fixed initial or terminal vertex is uniformly bounded: supmax♯𝑗^′∈Γ∶𝑗,𝑗^′,♯𝑗∈ℰ_′∈Γ∶𝑗^′,𝑗∈ℰ∶𝑗∈Γ<∞. (1) The bi-dimensionality property is expressed in the bound 10<sup𝑛♯Σ𝑗,𝑛∶𝑗∈Γ,𝑛=1,2,…<∞, (2) where Σ(𝑗,𝑛) stands for the set of vertices in Γ at the graph distance 𝑛 from 𝑗∈Γ (a sphere of radius 𝑛 around 𝑗): Σ𝑗,𝑛=𝑗^′∈Γ∶𝚍𝑗,𝑗^′=𝑛. (3) (The graph distance 𝚍(𝑗,𝑗^′)=𝚍_Γ,ℰ(𝑗,𝑗^′) between 𝑗,𝑗^′∈Γ is determined as the minimal length of a path on (Γ,ℰ) joining 𝑗 and 𝑗^′.) This implies that for any 𝑜∈Γ the cardinality ♯Λ(𝑜,𝑛) of the ball Λ(𝑜,𝑛)=𝑗^′∈Γ∶𝚍𝑜,𝑗^′≤𝑛 (4) grows at most quadratically with 𝑛.

A justification for putting a quantum system on a graph can be that graph-like structures become increasingly popular in rigorous Statistical Mechanics, for example, in quantum gravity. Namely, see [2–4]. On the other hand, a number of properties of Gibbs ensembles do not depend upon “regularity” of an underlying spatial geometry.

1.2. A Bosonic Model in the Fock Space

With each vertex 𝑖∈Γ we associate a copy of a compact manifold 𝑀 which we take in this paper to be a unit 𝑑-dimensional torus ℝ^𝑑/ℤ^𝑑 with a flat metric 𝜌 and the volume 𝑣. We also associate with 𝑖∈Γ a bosonic Fock-Hilbert space 𝓗(𝑖)≃𝓗. Here 𝓗=⊕_𝑘=0,1,…ℋ_𝑘 where ℋ_𝑘=L^sym₂(𝑀^𝑘) is the subspace in L₂(𝑀^𝑘) formed by functions symmetric under a permutation of the variables. Given a finite set Λ⊂Γ, we set 𝓗(Λ)=⊗_𝑖∈Λ𝓗(𝑖). An element 𝝓∈𝓗(Λ) is a complex function: 𝐱^∗_Λ∈𝑀^∗Λ⟼𝝓𝐱^∗_Λ. (5) Here 𝐱^∗_Λ is a collection {𝐱^∗(𝑗),𝑗∈Λ} of finite point sets 𝐱^∗(𝑗)⊂𝑀 associated with sites 𝑗∈Λ. Following [1], we call 𝐱^∗(𝑗) a particle configuration at site 𝑗 (which can be empty) and 𝐱^∗_Λ a particle configuration in, or over, Λ. The space 𝑀^∗Λ of particle configurations in Λ can be represented as the Cartesian product (𝑀^∗)^×Λ where 𝑀^∗ is the disjoint union ⋃_𝑘=0,1,…𝑀^(𝑘) and 𝑀^(𝑘) is the collection of (unordered) 𝑘-point subsets of 𝑀. (One can consider 𝑀^(𝑘) as the factor of the “off-diagonal” set 𝑀^𝑘_≠ in the Cartesian power 𝑀^𝑘 under the equivalence relation induced by the permutation group of order 𝑘.) The norm and the scalar product in 𝓗_Λ are given by ‖𝝓‖=_𝑀∗Λ𝝓(𝐱^∗_Λ)²d𝐱^∗_Λ^1/2,𝝓₁,𝝓₂=_𝑀∗Λ𝝓₁𝐱^∗_Λ𝝓₂𝐱^∗_Λd𝐱^∗_Λ, (6) where measure d𝐱^∗_Λ is the product ×_𝑗∈Λd𝐱^∗(𝑗) and d𝐱^∗(𝑗) is the Poissonian sum measure on 𝑀^∗: d𝐱^∗𝑗=_𝑘=0,1,…𝟏♯𝐱^∗𝑗1=𝑘𝑘!_{𝑥∈𝐱∗(𝑗)}d𝑣(𝑥)𝑒^{−𝑣(𝑀)}. (7) Here 𝑣(𝑀) is the volume of torus 𝑀.

As in [1], we assume that an action (𝚐,𝑥)∈𝙶×𝑀⟼𝚐𝑥∈𝑀 (8) is given, of a group 𝙶 that is a Euclidean space or a torus of dimension 𝑑^′≤𝑑. The action is written as 𝚐𝑥=𝑥+𝜃𝐴mod1. (9) Here vector 𝜃=(𝜃₁,…,𝜃_𝑑′) with components 𝜃_𝑙∈[0,1) and 𝜃𝐴 is the 𝑑-dimensional vector 𝜃=((𝜃𝐴)₁,…,(𝜃𝐴)_𝑑) representing the element 𝚐, where 𝐴 is a (𝑑^′×𝑑) matrix of column rank 𝑑^′ with rational entries. The action of 𝙶 is lifted to unitary operators 𝐔_Λ(𝚐) in 𝓗_Λ: 𝐔_Λ(𝚐)𝝓𝐱^∗_Λ=𝝓𝚐⁻¹𝐱^∗_Λ, (10) where 𝚐⁻¹𝐱^∗_Λ={𝚐⁻¹𝐱^∗(𝑗),𝑗∈Λ} and 𝚐⁻¹𝐱^∗(𝑗)={𝚐⁻¹𝑥,𝑥∈𝐱^∗(𝑗)}.

The generally accepted view is that the Hubbard model is a highly oversimplified model for strongly interacting electrons in a solid. The Hubbard model is a kind of minimum model which takes into account quantum mechanical motion of electrons in a solid, and nonlinear repulsive interaction between electrons. There is little doubt that the model is too simple to describe actual solids faithfully [5]. In our context the Hubbard Hamiltonian 𝐇_Λ of the system in Λ acts as follows: 𝐇_Λ𝝓𝐱^∗_Λ=−12_{𝑗∈Λ𝑥∈𝐱∗(𝑗)}Δ_𝑗^(𝑥)+_{𝑗∈Λ𝑥∈𝐱∗(𝑗)}𝑈⁽¹⁾(𝑥)+12_{𝑗∈Λ𝑥,𝑥′∈𝐱∗(𝑗)}𝟏𝑥≠𝑥^′𝑈⁽²⁾𝑥,𝑥^′+12_{𝑗,𝑗′∈Λ}𝟏𝑗≠𝑗^′𝐽𝚍𝑗,𝑗^′×_{𝑥∈𝐱∗(𝑗),𝑥′∈𝐱∗𝑗′}𝑉𝑥,𝑥^′𝝓𝐱^∗_Λ+_{𝑗,𝑗′∈Λ}𝜆_𝑗,𝑗′𝟏♯𝐱^∗𝑗≥1,♯𝐱^∗𝑗^′<𝜅×_{𝑥∈𝐱∗(𝑗)𝑀}𝑣×𝝓𝐱d𝑦_{∗(𝑗,𝑥)→(𝑗′Λ,𝑦)}−𝝓𝐱^∗_Λ. (11) Here Δ_𝑗^(𝑥) means the Laplacian in variable 𝑥∈𝐱^∗(𝑗). Next, ♯𝐱^∗ stands for the cardinality of the particle configuration 𝐱^∗ (i.e., ♯𝐱^∗=𝑘 when 𝐱^∗∈𝑀^(𝑘)), and the parameter 𝜅 is introduced in (17). (Symbol ♯ will be used for denoting the cardinality of a general (finite) set; for example, ♯Λ means the number of vertices in Λ.) Further, 𝐱^{∗(𝑗,𝑥)→(𝑗′}_Λ^,𝑦) denotes the particle configuration with the point 𝑥∈𝐱^∗(𝑗) removed and point 𝑦 added to 𝐱^∗(𝑗^′).

As in [1], we also consider a Hamiltonian 𝐇_Λ∣𝐱^∗_Γ⧵Λ in an external field generated by a configuration 𝐱^∗_Γ⧵Λ={𝐱^∗(𝑗^′),𝑗^′∈Γ⧵Λ}∈𝑀^∗Γ⧵Λ where Γ⊆Γ is a (finite or infinite) collection of vertices. More precisely, we only consider 𝐱^∗_Γ⧵Λ with ♯𝐱^∗(𝑗^′)≤𝜅 (see (17) below) and set 𝐇_Λ∣𝐱^∗_Γ⧵Λ𝝓𝐱^∗_Λ=−12_{𝑗∈Λ𝑥∈𝐱∗(𝑗)}Δ_𝑗^(𝑥)+_{𝑗∈Λ𝑥∈𝐱∗(𝑗)}𝑈⁽¹⁾(𝑥)+12_{𝑗∈Λ𝑥,𝑥′∈𝐱∗(𝑗)}𝟏𝑥≠𝑥^′𝑈⁽²⁾𝑥,𝑥^′+12_{𝑗,𝑗′∈Λ}𝟏𝑗≠𝑗^′𝐽𝚍𝑗,𝑗^′×_{𝑥∈𝐱∗(𝑗),𝑥′∈𝐱∗𝑗′}𝑉𝑥,𝑥^′𝝓𝐱^∗_Λ+_{𝑗∈Λ,𝑗′∈Γ⧵Λ}𝐽𝚍𝑗,𝑗^′×_{𝑥∈𝐱∗(𝑗),𝑥′∈𝐱∗𝑗′}𝑉𝑥,𝑥^′𝝓𝐱^∗_Λ+_{𝑗,𝑗′∈Λ}𝜆_𝑗,𝑗′𝟏♯𝐱^∗𝑗≥1,♯𝐱^∗𝑗^′<𝜅×_{𝑥∈𝐱∗𝑗𝑀}𝑣d𝑦𝝓𝐱^{∗(𝑗,𝑥)→(𝑗′}_Λ^,𝑦)−𝝓𝐱^∗_Λ. (12)

The novel elements in (11) and (12) compared with [1] are the presence of on-site potentials 𝑈⁽¹⁾ and 𝑈⁽²⁾ and the summand involving transition rates 𝜆_𝑗,𝑗′≥0 for jumps of a particle from site 𝑗 to 𝑗^′.

We will suppose that 𝜆_𝑗,𝑗′ vanishes if the graph distance 𝚍(𝑗,𝑗^′)>1. We will also assume uniform boundedness: sup𝜆_𝑗,𝑗′(𝑥,𝑀),𝑗,𝑗^′∈Γ,𝑥∈𝑀<∞; (13) in view of (1) it implies that the total exit rate ∑_{𝑗′∶𝚍(𝑗,𝑗′)=1}𝜆_𝑗,𝑗′(𝑥,𝑀) from site 𝑗 is uniformly bounded. These conditions are not sharp and can be liberalized.

The model under consideration can be considered as a generalization of the Hubbard model [6] (in its bosonic version). Its mathematical justification includes the following. (a) An opportunity to introduce a Fock space formalism incorporates a number of new features. For instance, a fermonic version of the model (not considered here) emerges naturally when the bosonic Fock space 𝓗(𝑖) is replaced by a fermonic one. Another opening provided by this model is a possibility to consider random potentials 𝑈⁽¹⁾, 𝑈⁽²⁾ and 𝑉 which would yield a sound generalization of the Mott-Anderson model. (b) Introducing jumps makes a step towards a treatment of a model of a quantum (Bose-) gas where particles “live” in a single Fock space. For example, a system of interacting quantum particles is originally confined to a “box” in a Euclidean space, with or without “internal” degrees of freedom. In the thermodynamical limit the box expands to the whole Euclidean space. In a two-dimensional model of a quantum gas one expects a phenomenon of invariance under space-translations; one hopes to be able to address this issue in future publications. (c) A model with jumps can be analysed by means of the theory of Markov processes which provides a developed methodology.

Physically speaking, the model with jumps covers a situation where “light” quantum particles are subject to a “random” force and change their “location.” This class of models is interesting from the point of view of transport phenomena that they may display. (An analogy with the famous Anderson model, in its multiparticle version, inevitably comes to mind; cf., e.g., [7].) Methodologically, such systems occupy an “intermediate” place between models where quantum particles are “fixed” forever to their designated locations (as in [1]) and models where quantum particles move in the same space (a Bose-gas, considered in [8, 9]). In particular, this work provides a bridge between [1, 8, 9]; reading this paper ahead of [8, 9] might help an interested reader to get through [8, 9] at a much quicker pace.

We would like to note an interesting problem of analysis of the small-mass limit (cf. [10]) from the point of Mermin-Wagner phenomena.

1.3. Assumptions on the Potentials

The between-sites potential 𝑉 is assumed to be of class 𝐶². Consequently, 𝑉 and its first and second derivatives satisfy uniform bounds. Namely, ∀𝑥^′,𝑥^′′∈𝑀−𝑉𝑥^′,𝑥^′′,∇_x𝑉𝑥^′,𝑥^′′,∇_x,x′𝑉𝑥^′,𝑥^′′≤𝑉. (14) Here 𝑥 and 𝑥^′ run through the pairs of variables 𝑥, 𝑥^′. A similar property is assumed for the on-site potential 𝑈⁽¹⁾ (here we need only a 𝐶¹ smoothness): −𝑈⁽¹⁾(𝑥),∇_x𝑈⁽¹⁾(𝑥)≤𝑈⁽¹⁾,𝑥∈𝑀. (15) Note that for 𝑉 and 𝑈⁽¹⁾ the bounds are imposed on their negative parts only.

As to 𝑈⁽²⁾, we suppose that (a) 𝑈⁽²⁾𝑥,𝑥^′=+∞when𝑥−𝑥^′≤𝜌, (16) and (b) ∃ a 𝐶¹-function (𝑥,𝑥^′)↦𝑈⁽²⁾(𝑥,𝑥^′)∈ℝ such that 𝑈⁽²⁾(𝑥,𝑥^′)=𝑈⁽²⁾(𝑥,𝑥^′) whenever 𝜌(𝑥,𝑥^′)>𝜌. Here 𝜌(𝑥,𝑥^′) stands for the (flat) Riemannian distance between points 𝑥,𝑥^′∈𝑀. As a result of (16), there exists a “hard core” of diameter 𝜌, and a given atom cannot “hold” more than 𝑣(𝑀)𝜅=𝑣𝐵𝜌 (17) particles where 𝑣(𝐵(𝜌)) is the volume of a 𝑑-dimensional ball of diameter 𝜌. We will also use the bound −𝑈⁽²⁾𝑥,𝑥^′,∇_x𝑈⁽²⁾𝑥,𝑥^′≤𝑈⁽²⁾,𝑥,𝑥^′∈𝑀. (18) Formally, (16) means that the operators in (11) and (12) are considered for functions 𝝓(𝐱^∗_Λ) vanishing when in the particle configuration 𝐱^∗_Λ={𝐱^∗(𝑗),𝑗∈Λ}, the cardinality ♯𝐱^∗(𝑗)>𝜅 for some 𝑗∈Λ.

The function 𝐽∶𝑟∈(0,∞)↦𝐽(𝑟)≥0 is assumed monotonically nonincreasing with 𝑟 and obeying the relation 𝐽(𝑙)→0 as 𝑙→∞, where 𝐽(𝑙)=sup_{𝑗′′∈Γ}𝐽𝚍𝑗^′,𝑗^′′𝟏𝚍𝑗^′,𝑗^′′≥𝑙∶𝑗^′∈Γ<∞. (19) Additionally, let 𝐽(𝑟) be such that 𝐽^∗=sup_𝑗′∈Γ𝐽𝚍𝑗,𝑗^′𝚍𝑗,𝑗^′2∶𝑗∈Γ<∞. (20) Next, we assume that the functions 𝑈⁽¹⁾, 𝑈⁽²⁾, and 𝑉 are 𝚐-invariant: ∀𝑥,𝑥^′∈𝑀 and 𝚐∈𝙶, 𝑈⁽¹⁾(𝑥)=𝑈⁽¹⁾(),𝑈𝚐𝑥₍₂₎𝑥,𝑥^′=𝑈⁽²⁾𝚐𝑥,𝚐𝑥^′,𝑉𝑥,𝑥^′=𝑉𝚐𝑥,𝚐𝑥^′. (21) In the following we will need to bound the fugacity (or activity, cf. (25)) 𝑧 in terms of the other parameters of the model 𝑧𝑒^Θ<1,whereΘ=𝜅𝛽𝑈⁽¹⁾+𝜅𝑈⁽²⁾+𝜅𝐽(1)𝑉. (22)

1.4. The Gibbs State in a Finite Volume

Define the particle number operator 𝐍_Λ, with the action 𝐍_Λ𝝓𝐱^∗_Λ=♯𝐱^∗_Λ𝝓𝐱^∗_Λ,𝐱^∗_Λ∈𝑀^∗Λ. (23) Here, for a given 𝐱^∗_Λ={𝐱^∗(𝑗),𝑗∈Λ}, ♯𝐱^∗_Λ stands for the total number of particles in configuration 𝐱^∗_Λ: ♯𝐱^∗_Λ=_𝑗∈Λ♯𝐱^∗𝑗. (24) The standard canonical variable associated with 𝐍_Λ is activity 𝑧∈(0,∞).

The Hamiltonians (11) and (12) are self-adjoint (on the natural domains) in 𝓗(Λ). Moreover, they are positive definite and have a discrete spectrum, cf. [14]. Furthermore, ∀𝑧,𝛽>0, 𝐇_Λ and 𝐇_Λ∣𝐱^∗_Γ⧵Λ give rise to positive-definite trace-class operators 𝐆_Λ=𝐆_𝑧,𝛽,Λ and 𝐆_Λ∣𝐱^∗_Γ⧵Λ=𝐆_{𝑧,𝛽,Λ∣𝐱}^∗_Γ⧵Λ: 𝐆_Λ=𝑧^𝐍Λexp−𝛽𝐇_Λ,𝐆_Λ∣𝐱^∗_Γ⧵Λ=𝑧^𝐍Λexp−𝛽𝐇_Λ∣𝐱^∗_Γ⧵Λ. (25) We would like to stress that the full range of variables 𝑧,𝛽>0 is allowed here because of the hard-core condition (16): it does not allow more than 𝜅♯Λ particles in Λ where ♯Λ stands for the number of vertices in Λ. However, while passing to the thermodynamic limit, we will need to control 𝑧 and 𝛽.

Definition 1. We will call 𝐆_Λ and 𝐆_Λ∣𝐱^∗_Γ⧵Λ the Gibbs operators in volume Λ, for given values of 𝑧 and 𝛽 (and—in the case of 𝐆_Λ∣𝐱^∗_Γ⧵Λ—with the boundary condition 𝐱^∗_Γ⧵Λ).
The Gibbs operators in turn give rise to the Gibbs states 𝝋_Λ=𝝋_𝛽,𝑧,Λ and 𝝋_{Λ∣𝐱Γ⧵Λ}=𝝋_{𝛽,𝑧,Λ∣𝐱Γ⧵Λ}, at temperature 𝛽⁻¹ and activity 𝑧 in volume Λ. These are linear positive normalized functionals on the 𝐶^∗-algebra 𝕭_Λ of bounded operators in space 𝓗_Λ: 𝝋_Λ(𝐀)=tr_𝓗Λ𝐑_Λ𝐀,𝝋_{Λ∣𝐱Γ⧵Λ}(𝐀)=tr_𝓗Λ𝐑_{Λ∣𝐱Γ⧵Λ}𝐀,𝐀∈𝕭_Λ, (26) where 𝐑_Λ=𝐆_ΛΞ(Λ),withΞ(Λ)=Ξ_𝑧,𝛽(Λ)=tr_𝓗Λ𝐆_Λ,𝐑_Λ∣𝐱^∗_Γ⧵Λ=𝐆_{Λ∣𝐱Γ⧵Λ}ΞΛ∣𝐱^∗_Γ⧵Λ,withΞΛ∣𝐱^∗_Γ⧵Λ=Ξ_𝑧,𝛽Λ∣𝐱^∗_Γ⧵Λ=tr_𝓗Λ𝑧^𝐍Λexp−𝛽𝐇_Λ∣𝐱^∗_Γ⧵Λ. (27)(28)

The hard-core assumption (16) yields that the quantities Ξ(Λ) and Ξ_𝑧,𝛽(Λ∣𝐱^∗_Γ⧵Λ) are finite; formally, these facts will be verified by virtue of the Feynman-Kac representation.

Definition 2. Whenever Λ⁰⊂Λ, the 𝐶^∗-algebra 𝕭_Λ⁰ is identified with the 𝐶^∗ subalgebra in 𝕭_Λ formed by the operators of the form 𝐀₀⊗𝐈_Λ⧵Λ⁰. Consequently, the restriction 𝝋^Λ0_Λ of state 𝝋_Λ to 𝐶^∗-algebra 𝕭_Λ⁰ is given by 𝝋^Λ0_Λ𝐀₀=tr_𝓗Λ0𝐑^Λ0_Λ𝐀₀,𝐀₀∈𝕭_Λ⁰, (29) where 𝐑^Λ0_Λ=tr_𝓗0Λ⧵Λ𝐑_Λ. (30) Operators 𝐑^Λ0_Λ (we again call them RDMs) are positive definite and have tr_𝓗Λ0𝐑^Λ0_Λ=1. They also satisfy the compatibility property: ∀Λ⁰⊂Λ¹⊂Λ, 𝐑^Λ0_Λ=tr_{𝓗Λ10⧵Λ}𝐑^Λ1_Λ. (31) In a similar fashion one defines functionals 𝝋^Λ0_Λ∣𝐱^∗_Γ⧵Λ and operators 𝐑^Λ0_Λ∣𝐱^∗_Γ⧵Λ, with the same properties.

1.5. Limiting Gibbs States

The concept of a limiting Gibbs state is related to notion of a quasilocal 𝐶^∗-algebra; see [14]. For the class of systems under consideration, the construction of the quasilocal 𝐶^∗-algebra 𝕭_Γ is done along the same lines as in [1]: 𝕭_Γ is the norm completion of the 𝐶^∗ algebra (𝕭⁰_Γ)=limind_𝑛→∞𝕭_Λ𝑛. Any family of positive-definite operators 𝐑^Λ0 in spaces 𝓗_Λ⁰ of trace one, where Λ⁰ runs through finite subsets of Γ, with the compatibility property 𝐑^Λ1=tr_{𝓗Λ01⧵Λ}𝐑^Λ0,Λ¹⊂Λ⁰, (32) determines a state of 𝕭_Γ, see [12, 13].

Finally, we introduce unitary operators 𝐔_Λ⁰(𝚐), 𝚐∈𝙶, in 𝓗_Λ⁰: 𝐔_Λ𝝓𝐱^∗_Λ0=𝝓𝚐⁻¹𝐱^∗_Λ0, (33) where 𝚐⁻¹𝐱^∗_Λ0=𝚐⁻¹𝐱^∗𝑗,𝑗∈Λ⁰,𝚐⁻¹𝐱^∗𝑗=𝚐⁻¹𝑥∶𝑥∈𝐱^∗𝑗. (34)

Theorem 3. Assuming the conditions listed above, for all 𝑧,𝛽∈(0,+∞) satisfying (22) and a finite Λ⁰⊂Γ, operators 𝐑^Λ0_Λ form a compact sequence in the trace-norm topology in 𝓗_Λ⁰ as Λ↗Γ. Furthermore, given any family of (finite or infinite) sets Γ=Γ(Λ)⊆Γ and configurations 𝐱^∗_Γ⧵Λ={𝐱^∗(𝑖),𝑖∈Γ⧵Λ} with ♯𝐱^∗(𝑖)<𝜅, operators 𝐑^Λ0_{Λ∣𝐱Γ⧵Λ} also form a compact sequence in the trace-norm topology. Any limit point, 𝐑^Λ0, for {𝐑^Λ0_Λ} or {𝐑^Λ0_Λ∣𝐱^∗_Γ⧵Λ} as Λ↗Γ, is a positive-definite operator in ℋ(Λ⁰) of trace one. Moreover, if Λ¹⊂Λ⁰ and 𝐑^Λ0 and 𝐑^Λ1 are the limits for 𝐑^Λ0_Λ and 𝐑^Λ1_Λ or for 𝐑^Λ0_Λ∣𝐱^∗_Γ⧵Λ and 𝐑^Λ1_Λ∣𝐱^∗_Γ⧵Λ along the same subsequence Λ_𝑠↗Γ, then the property (32) holds true.
Consequently, the Gibbs states 𝝋_Λ and 𝝋_Λ∣𝐱^∗_Γ⧵Λ form compact sequences as Λ↗Γ.

Remark 4. In fact, the assertion of Theorem 3 holds without assuming the bidimensionality condition on graph (Γ,ℰ), only under an assumption that the degree of the vertices in Γ is uniformy bounded.

Definition 5. Any limit point 𝝋 for states 𝝋_Λ and 𝝋_Λ∣𝐱^∗_Γ⧵Λ is called a limiting Gibbs state (for given 𝑧,𝛽∈(0,+∞)).

Theorem 6. Under the condition (22), any limiting point, 𝐑^Λ0, for {𝐑^Λ0_Λ} or {𝐑^Λ0_Λ∣𝐱^∗_Γ⧵Λ}, as Λ↗Γ, is a positive-definite operator of trace one commuting with operators 𝐔_Λ⁰(𝚐): ∀𝚐∈𝙶, 𝐔_Λ⁰(𝚐)⁻¹𝐑^Λ0𝐔_Λ⁰(𝚐)=𝐑^Λ0. (35) Accordingly, any limiting Gibbs state 𝝋 of 𝕭 determined by a family of limiting operators 𝐑^Λ0 obeying (35) satisfies the corresponding invariance property: ∀ finite Λ⁰⊂Γ, any 𝐀∈𝕭_Λ⁰, and 𝚐∈𝙶, 𝝋(𝐀)=𝝋𝐔_Λ⁰(𝚐)⁻¹𝐀𝐔_Λ⁰(𝚐). (36)

Remarks. (1) Condition (22) does not imply the uniqueness of an infinite-volume Gibbs state (i.e., absence of phase transitions).

(2) Properties (35) and (36) are trivially fulfilled for the limiting points 𝐑^Λ0 and 𝝋 of families {𝐑^Λ0_Λ} and {𝝋_Λ}. However, they require a proof for the limit points of the families {𝐑^Λ0_Λ∣𝐱^∗_Γ⧵Λ} and {𝝋_Λ∣𝐱^∗_Γ⧵Λ}.

The set of limiting Gibbs states (which is nonempty due to Theorem 3) is denoted by 𝕲⁰. In Section 3 we describe a class 𝕲⊃𝕲⁰ of states of 𝐶^∗-algebra 𝕭 satisfying the FK-DLR equation, similar to that in [1].

2. Feynman-Kac Representations for the RDM Kernels in a Finite Volume

2.1. The Representation for the Kernels of the Gibbs Operators

A starting point for the forthcoming analysis is the Feynman-Kac (FK) representation for the kernels 𝐊_Λ(𝐱^∗_Λ,𝐲^∗_Λ)=𝐊_𝛽,𝑧,Λ(𝐱^∗_Λ,𝐲^∗_Λ) and 𝐅_Λ(𝐱^∗_Λ,𝐲^∗_Λ)=𝐅_𝛽,𝑧,Λ(𝐱^∗_Λ,𝐲^∗_Λ) of operators 𝐆_Λ and 𝐑_Λ.

Definition 7. Given (𝑥,𝑖),(𝑦,𝑗)∈𝑀×Γ, 𝑊^𝛽_{(𝑥,𝑖),(𝑦,𝑗)} denotes the space of path, or trajectories, 𝜔=𝜔_{(𝑥,𝑖),(𝑦,𝑗)} in 𝑀×Γ, of time-length 𝛽, with the end-points (𝑥,𝑖) and (𝑦,𝑗). Formally, 𝜔∈𝑊^𝛽_{(𝑥,𝑖),(𝑦,𝑗)} is defined as follows: 𝜔∶𝜏∈0,𝛽⟼𝜔(𝜏)=(𝑢(𝜔,𝜏),𝑙(𝜔,𝜏))∈𝑀×Γ,𝜔iscad́laǵ;𝜔(0)=(𝑥,𝑖),𝜔𝛽−=𝑦,𝑗,;[𝑙(𝜔hasfinitelymanyjumpson0,𝛽ifajumpoccursattime𝜏,then𝚍𝜔,𝜏−),𝑙(𝜔,𝜏)]=1. (37) The notation 𝜔(𝜏) and its alternative, (𝑢(𝜔,𝜏),𝑙(𝜔,𝜏)), for the position and the index of trajectory 𝜔 at time 𝜏 will be employed as equal in rights. We use the term the temporal section (or simply the section) of path 𝜔 at time 𝜏.

Definition 8. Let 𝐱^∗_Λ={𝐱^∗(𝑖),𝑖∈Λ}∈𝑀^∗Λ, and 𝐲^∗_Λ={𝐲^∗(𝑗),𝑗∈Λ}∈𝑀^∗Λ be particle configurations over Λ, with ♯𝐱^∗_Λ=♯𝐲^∗_Λ. A matching (or pairing) 𝛾 between 𝐱^∗_Λ and 𝐲^∗_Λ is defined as a collection of pairs [(𝑥,𝑖),(𝑦,𝑗)]_𝛾, with 𝑖,𝑗∈Λ, 𝑥∈𝐱^∗(𝑖), and 𝑦∈𝐲^∗(𝑗), with the properties that (i) ∀𝑖∈Λ and 𝑥∈𝐱^∗(𝑖)∶ there exist unique 𝑗∈Λ and 𝑦∈𝐲^∗(𝑗) such that (𝑥,𝑖) and (𝑦,𝑗) form a pair, and (ii) ∀𝑗∈Λ and 𝑦∈𝐲^∗(𝑗)∶ there exist unique 𝑖∈Λ and 𝑥∈𝐱^∗(𝑖) such that (𝑥,𝑖) and (𝑦,𝑗) form a pair. (Owing to the condition ♯𝐱^∗_Λ=♯𝐲^∗_Λ, these properties are equivalent.) It is convenient to write [(𝑥,𝑖),(𝑦,𝑗)]_𝛾=[(𝑥,𝑖),𝛾(𝑥,𝑖)].
Next, consider the Cartesian product 𝑊^𝛽_𝐱^∗_Λ,𝐲^∗_Λ,𝛾=×_𝑖∈Λ×_{𝑥∈𝐱∗(𝑖)}𝑊^𝛽_{(𝑥,𝑖),𝛾(𝑥,𝑖)}, (38) and the disjoint union 𝑊^𝛽_𝐱^∗_Λ,𝐲^∗_Λ=_𝛾𝑊^𝛽_𝐱^∗_Λ,𝐲^∗_Λ,𝛾. (39) Accordingly, an element 𝝎_Λ∈𝑊^𝛽_𝐱^∗_Λ,𝐲^∗_Λ,𝛾 in (38) represents a collection of paths 𝜔_𝑥,𝑖, 𝑥∈𝐱^∗(𝑖), 𝑖∈Λ, of time-length 𝛽, starting at (𝑥,𝑖) and ending up at 𝛾(𝑥,𝑖). We say that 𝝎_Λ is a path configuration in (or over) Λ.

The presence of matchings in the above construction is a feature of the bosonic nature of the systems under consideration.

We will work with standard sigma algebras (generated by cylinder sets) in 𝑊^𝛽_{(𝑥,𝑖),(𝑦,𝑗)}, 𝑊^𝛽_𝐱^∗_Λ,𝐲^∗_Λ,𝛾, and 𝑊^𝛽_𝐱^∗_Λ,𝐲^∗_Λ.

Definition 9. In what follows, 𝜉(𝜏), 𝜏≥0, stands for the Markov process on 𝑀×Γ, with cádlág trajectories, determined by the generator ℒ acting on a function (𝑥,𝑖)∈𝑀×Λ↦𝜓(𝑥,𝑖) by ()1ℒ𝜓𝑥,𝑖=−2Δ𝜓(𝑥,𝑖)+_{𝑗∶𝚍(𝑖,𝑗)=1}𝜆_{𝑖,𝑗𝑀}𝑣d𝑦𝜓𝑦,𝑗−𝜓(𝑥,𝑖). (40) In the probabilistic literature, such processes are referred to as Lévy processes; see, for example, [14].

Pictorially, a trajectory of process 𝜉 moves along 𝑀 according to the Brownian motion with the generator −Δ/2 and changes the index 𝑖∈Γ from time to time in accordance with jumps occurring in a Poisson process of rate ∑_{𝑗∶𝚍(𝑖,𝑗)=1}𝜆_𝑖.𝑗. In other words, while following a Brownian motion rule on 𝑀, having index 𝑖∈Γ and being at point 𝑥∈𝑀, the moving particle experiences an urge to jump from 𝑖 to a neighboring vertex 𝑗 and to a point 𝑦 at rate 𝜆_𝑖,𝑗𝑣(d𝑦). After a jump, the particle continues the Brownian motion on 𝑀 from 𝑦 and keeps its new index 𝑗 until the next jump, and so on.

For a given pairs (𝑥,𝑖),(𝑦,𝑗)∈𝑀×Γ, we denote by ℙ^𝛽_{(𝑥,𝑖),(𝑦,𝑗)} the nonnormalised measure on 𝑊^𝛽_{(𝑥,𝑖),(𝑦,𝑗)} induced by 𝜉. That is, under measure ℙ^𝛽_{(𝑥,𝑖),(𝑦,𝑗)} the trajectory at time 𝜏=0 starts from the point 𝑥 and has the initial index 𝑖 while at time 𝜏=𝛽 it is at the point 𝑦 and has the index 𝑗. The value 𝑝_{(𝑥,𝑖),(𝑦,𝑗)}=ℙ^𝛽_{(𝑥,𝑖),(𝑦,𝑗)}(𝑊^𝛽_{(𝑥,𝑖),(𝑦,𝑗)}) is given by 𝑝_{(𝑥,𝑖),(𝑦,𝑗)}=𝟏𝑖=𝑗𝑝^𝛽_𝑀𝑗̃𝑥,𝑦exp−𝛽_∶𝚍𝑖,𝑗̃₌₁𝜆_𝑖,𝑗̃+_{𝑘≥1𝑙0=𝑖,𝑙1,…,𝑙𝑘,𝑙𝑘+1=𝑗0≤𝑠≤𝑘}𝟏𝚍𝑙_𝑠,𝑙_𝑠+1=1×𝜆_{𝑙𝑠,𝑙𝑠+1}^𝛽₀d𝜏_𝑠exp−𝜏_𝑠+1−𝜏_𝑠𝑗̃_{∶𝚍(𝑙𝑠,}𝑗̃₎₌₁𝜆_𝑙𝑠,𝑗̃×𝟏0=𝜏₀<𝜏₁<⋯<𝜏_𝑘<𝜏_𝑘+1=𝛽, (41) where 𝑝^𝛽_𝑀(𝑥,𝑦) denotes the transition probability density for the Brownian motion to pass from 𝑥 to 𝑦 on 𝑀 in time 𝛽: 𝑝^𝛽_𝑀=1𝑥,𝑦2𝜋𝛽^𝑑/2×_{𝑛=𝑛1,…,𝑛𝑑∈ℤ𝑑}−exp𝑥−𝑦+𝑛².2𝛽 (42) In view of (13), the quantity 𝑝_{(𝑥,𝑖),(𝑦,𝑗)} and its derivatives are uniformly bounded: 𝑝_{(𝑥,𝑖),(𝑦,𝑗)},∇_𝑥𝑝_{(𝑥,𝑖),(𝑦,𝑗)},∇_𝑦𝑝_{(𝑥,𝑖),(𝑦,𝑗)}≤𝑝_𝑀,𝑥,𝑦∈𝑀,𝑖,𝑗∈Γ, (43) where 𝑝_𝑀=𝑝_𝑀(𝛽)∈(0,+∞) is a constant.

We suggest a term “non-normalised Brownian bridge with jumps” for the measure but expect that a better term will be proposed in future.

Definition 10. Suppose that 𝐱^∗_Λ={𝐱^∗(𝑖),𝑖∈Λ}∈𝑀^∗Λ and 𝐲^∗_Λ={𝐲^∗(𝑗),𝑗∈Λ}∈𝑀^∗Λ are particle configurations over Λ, with ♯𝐱^∗_Λ=♯𝐲^∗_Λ. Let 𝛾 be a pairing between 𝐱^∗_Λ and 𝐲^∗_Λ. Then ℙ^∗_𝐱^∗_Λ,𝐲^∗_Λ,𝛾 denotes the product measure on 𝑊^𝛽_𝐱^∗_Λ,𝐲^∗_Λ,𝛾: ℙ^𝛽_𝐱^∗_Λ,𝐲^∗_Λ,𝛾=×_𝑖∈Λ×_{𝑥∈𝐱∗(𝑖)}ℙ^𝛽_{(𝑥,𝑖),𝛾(𝑥,𝑖)}. (44) Furthermore, ℙ^𝛽_𝐱^∗_Λ,𝐲^∗_Λ stands for the sum measure on 𝑊^𝛽_𝐱^∗_Λ,𝐲^∗_Λ: ℙ^𝛽_𝐱^∗_Λ,𝐲^∗_Λ=_𝛾ℙ^𝛽_𝐱^∗_Λ,𝐲^∗_Λ,𝛾. (45)

According to Definition 10, under the measure ℙ^𝛽_𝐱^∗_Λ,𝐲^∗_Λ,𝛾, the trajectories 𝜔_𝑥,𝑖∈𝑊^𝛽_{(𝑥,𝑖),𝛾(𝑥,𝑖)} constituting 𝝎_Λ are independent components. (Here the term independence is used in the measure-theoretical sense.)

As in [1], we will work with functionals on 𝑊^𝛽_𝐱^∗_Λ,𝐲^∗_Λ,𝛾 representing integrals along trajectories. The first such functional, 𝐡^Λ(𝝎_Λ), is given by 𝐡^Λ𝝎_Λ=_{𝑖∈Λ𝑥∈𝐱∗(𝑖)}𝐡^𝑥,𝑖𝜔_𝑥,𝑖+12_{(𝑖,𝑖′)∈Λ×Λ𝑥∈𝐱∗(𝑖),𝑥′∈𝐱∗(𝑖′)}𝐡^{(𝑥,𝑖),(𝑥′,𝑖′)}×𝜔_𝑥,𝑖,𝜔_{𝑥′,𝑖′}. (46) Here, introducing the notation 𝑢_𝑥,𝑖(𝜏)=𝑢(𝜔_𝑥,𝑖,𝜏) and 𝑢_{𝑥′,𝑖′}(𝜏)=𝑢(𝜔_{𝑥′,𝑖′},𝜏) for the positions in 𝑀 of paths 𝜔_𝑥,𝑖∈𝑊^𝛽_{(𝑥,𝑖),𝛾(𝑥,𝑖)} and 𝜔_{𝑥′,𝑖′}∈𝑊^𝛽_{(𝑥′,𝑖′),𝛾(𝑥′,𝑖′)} at time 𝜏, we define 𝐡^𝑥,𝑖𝜔_𝑥,𝑖=^𝛽₀d𝜏𝑈⁽¹⁾𝑢_𝑖,𝑥(𝜏). (47) Next, with 𝑙_𝑥,𝑖(𝜏) and 𝑙_{𝑥′,𝑖′}(𝜏) standing for the indices of 𝜔_𝑥,𝑖 and 𝜔_{𝑥′,𝑖′} at time 𝜏, 𝐡^{(𝑥,𝑖),(𝑥′,𝑖′)}𝜔_𝑥,𝑖,𝜔_{𝑥′,𝑖′}=^𝛽₀d𝜏_𝑗′∈Γ𝑈⁽²⁾𝑢_𝑖,𝑥(𝜏),𝑢_{𝑖′,𝑥′}(𝜏𝑙)×𝟏_𝑥,𝑖(𝜏)=𝑗^′=𝑙_{𝑥′,𝑖′}(𝜏)+12_{(𝑗′,𝑗′′)∈Γ×Γ}𝐽𝚍𝑗^′,𝑗^′′×𝑉𝑢_𝑖,𝑥(𝜏),𝑢_{𝑖′,𝑥′}(𝜏)×𝟏𝑙_𝑥,𝑖(𝜏)=𝑗^′≠𝑗^′′=𝑙_{𝑥′,𝑖′}(𝜏). (48)

Next, consider the functional 𝐡^Λ(𝝎_Λ∣𝐱^∗_Γ⧵Λ): for 𝐱^∗_Γ⧵Λ={𝐱^∗(𝑗),𝑗∈Γ⧵Λ}. As before, we assume that ♯𝐱^∗(𝑗)≤𝜅. Define 𝐡^Λ𝝎_Λ∣𝐱^∗_Γ⧵Λ=𝐡^Λ𝝎_Λ+𝐡^Λ𝝎_Λ∣∣𝐱^∗_Γ⧵Λ. (49) Here 𝐡^Λ(𝝎_Λ) is as in (46) and 𝐡^Λ𝝎_Λ∣∣𝐱^∗_Γ⧵Λ=_{(𝑖,𝑖′)∈Λ×(Γ⧵Λ)𝑥∈𝐱∗(𝑖),𝑥′∈𝐱∗(𝑖′)}𝐡^{(𝑥,𝑖),(𝑥′,𝑖′)}×𝜔_𝑥,𝑖,𝑥^′,𝑖^′, (50) where, in turn, 𝐡^{(𝑥,𝑖),(𝑥′,𝑖′)}𝜔_𝑥,𝑖,𝑥^′,𝑖^′=^𝛽₀d𝜏𝑈⁽²⁾𝑢_𝑖,𝑥(𝜏),𝑥^′𝟏𝑙_𝑥,𝑖(𝜏)=𝑖^′+_{𝑗∈Γ∶𝑗≠𝑖′}𝐽𝚍𝑗,𝑖^′×𝑉𝑢_𝑖,𝑥(𝜏),𝑥^′𝟏𝑙_𝑥,𝑖(𝜏)=𝑗. (51) The functionals 𝐡^Λ(𝝎_Λ) and 𝐡^Λ(𝝎_Λ∣∣𝐱^∗_Γ⧵Λ) are interpreted as energies of path configurations. Compare (2.1.4) and (2.3.8) in [1].

Finally, we introduce the indicator functional 𝛼_Λ(𝝎_Λ): 𝛼_Λ𝝎_Λ=1,ifindex𝑙_𝑥,𝑖(𝜏)∈Λ,∀𝜏∈0,𝛽,𝑖∈Λ,𝑥∈𝐱^∗(𝑖),0,otherwise. (52) It can be derived from known results [11, 15–17] (for a direct argument, see [18]) that the following assertion holds true.

Lemma 11. For all 𝑧,𝛽>0 and a finite Λ, the Gibbs operators 𝐆_Λ and 𝐆_Λ∣𝐱^∗_Γ⧵Λ act as integral operators in ℋ(Λ): 𝐆_Λ𝝓𝐱^∗_Λ=_{𝑀∗Λ𝑗∈Λ𝑦∈𝐲∗𝑗}𝑣d𝑦𝐊_Λ𝐱^∗_Λ,𝐲^∗_Λ𝝓𝐲^∗_Λ,𝐆_Λ∣𝐱^∗_Γ⧵Λ𝝓𝐱^∗_Λ=_{𝑀∗Λ𝑗∈Λ𝑦∈𝐲∗(𝑗)}𝑣d𝑦𝐊_Λ𝐱^∗_Λ,𝐲^∗_Λ∣𝐱^∗_Γ⧵Λ𝝓𝐲^∗_Λ. (53) Moreover, the integral kernels 𝐊_Λ(𝐱^∗_Λ,𝐲^∗_Λ) and 𝐊_Λ(𝐱^∗_Λ,𝐲^∗_Λ∣𝐱^∗_Γ⧵Λ) vanish if ♯𝐱^∗_Λ≠♯𝐲^∗_Λ. On the other hand, when ♯𝐱^∗_Λ=♯𝐲^∗_Λ, the kernels 𝐊_Λ(𝐱^∗_Λ,𝐲^∗_Λ) and 𝐊_Λ(𝐱^∗_Λ,𝐲^∗_Λ∣𝐱^∗_Γ⧵Λ) admit the following representations: 𝐊_Λ𝐱^∗_Λ,𝐲^∗_Λ=𝑧^♯𝐱∗Λ_{𝑊∗𝐱∗Λ∗Λ,𝐲}ℙ^𝛽_𝐱^∗_Λ,𝐲^∗_Λd𝝎_Λ𝛼_Λ𝝎_Λ×exp−𝐡^Λ𝝎_Λ,𝐊_Λ𝐱^∗_Λ,𝐲^∗_Λ∣𝐱^∗_Γ⧵Λ=𝑧^♯𝐱∗Λ_{𝑊∗𝐱∗Λ∗Λ,𝐲}ℙ^𝛽_𝐱^∗_Λ,𝐲^∗_Λd𝝎_Λ𝛼_Λ𝝎_Λ×exp−𝐡^Λ𝝎_Λ∣𝐱^∗_Γ⧵Λ. (54)(55) The ingredients of these representations are determined in (46)–(51).

Remark 12. As before, we stress that, owing to (16) and (17), a nonzero contribution to the integral in the RHS of (54) can only come from a path configuration 𝝎_Λ={𝜔_𝑥,𝑖} such that ∀𝜏∈[0,𝛽] and ∀𝑗∈Γ, the number of paths 𝜔_𝑥,𝑖 with index 𝑙_𝑥,𝑖(𝜏)=𝑗 is less than or equal to 𝜅. Likewise, the integral in the RHS of (55) receives a non-zero contribution only from configurations 𝝎_Λ={𝜔^∗_𝑥,𝑖} such that, ∀ site 𝑗∈Γ, the number of paths 𝜔_𝑥,𝑖 with index 𝑙_𝑥,𝑖(𝜏)=𝑗 plus the cardinality ♯𝐱^∗(𝑗) does not exceed 𝜅.

2.2. The Representation for the Partition Function

The FK representations of the partition functions Ξ(Λ)=Ξ_𝛽,𝑧(Λ) in (27) and Ξ(Λ∣𝐱^∗_Γ⧵Λ) in (1.4.6) reflect a specific character of the traces tr𝐆_Λ and tr𝐆_Λ∣𝐱^∗_Γ⧵Λ in ℋ(Λ). The source of a complication here is the jump terms in the Hamiltonians 𝐻_Λ and 𝐻_Λ∣𝐱^∗_Γ⧵Λ in (11) and (12), respectively. In particular, we will have to pass from trajectories of fixed time-length 𝛽 to loops of a variable time length. To this end, a given matching 𝛾 is decomposed into a product of cycles, and the trajectories associated with a given cycle are merged into closed paths (loops) of a time-length multiple of 𝛽. (A similar construction has been performed in [18].)

To simplify the notation, we omit, wherever possible, the index 𝛽.

Definition 13. For given (𝑥,𝑖),(𝑦,𝑗)∈𝑀×Γ, the symbol 𝑊^∗_{(𝑥,𝑖),(𝑦,𝑗)} denotes the disjoint union: 𝑊^∗_{(𝑥,𝑖),(𝑦,𝑗)}=_𝑘=0,1,…𝑊^𝑘𝛽_{(𝑥,𝑖),(𝑦,𝑗)}. (56) In other words, 𝑊^∗_{(𝑥,𝑖),(𝑦,𝑗)} is the space of paths Ω^∗=Ω^∗_{(𝑥,𝑖),(𝑦,𝑗)} in 𝑀×Γ, of a variable time-length 𝑘𝛽, where 𝑘=𝑘(Ω^∗) takes values 1,2,… and called the length multiplicity, with the end-points (𝑥,𝑖) and (𝑦,𝑗). The formal definition follows the same line as in (37), and we again use the notation Ω^∗(𝜏) and the notation (𝑢(Ω^∗,𝜏),𝑙(Ω^∗,𝜏)) for the pair of the position and the index of path Ω^∗ at time 𝜏. Next, we call the particle configuration {Ω^∗(𝜏+𝛽𝑚),0≤𝑚<𝑘(Ω^∗)} the temporal section (or simply the section) of Ω^∗ at time 𝜏∈[0,𝛽]. We also call Ω^∗_{(𝑥,𝑖),(𝑦,𝑗)}∈𝑊^∗_{(𝑥,𝑖),(𝑦,𝑗)} a path (from (𝑥,𝑖) to (𝑦,𝑗)).
A particular role will be played by closed paths (loops), with coinciding endpoints (where (𝑥,𝑖)=(𝑦,𝑗)). Accordingly, we denote by 𝑊^∗_𝑥,𝑖 the set 𝑊^∗_{(𝑥,𝑖),(𝑥,𝑖)}. An element of 𝑊^∗_𝑥,𝑖 is denoted by Ω^∗_𝑥,𝑖 or, in short, by Ω^∗ and called a loop at vertex 𝑖. (The upper index ∗ indicates that the length multiplicity is unrestricted.) The length multiplicity of a loop Ω^∗_𝑥,𝑖∈𝑊^∗_𝑥,𝑖 is denoted by 𝑘(𝛀^∗_𝑥,𝑖) or 𝑘_𝑥,𝑖. It is instructive to note that, as topological object, a given loop 𝛀^∗ admits a multiple choice of the initial pair (𝑥,𝑖): it can be represented by any pair (𝑢(Ω^∗,𝜏),𝑙(Ω^∗,𝜏)) at a time 𝜏=𝑙𝛽 where 𝑙=1,…,𝑘(𝛀^∗). As above, we use the term the temporal section at time 𝜏∈[0,𝛽] for the particle configuration {Ω^∗_𝑥,𝑖(𝜏+𝛽𝑚),0≤𝑚<𝑘_𝑥,𝑖} and employ the alternative notation (𝑢(𝜏+𝛽𝑚;Ω^∗),𝑙(𝜏+𝛽𝑚;Ω^∗)) addressing the position and the index of Ω^∗ at time 𝜏+𝛽𝑚∈[0,𝛽𝑘(Ω^∗)].

Definition 14. Suppose 𝐱^∗_Λ={𝐱^∗(𝑖),𝑖∈Λ}∈𝑀^∗Λ and 𝐲^∗_Λ={𝐲^∗(𝑗),𝑗∈Λ}∈𝑀^∗Λ are particle configurations over Λ, with ♯𝐱^∗_Λ=♯𝐲^∗_Λ. Let 𝛾 be a matching between 𝐱^∗_Λ and 𝐲^∗_Λ. We consider the Cartesian product: 𝑊^∗_𝐱^∗_Λ,𝐲^∗_Λ,𝛾=×_𝑖∈Λ×_{𝑥∈𝐱∗(𝑖)}𝑊^∗_{(𝑥,𝑖),𝛾(𝑥,𝑖)}, (57) and the disjoint union: 𝑊^∗_𝐱^∗_Λ,𝐲^∗_Λ=_𝛾𝑊^∗_𝐱^∗_Λ,𝐲^∗_Λ,𝛾. (58) Accordingly, an element 𝛀^∗_Λ∈𝑊^∗_𝐱^∗_Λ,𝐲^∗_Λ,𝛾 in (58) represents a collection of paths 𝛀^∗_𝑥,𝑖, 𝑥∈𝐱^∗(𝑖), 𝑖∈Λ, of time-length 𝑘𝛽, starting at (𝑥,𝑖) and ending up at (𝑦,𝑗)=𝛾(𝑥,𝑖). We say that 𝛀^∗_Λ∈𝑊^∗_𝐱^∗_Λ,𝐲^∗_Λ is a path configuration in (or over) Λ.
Again, loops play a special role and deserve a particular notation. Namely, 𝑊^∗_𝐱^∗_Λ denotes the Cartesian product: 𝑊^∗_𝐱^∗_Λ=×_𝑖∈Λ×_{𝑥∈𝐱∗(𝑖)}𝑊^∗_𝑥,𝑖, (59) and 𝑊^∗_Λ stands for the disjoint union (or equivalently, the Cartesian power): 𝑊^∗_Λ=_{𝐱∗Λ∈𝑀∗Λ}𝑊^∗_𝐱^∗_Λ=×_𝑖∈Λ𝑊^∗_{𝑖},where𝑊^∗_{𝑖}=_{𝐱∗∈𝑀∗}×_{𝑥∈𝐱∗}𝑊^∗_𝑥,𝑖. (60)
Denote by 𝛀^∗={𝛀^∗(𝑖),𝑖∈Λ}∈𝑊^∗_Λ a collection of loop configurations at vertices 𝑖∈Λ starting and ending up at particle configurations 𝐱^∗(𝑖)∈𝑀^∗ (note that some of the 𝛀^∗(𝑖)’s may be empty). The temporal section (or, in short, the section), 𝛀^∗(𝜏), of 𝛀^∗ at time 𝜏 is defined as the particle configuration formed by the points Ω^∗_𝑥,𝑖(𝜏+𝛽𝑚) where 𝑖∈Λ, 𝑥∈𝐱^∗(𝑖), and 0≤𝑚<𝑘_𝑥,𝑖.

As before, consider the standard sigma algebras of subsets in the spaces 𝑊^∗_{(𝑥,𝑖),(𝑦,𝑗)}, 𝑊_𝑥,𝑖, 𝑊^∗_𝐱^∗_Λ,𝐲^∗_Λ,𝛾, 𝑊^∗_𝐱^∗_Λ,𝐲^∗_Λ, 𝑊^∗_𝐱^∗_Λ, and 𝑊^∗_Λ introduced in Definitions 13 and 14. In particular, the sigma algebra of subsets in 𝑊^∗_Λ will be denoted by 𝖂_Λ; we comment on some of its specific properties in Section 3.1. (An infinite-volume version 𝑊^∗_Γ of 𝑊^∗_Λ is treated in Section 3.2 and after.)

Definition 15. Given points (𝑥,𝑖),(𝑦,𝑗)∈𝑀×Γ, we denote by ℙ^∗_{(𝑥,𝑖),(𝑦,𝑗)} the sum measure on 𝑊^∗_{(𝑥,𝑖),(𝑦,𝑗)}: ℙ^∗_{(𝑥,𝑖),(𝑦,𝑗)}=_𝑘=0,1,…ℙ^𝑘𝛽_{(𝑥,𝑖),(𝑦,𝑗)}. (61)
Further, ℙ^∗_𝑥,𝑖 denotes the similar measure on 𝑊^∗_𝑥,𝑖: ℙ^∗_𝑥,𝑖=_𝑘=0,1,…ℙ^𝑘𝛽_𝑥,𝑖. (62)

Definition 16. Let 𝐱^∗_Λ={𝐱^∗(𝑖),𝑖∈Λ}∈𝑀^∗Λ and 𝐲^∗_Λ={𝐲^∗(𝑗),𝑗∈Λ}∈𝑀^∗Λ be particle configurations over Λ, with ♯𝐱^∗_Λ=♯𝐲^∗_Λ. Let 𝛾 be a matching between 𝐱^∗_Λ and 𝐲^∗_Λ, and we define the product measure ℙ^∗_𝐱^∗_Λ,𝐲^∗_Λ,𝛾: ℙ^∗_𝐱^∗_Λ,𝐲^∗_Λ,𝛾=×_𝑖∈Λ×_{𝑥∈𝐱∗(𝑖)}ℙ^∗_{(𝑥,𝑖),𝛾(𝑥,𝑖)} (63) and the sum measure ℙ^∗_𝐱^∗_Λ,𝐲^∗_Λ=_𝛾ℙ^∗_𝐱^∗_Λ,𝐲^∗_Λ,𝛾. (64)
Next, symbol ℙ^∗_𝐱^∗_Λ stands for the product measure on 𝑊^∗_𝐱^∗_Λ: ℙ^∗_𝐱^∗_Λ=×_𝑖∈Λ×_{𝑥∈𝐱∗(𝑖)}ℙ^∗_𝑥,𝑖. (65)
Finally, d𝛀^∗_Λ yields the measure on 𝑊^∗_Λ: d𝛀^∗_Λ=d𝐱^∗_Λ×ℙ^∗_𝐱^∗_Λd𝛀^∗_Λ. (66) Here, for 𝐱^∗_Λ={𝐱^∗(𝑖),𝑖∈Λ}, we set: d𝐱^∗_Λ=∏_𝑖∈Λ∏_{𝑥∈𝐱∗(𝑖)}𝑣(d𝑥). For sites 𝑖 with 𝐱^∗(𝑖)=∅, the corresponding factors are trivial measures sitting on the empty configurations.

We again need to introduce energy-type functionals represented by integrals along loops. More precisely, we define the functionals 𝐡^Λ(𝛀^∗_Λ) and 𝐡^Λ(𝛀^∗_Λ∣𝐱^∗_Γ⧵Λ) which are modifications of the above functionals 𝐡^Λ(𝝎_Λ) and 𝐡^Λ(𝝎_Λ∣𝐱^∗_Γ⧵Λ); confer (46) and (50). Say, for a loop configuration 𝛀^∗_Λ={Ω^∗_𝑥,𝑖} over Λ with an initial and final particle configuration 𝐱^∗_Λ={𝐱^∗(𝑖),𝑖∈Λ}, 𝐡^Λ𝛀^∗_Λ=_{𝑖∈Λ𝑥∈𝐱∗(𝑖)}𝐡^𝑥,𝑖Ω^∗_𝑥,𝑖+12_{(𝑖,𝑖′)∈Λ×Λ𝑥∈𝐱∗(𝑖),𝑥′∈𝐱∗(𝑖′)}×𝟏(𝑥,𝑖)≠𝑥^′,𝑖^′𝐡^{(𝑥,𝑖),(𝑥′,𝑖′)}Ω^∗_x,𝑖,Ω^∗_{𝑥′,𝑖′}. (67) To determine the functionals 𝐡^𝑥,𝑖(Ω^∗_𝑥,𝑖) and 𝐡^{(𝑥,𝑖),(𝑥′,𝑖′)}(Ω^∗_𝑥,𝑖,Ω^∗_{𝑥′,𝑖′}), we set, for given 𝑚=0,1,…,𝑘_𝑥,𝑖−1 and 𝑚^′=0,1,…,𝑘_{𝑥′,𝑖′}−1: 𝑢_𝑖,𝑥𝜏+𝛽𝑚=𝑢𝜏+𝛽𝑚;Ω^∗_𝑥,𝑖,𝑙_𝑖,𝑥𝜏+𝛽𝑚=𝑙𝜏+𝛽𝑚;Ω^∗_𝑥,𝑖,𝑢_{𝑖′,𝑥′}𝜏+𝛽𝑚^′=𝑢𝜏+𝛽𝑚^′;Ω^∗_{𝑥′,𝑖′},𝑙_{𝑖′,𝑥′}𝜏+𝛽𝑚^′=𝑙𝜏+𝛽𝑚^′;Ω^∗_{𝑥′,𝑖′}. (68) A (slightly) shortened notation 𝑙_𝑖,𝑥(𝜏+𝛽𝑚) is used for the index 𝑙_𝑥,𝑖(𝜏+𝛽𝑚;Ω^∗_𝑥,𝑖) and 𝑢_𝑖,𝑥(𝜏+𝛽𝑚) for the position 𝑢(𝜏+𝛽𝑚;Ω^∗_𝑥,𝑖) for Ω^∗_𝑥,𝑖(𝜏)∈𝑀×Γ, of the section Ω^∗_𝑥,𝑖(𝜏) of the loop Ω^∗_𝑥,𝑖 at time 𝜏, and similarly with 𝑙_{𝑖′,𝑥′}(𝜏+𝛽𝑚^′) and 𝑢_{𝑖′,𝑥′}(𝜏+𝛽𝑚^′). (Note that the pairs (𝑥,𝑖) and (𝑥^′,𝑖^′) may coincide.) Then 𝐡^(𝑥,𝑖)Ω^∗_𝑥,𝑖=^𝛽₀d𝜏_{0≤𝑚<𝑘𝑥,𝑖}𝑈⁽¹⁾𝑢_𝑥,𝑖𝜏+𝛽𝑚+_{0≤𝑚<𝑚′<𝑘𝑥,𝑖𝑗∈Γ}𝟏𝑙_𝑥,𝑖𝜏+𝛽𝑚=𝑗=𝑙_𝑥,𝑖𝜏+𝛽𝑚^′×𝑈⁽²⁾𝑢_𝑥,𝑖𝜏+𝛽𝑚,𝑢_𝑥,𝑖𝜏+𝛽𝑚^′,𝐡^{(𝑥,𝑖),(𝑥′,𝑖′)}Ω^∗_𝑥,𝑖,Ω^∗_{𝑥′,𝑖′}=_{0≤𝑚<𝑘𝑥,𝑖0≤𝑚′<𝑘𝑥′′,𝑖}^𝛽₀×d𝜏_𝑗∈Γ𝑈⁽²⁾𝑢_𝑥,𝑖𝜏+𝛽𝑚,𝑢_{𝑥′,𝑖′}𝜏+𝛽𝑚^′×𝟏𝑙_𝑥,𝑖𝜏+𝛽𝑚=𝑗=𝑙_{𝑥′,𝑖′}𝜏+𝛽𝑚^′+_{𝑗,𝑗′∈Γ×Γ}𝐽𝚍𝑗,𝑗^′𝑉×𝑢_𝑖,𝑥𝜏+𝛽𝑚,𝑢_{𝑖′,𝑥′}𝜏+𝛽𝑚^′×𝟏𝑙_𝑥,𝑖𝜏+𝛽𝑚=𝑗≠𝑗^′=𝑙_{𝑥′,𝑖′}𝜏+𝛽𝑚^′. (69)

Next, the functional 𝐵(𝛀^∗_Λ) takes into account the bosonic character of the model: 𝐵𝛀^∗_Λ=_{𝑖∈Λ𝑥∈𝐱∗(𝑖)}𝑧^{𝑘𝑥,𝑖}𝑘_𝑥,𝑖. (70) The factor 𝑘⁻¹_𝑥,𝑖 in (70) reflects the fact that the starting point of a loop Ω^∗_𝑥,𝑖 may be selected among points 𝑢(𝛽𝑚,Ω^∗_𝑥,𝑖) arbitrarily.

Next, we define the functional 𝐡^Λ(𝛀^∗_Λ∣𝐱^∗_Γ⧵Λ): for 𝐱^∗_Γ⧵Λ={𝐱^∗(𝑗),𝑗∈Γ⧵Λ}, again assuming that ♯𝐱^∗(𝑗)≤𝜅. Set 𝐡^Λ𝛀^∗_Λ∣𝐱^∗_Γ⧵Λ=𝐡^Λ𝛀^∗_Λ+𝐡^Λ𝛀^∗_Λ∣∣𝐱^∗_Γ⧵Λ. (71) Here 𝐡^Λ(𝛀^∗_Λ) is as in (67) and 𝐡^Λ𝛀^∗_Λ∣∣𝐱^∗_Γ⧵Λ=_{(𝑖,𝑖′)∈Λ×(Γ⧵Λ)𝑥∈𝐱∗(𝑖),𝑥′∈𝐱∗𝑖′}𝐡^{(𝑥,𝑖),(𝑥′,𝑖′)}Ω^∗_𝑥,𝑖,𝑥^′,𝑖^′, (72) where, in turn, 𝐡^{(𝑥,𝑖),(𝑥′,𝑖′)}Ω^∗_𝑥,𝑖,𝑥^′,𝑖^′=_{0≤𝑚<𝑘𝑥,𝑖}^𝛽₀×𝑈d𝜏₍₂₎𝑢_𝑥,𝑖𝜏+𝛽𝑚,𝑥^′×𝟏𝑙_𝑥,𝑖𝜏+𝛽𝑚=𝑖^′+_𝑗∈Γ𝐽𝚍𝑗,𝑖^′𝑉𝑢_𝑖,𝑥𝜏+𝛽𝑚,𝑥^′×𝟏𝑙_𝑥,𝑖𝜏+𝛽𝑚=𝑗. (73) As before, the functionals 𝐡^Λ(𝛀^∗_Λ) and 𝐡^Λ(𝛀^∗_Λ∣∣𝐱^∗_Γ⧵Λ) have a natural interpretation as energies of loop configurations.

Finally, as before, the functional 𝛼_Λ(𝛀^∗_Λ) is the indicator that the collection of loops 𝛀^∗_Λ={Ω^∗_𝑥,𝑖} does not quit Λ: 𝛼_Λ𝛀^∗_Λ=1,ifΩ^∗_𝑥,𝑖(𝜏)∈𝑀×Λ,∀𝑖∈Λ,𝑥∈𝐱^∗(𝑖),0≤𝜏≤𝛽𝑘_𝑥,𝑖,0,otherwise. (74)

Like above, we invoke known results [11, 15–17] to establish the following statement (again a direct argument can be found in [18]).

Lemma 17. For all finite Λ⊂Γ and 𝑧,𝛽>0 satisfying (22), the partition functions Ξ(Λ) in (27) and Ξ(Λ∣𝐱^∗_Γ⧵Λ) in (27) and (28) admit the representations as converging integrals: Ξ(Λ)=_𝑊∗Λd𝛀^∗_Λ𝐵𝛀^∗_Λ𝛼_Λ𝛀^∗_Λexp−𝐡^Λ𝛀^∗_Λ,ΞΛ∣𝐱^∗_Γ⧵Λ=_𝑊∗Λd𝛀^∗_Λ𝐵𝛀^∗_Λ𝛼_Λ𝛀^∗_Λexp−𝐡^Λ𝛀^∗_Λ∣∣𝐱^∗_Γ⧵Λ (75)(76) with the ingredients introduced in (60)–(74).

Again, we emphasis that the non-zero contribution to the integral in (75) can only come from loop configurations 𝛀^∗_Λ={Ω^∗_𝑥,𝑖,𝑥∈𝐱^∗(𝑖),𝑖∈Λ} such that ∀ vertex 𝑗∈Λ and 𝜏∈[0,𝛽], the total number of pairs (𝑢_𝑥,𝑖(𝜏+𝑚𝛽),𝑙_𝑥,𝑖(𝜏+𝑚𝛽)) with 0≤𝑚<𝑘_𝑥,𝑖, and 𝑙(𝜏+𝑚𝛽)=𝑗 does not exceed 𝜅.

Remark 18. The integrals in (75) and (76) represent examples of partition functions which will be encountered in the forthcoming sections. See (96), (97), (101), (103), (105), (107), (111), and (113) below. A general form of such a partition function treated as an integral over a set of loop configurations rather than a trace in a Hilbert space is given in (96) and (97).

2.3. The Representation for the RDM Kernels

Let Λ⁰,Λ be finite sets, Λ⁰⊂Λ⊂Γ. The construction developed in Section 2.2 also allows us to write a convenient representation for the integral kernels of the RDMs 𝐑^Λ0_Λ (see (31)) and 𝐑^Λ0_Λ∣𝐱^∗_Γ⧵Λ. In accordance with Lemma 11 and the definition of 𝐑^Λ0_Λ in (31), the operator 𝐑^Λ0_Λ acts as an integral operator in ℋ(Λ⁰): 𝐑^Λ0_Λ𝝓𝐱^∗0=_{𝑀0∗Λ𝑗∈Λ0𝑦∈𝐲∗(𝑗)}𝑣d𝑦𝐅^Λ0_Λ𝐱^∗0,𝐲^∗0𝝓𝐲^∗0, (77) where 𝐅^Λ0_Λ𝐱^∗0,𝐲^∗0∶=_{𝑀0∗Λ⧵Λ𝑗∈Λ⧵Λ0𝑧∈𝐳∗(𝑗)}𝑣(d𝑧)×𝐅_Λ𝐱^∗0∨𝐳^∗_Λ⧵Λ0,𝐲^∗0∨𝐳^∗_Λ⧵Λ0Ξ∶=^Λ0Λ𝐱^∗0,𝐲^∗0;Λ⧵Λ⁰Ξ(Λ). (78)

We employ here and below the notation 𝐱^∗0 and 𝐲^∗0 for particle configurations 𝐱^∗_Λ0={𝐱^∗(𝑖),𝑖∈Λ⁰} and 𝐲^∗_Λ0={𝐲^∗(𝑗),𝑗∈Λ⁰} over Λ⁰. Next, 𝐱^∗0∨𝐳^∗_Λ⧵Λ0; 𝐲^∗0∨𝐳^∗_Λ⧵Λ0 denotes the concatenated configurations over Λ.

Similarly, the RDM 𝐑^Λ0_Λ∣𝐱^∗_Γ⧵Λ is determined by its integral kernel 𝐅^Λ0_Λ∣𝐱^∗_Γ⧵Λ(𝐱^∗0,𝐲^∗0), again admitting the representation 𝐅^Λ0_Λ∣𝐱^∗_Γ⧵Λ𝐱^∗0,𝐲^∗0Ξ∶=^Λ0Λ𝐱^∗0,𝐲^∗0;Λ⧵Λ⁰∣𝐱^∗_Γ⧵ΛΞ(Λ). (79)

As in [1], we call 𝐅^Λ0_Λ and 𝐅^Λ0_Λ∣𝐱^∗_Γ⧵Λ the RDM kernels (in short, RDMKs). The focus of our interest is the numerators Ξ^Λ0_Λ(𝐱^∗0,𝐲^∗0,Λ⧵Λ⁰) and Ξ^Λ0_Λ(𝐱^∗0,𝐲^∗0;Λ⧵Λ⁰∣𝐱^∗_Γ⧵Λ) in (78) and (79). To introduce the appropriate representation for these quantities, we need some additional definitions.

Definition 19. Repeating (57)-(58), symbol 𝑊^∗_{𝐱∗0,𝐲∗0} denotes the disjoint union ⋃_𝛾0𝑊^∗_{𝐱∗0,𝐲∗0,𝛾0} over matchings 𝛾⁰ between 𝐱^∗0 and 𝐲^∗0. Accordingly, element 𝛀^∗0=𝛀^∗_{𝐱∗0,𝐲∗0,𝛾0}∈𝑊^∗_{𝐱∗0,𝐲∗0,𝛾0} yields a collection of paths Ω^∗_{(𝑥,𝑖),𝛾0(𝑥,𝑖)}∈𝑊^∗_{(𝑥,𝑖),𝛾0(𝑥,𝑖)} lying in 𝑀×Γ. Each path Ω^∗_{(𝑥,𝑖),𝛾0(𝑥,𝑖)} has time lengths 𝛽𝑘_{(𝑥,𝑖),(𝑦,𝑗)}, begins at (𝑥,𝑖), and ends up at (𝑦,𝑗)=𝛾⁰(𝑥,𝑖), where 𝑥∈𝐱^∗(𝑖), 𝑦∈𝐲^∗(𝑗). Like above, we will use for 𝛀^∗0 the term a path configuration over Λ⁰. Repeating (63)-(64), we obtain the measures ℙ^∗_{𝐱∗0,𝐲∗0,𝛾0} on 𝑊^∗_{𝐱∗0,𝐲∗0,𝛾0} and ℙ^∗_{𝐱∗0,𝐲∗0} on 𝑊^∗_{𝐱∗0,𝐲∗0}.

The assertion of Lemma 20 below again follows directly from known results, in conjunction with calculations of the partial trace tr_ℋ0Λ⧵Λ in ℋ_Λ. The meaning of new ingredients in (79)–(82) is explained below.

Lemma 20. The quantity Ξ^Λ0_Λ(𝐱^∗0,𝐲^∗0;Λ⧵Λ⁰) emerging in (78) is set to be 0 when ♯𝐱^∗0≠♯𝐲^∗0. On the other hand, for ♯𝐱^∗0=♯𝐲^∗0, Ξ^Λ0_Λ𝐱^∗0,𝐲^∗0;Λ⧵Λ⁰=_{𝑊∗𝐱∗0,𝐲∗0}ℙ^∗_{𝐱∗0,𝐲∗0}d𝛀^∗0𝐵𝛀^∗0×𝛼_Λ𝛀^∗0𝟏𝛀^∗0∈ℱ^Λ0×exp−𝐡^Λ0𝛀^∗0Ξ^Λ0_ΛΛ⧵Λ⁰∣𝛀^∗0, (80) where Ξ^Λ0_ΛΛ⧵Λ⁰∣𝛀^∗0=_{𝑊∗0Λ⧵Λ}d𝛀^∗_Λ⧵Λ0𝐵𝛀^∗_Λ⧵Λ0×𝛼_Λ𝛀^∗_Λ⧵Λ0𝟏𝛀^∗_Λ⧵Λ0∈ℱ^Λ0×exp−𝐡^Λ⧵Λ0𝛀^∗_Λ⧵Λ0∣𝛀^∗0. (81) Similarly, the quantity Ξ^Λ0_Λ(𝐱^∗0,𝐲^∗0;Λ⧵Λ⁰∣𝐱^∗_Γ⧵Λ) from (79) vanishes when ♯𝐱^∗0≠♯𝐲^∗0. For ♯𝐱^∗0=♯𝐲^∗0, Ξ^Λ0_Λ𝐱^∗0,𝐲^∗0;Λ⧵Λ⁰∣𝐱^∗_Γ⧵Λ=_{𝑊∗𝐱∗0,𝐲∗0}ℙ^∗_{𝐱∗0,𝐲∗0}d𝛀^∗0𝐵𝛀^∗0×𝛼_Λ𝛀^∗0𝟏𝛀^∗0∈ℱ^Λ0×exp−𝐡^Λ0𝛀^∗0∣𝐱^∗_Γ⧵Λ×Ξ^Λ0_ΛΛ⧵Λ⁰∣𝛀^∗0∨𝐱^∗_Γ⧵Λ, (82) where Ξ^Λ0_ΛΛ⧵Λ⁰∣𝛀^∗0∨𝐱^∗_Γ⧵Λ=_{𝑊∗0Λ⧵Λ}d𝛀^∗_Λ⧵Λ0𝐵𝛀^∗_Λ⧵Λ0𝛼_Λ𝛀^∗_Λ⧵Λ0×𝟏𝛀^∗_Λ⧵Λ0∈ℱ^Λ0×exp−𝐡^Λ⧵Λ0𝛀^∗_Λ⧵Λ0∣𝛀^∗0∨𝐱^∗_Γ⧵Λ. (83) These representations hold ∀𝑧,𝛽>0 and finite Λ⁰⊂Λ⊂Γ.

Let us define the functionals 𝐵(𝛀^∗0), 𝛼_Λ(𝛀^∗0), 𝟏(𝛀^∗0∈ℱ^Λ0), 𝟏(𝛀^∗_Λ⧵Λ0∈ℱ^Λ0)𝐡^Λ0(𝛀^∗0), 𝐡^Λ⧵Λ0(𝛀^∗_Λ⧵Λ0∣𝛀^∗0), 𝐡^Λ0(𝛀^∗0𝐱^∗_Γ⧵Λ), and 𝐡^Λ⧵Λ0(𝛀^∗_Λ⧵Λ0∣𝛀^∗0∨𝐱^∗_Γ⧵Λ) in (79)–(83). (The functionals 𝐵(𝛀^∗_Λ⧵Λ0) and 𝛼_Λ(𝛀^∗_Λ⧵Λ0) are defined as (70) and (74), respectively, replacing Λ with Λ⧵Λ⁰.)

To this end, let 𝛀^∗0=𝛀^∗_{𝐱∗0,𝐲∗0,𝛾0}∈𝑊^∗_{𝐱∗0,𝐲∗0,𝛾0} be a path configuration represented by a collection of paths Ω^∗_{(𝑥,𝑖),𝛾0(𝑥,𝑖)}∈𝑊^∗_{(𝑥,𝑖),𝛾∗0(𝑥,𝑖)} (Ω^∗_𝑥,𝑖 in short), with end points (𝑥,𝑖) and (𝑦,𝑗)=𝛾⁰(𝑥,𝑖), of time-length 𝛽𝑘_{(𝑥,𝑖),(𝑦,𝑗)}. The functional 𝐵(𝛀^∗0) is given by 𝐵𝛀^∗0=_{𝑖∈Λ𝑥∈𝐱∗(𝑖)}𝑧^{𝑘(𝑥,𝑖),(𝑦,𝑗)}. (84) The functional 𝛼_Λ(𝛀^∗0) is again an indicator: 𝛼_Λ𝛀^∗0=1,ifΩ^∗_{(𝑥,𝑖),𝛾0(𝑥,𝑖)}(𝜏)∈𝑀×Λ,∀𝑖∈Λ,𝑥∈𝐱^∗(𝑖),0≤𝜏≤𝛽𝑘_{(𝑥,𝑖),(𝑦,𝑗)},0,otherwise. (85)

Now let us define the indicator function 𝟏(⋅∈ℱ^Λ0) in (80)–(83). The factor 𝟏(𝛀^∗0∈ℱ^Λ0) equals one if and only if every path Ω^∗_{(𝑥,𝑖),(𝑦,𝑗)} from 𝛀^∗0, of time-length 𝛽𝑘_{(𝑥,𝑖),(𝑦,𝑗)}, starting at (𝑥,𝑖)∈𝑀×Λ⁰ and ending up at (𝑦,𝑗)=𝛾⁰(𝑥,𝑖)∈𝑀×Λ⁰ remains in 𝑀×(Λ⧵Λ⁰) at the intermediate times 𝛽𝑙 for 𝑙=1,…,𝑘_{(𝑥,𝑖),(𝑦,𝑗)}−1: Ω^∗_{(),𝑥,𝑖𝑦,𝑗}𝑙𝛽∉𝑀×Λ⁰,∀𝑙=1,…,𝑘_{(𝑥,𝑖),(𝑦,𝑗)}−1, (86) (when  𝑘_{(𝑥,𝑖),(𝑦,𝑗)}=1, this is not a restriction).

Furthermore, suppose that 𝛀^∗_Λ⧵Λ0=𝛀^∗_𝐱^∗_0Λ⧵Λ is a loop configuration over Λ⧵Λ⁰, with the initial/end configuration 𝐱^∗_Λ⧵Λ0={𝐱^∗(𝑖),𝑖∈Λ⧵Λ⁰}, represented by a collection of loops Ω^∗_𝑥,𝑖,𝑖∈Λ⧵Λ⁰,𝑥∈𝐱^∗(𝑖). Then 𝟏(𝛀^∗_Λ⧵Λ0∈ℱ^Λ0)=1 if and only if each loop Ω^∗_𝑥,𝑖 of time-length 𝛽𝑘_𝑥,𝑖, beginning and finishing at (𝑥,𝑖)∈𝑀×(Λ⧵Λ⁰), does not enter the set 𝑀×Λ⁰ at times 𝛽𝑙 for 𝑙=1,…,𝑘_𝑥,𝑖−1: Ω^∗_𝑥,𝑖𝑙𝛽∉𝑀×Λ⁰,∀𝑙=1,…,𝑘_𝑥,𝑖−1 (87) (again, if 𝑘_𝑥,𝑖=1, this is not a restriction).

The functional 𝐡^Λ0(𝛀^∗0) in (80) gives the energy of the path configuration 𝛀^∗0 and is introduced similarly to (67), mutatis mutandis. Next, the functional 𝐡^Λ⧵Λ0(𝛀^∗_Λ⧵Λ0∣𝛀^∗0) in (81) represents the energy of the loop configuration 𝛀^∗_Λ⧵Λ0 in the potential field generated by the path configuration 𝛀^∗0: 𝐡^Λ⧵Λ0𝛀^∗_Λ⧵Λ0∣𝛀^∗0=𝐡^Λ⧵Λ0𝛀^∗_Λ⧵Λ0+𝐡𝛀^∗0∣∣𝛀^∗_Λ⧵Λ0. (88) Here, the summand 𝐡^Λ⧵Λ0(𝛀^∗_Λ⧵Λ0) yields the energy of the loop configuration 𝛀^∗_Λ⧵Λ0; again confer (67). Further, the term 𝐡(𝛀^∗0∣∣𝛀^∗_Λ⧵Λ0) yields the energy of interaction between 𝛀^∗0 and 𝛀^∗_Λ⧵Λ0: for a path/loop configurations 𝛀^∗0={Ω^∗_𝑥,𝑖}∈𝑊^∗_{𝐱∗0,𝐲∗0,𝛾0} and a 𝛀^∗_Λ⧵Λ0={Ω^∗_{𝑥′,𝑖′}}∈𝑊^∗_Λ⧵Λ0 we set 𝐡𝛀^∗0∣∣𝛀^∗_Λ⧵Λ0=_{𝑖,𝑖′∈Λ0×Λ⧵Λ0𝑥∈𝐱∗(𝑖),𝑥′∈𝐱∗(𝑖′)}𝐡^{(𝑥,𝑖),(𝑥′,𝑖′)}Ω^∗_𝑥,𝑖,Ω^∗_{𝑥′,𝑖′}. (89) Here, for a path Ω^∗_𝑥,𝑖=Ω^∗_{(𝑥,𝑖),𝛾0(𝑥,𝑖)}, of time-length 𝛽𝑘_{(𝑥,𝑖),𝛾0(𝑥,𝑖)}, and a loop Ω^∗_{𝑥′,𝑖′}, of time-length 𝛽𝑘_𝑥,𝑖, 𝐡^{(𝑥,𝑖),(𝑥′,𝑖′)}Ω^∗_𝑥,𝑖,Ω^∗_{𝑥′,𝑖′}=_{0≤𝑚<𝑘0(𝑥,𝑖),𝛾(𝑥,𝑖)0≤𝑚′<𝑘𝑥′′,𝑖}^𝛽₀×d𝜏_𝑗∈Γ𝑈⁽²⁾𝑢𝜏+𝛽𝑚;Ω^∗_{()𝑥,𝑖,𝛾0()𝑥,𝑖},𝑢𝜏+𝛽𝑚^′;Ω^∗_{𝑥′,𝑖′}×𝟏𝑙_{(𝑥,𝑖),𝛾0(𝑥,𝑖)}𝜏+𝛽𝑚=𝑗=𝑙_{𝑥′,𝑖′}𝜏+𝛽𝑚^′+_{𝑗,𝑗′∈Γ×Γ}𝐽𝚍𝑗,𝑗^′𝑉𝑢_𝑖,𝑥𝜏+𝛽𝑚,𝑢_{𝑖′,𝑥′}𝜏+𝛽𝑚^′×𝟏𝑙_{(𝑥,𝑖),𝛾0(𝑥,𝑖)}𝜏+𝛽𝑚=𝑗≠𝑗^′=𝑙_{𝑥′,𝑖′}𝜏+𝛽𝑚^′. (90) Here, in turn, we employ the shortened notation for the positions and indices of the sections Ω^∗_{(𝑥,𝑖),𝛾0(𝑥,𝑖)}(𝜏+𝛽𝑚) and Ω^∗_{𝑥′,𝑖′}(𝜏+𝛽𝑚^′) of Ω^∗_{(𝑥,𝑖),𝛾0(𝑥,𝑖)} and Ω^∗_{𝑥′,𝑖′} at times 𝜏+𝛽𝑚 and 𝜏+𝛽𝑚^′, respectively: 𝑢_𝑖,𝑥𝜏+𝛽𝑚=𝑢𝜏+𝛽𝑚;Ω^∗_{(𝑥,𝑖),𝛾0(𝑥,𝑖)},𝑙_𝑖,𝑥𝜏+𝛽𝑚=𝑙𝜏+𝛽𝑚;Ω^∗_{(𝑥,𝑖),𝛾0(𝑥,𝑖)},𝑢_{𝑖′,𝑥′}𝜏+𝛽𝑚^′=𝑢𝜏+𝛽𝑚^′;Ω^∗_{𝑥′,𝑖′},𝑙_{𝑖′,𝑥′}𝜏+𝛽𝑚^′=𝑙𝜏+𝛽𝑚^′;Ω^∗_{𝑥′,𝑖′}. (91)

Further, the functional 𝐡^Λ0(𝛀^∗0∣𝐱^∗_Γ⧵Λ) in (82) is determined as in (71)–(73), with Ω^∗_{(𝑥,𝑖),𝛾0(𝑥,𝑖)} instead of Ω^∗_𝑥,𝑖. Next, for 𝐡^Λ⧵Λ0(𝛀^∗_Λ⧵Λ0∣𝛀^∗0∨𝐱^∗_Γ⧵Λ) in (83), we set 𝐡^Λ⧵Λ0𝛀^∗_Λ⧵Λ0∣𝛀^∗0∨𝐱^∗_Γ⧵Λ=𝐡^Λ⧵Λ0𝛀^∗_Λ⧵Λ0+𝐡𝛀^∗_Λ⧵Λ0∣∣𝛀^∗0∨𝐱^∗_Γ⧵Λ. (92) Here again, the summand 𝐡^Λ⧵Λ0(𝛀^∗_Λ⧵Λ0) is determined as in (67). Next, the term 𝐡^Λ⧵Λ0(𝛀^∗_Λ⧵Λ0∣∣𝛀^∗0∨𝐱^∗_Γ⧵Λ) is defined similarly to (72)-(73): 𝐡^Λ⧵Λ0𝛀^∗_Λ⧵Λ0∣∣𝛀^∗0∨𝐱^∗_Γ⧵Λ=_{(𝑖,𝑖′)∈Λ⧵Λ0×Λ0𝑥∈𝐱∗(𝑖),𝑥′∈𝐱∗(𝑖′)}𝐡^{(𝑥,𝑖),(𝑥′,𝑖′)}Ω^∗_𝑥,𝑖,Ω^∗_𝑥′,𝑖^′+_{𝑖∈Λ⧵Λ0𝑥∈𝐱∗(𝑖),𝑥′∈𝐱∗(𝑖′)}𝐡^{(𝑥,𝑖),(𝑥′,𝑖′)}Ω^∗_𝑥,𝑖,𝑥^′,𝑖^′ (93) with 𝐡^{(𝑥,𝑖),(𝑥′,𝑖′)}Ω^∗_𝑥,𝑖,Ω^∗_𝑥′,𝑖^′=_{0≤𝑚<𝑘𝑥,𝑖0≤𝑚′<𝑘′′0′′)(𝑥,𝑖),𝛾(𝑥,𝑖}×^𝛽₀×d𝜏_𝑗∈Γ𝑈⁽²⁾𝑢_𝑥,𝑖𝜏+𝛽𝑚,𝑢_{𝑥′,𝑖′}𝜏+𝛽𝑚^′×𝟏𝑙_𝑥,𝑖𝜏+𝛽𝑚=𝑗=𝑙_{𝑥′,𝑖′}𝜏+𝛽𝑚^′+_{𝑗,𝑗′∈Γ×Γ}𝐽𝚍𝑗,𝑗^′𝑉×𝑢_𝑖,𝑥𝜏+𝛽𝑚,𝑢_{𝑥′,𝑖′}𝜏+𝛽𝑚^′×𝟏𝑙_𝑥,𝑖𝜏+𝛽𝑚=𝑗≠𝑗^′=𝑙_{𝑥′,𝑖′}𝜏+𝛽𝑚^′,𝐡^{(𝑥,𝑖),(𝑥′,𝑖′)}Ω^∗_𝑥,𝑖,𝑥^′,𝑖^′=_{0≤𝑚<𝑘𝑥,𝑖}^𝛽₀d𝜏𝑈⁽²⁾𝑢_𝑥,𝑖𝜏+𝛽𝑚,𝑥^′×𝟏𝑙_𝑥,𝑖𝜏+𝛽𝑚=𝑖^′+_𝑗∈Γ𝐽𝚍𝑗,𝑖^′𝑉𝑢_𝑖,𝑥𝜏+𝛽𝑚,𝑥^′×𝟏𝑙_𝑥,𝑖𝜏+𝛽𝑚=𝑗. (94)

As before, the functionals 𝐡^Λ(𝛀^∗_Λ) and 𝐡^Λ(𝛀^∗_Λ∣∣𝐱^∗_Γ⧵Λ) have a natural interpretation as energies of loop configurations.

Repeating the above observation, non-zero contributions to the integral in (80) come only from pairs (𝛀^∗0,𝛀^∗_Λ⧵Λ0) such that ∀𝑗∈Γ and 𝜏∈[0,𝛽], the total number of pairs (𝑢(𝜏+𝛽𝑚;Ω^∗_{(𝑥,𝑖),𝛾0(𝑥,𝑖)}),𝑙(𝜏+𝛽𝑚;Ω^∗_{(𝑥,𝑖),𝛾0(𝑥,𝑖)})) with 0≤𝑚<𝑘_{(𝑥,𝑖),𝛾0(𝑥,𝑖)}, 𝑖∈Λ⁰ and 𝑥∈𝐱^∗(𝑖) incident to the paths of the configuration 𝛀^∗0 and pairs (𝑢(𝜏+𝛽𝑚^′;Ω^∗_{𝑥′,𝑖′}),𝑙(𝜏+𝛽𝑚^′;Ω^∗_{𝑥′,𝑖′})) with 0≤𝑚^′<𝑘_{(𝑥′,𝑖′)}, 𝑖^′∈Λ⧵Λ⁰ and 𝑥^′∈𝐱^∗(𝑖^′) incident to the loops of the configuration 𝛀^∗_Λ⧵Λ⁰ does not exceed 𝜅. Similarly, non-zero contributions to the integral in (82) come only from pairs (𝛀^∗0,𝛀^∗_Λ⧵Λ0) such that the above inequality holds when we additionally count points 𝑥∈𝐱^∗(𝑗).

The integral Ξ^Λ0_Λ(Λ⧵Λ⁰∣𝛀^∗0) defined in (81) can be considered as a particular (although important) example of a partition function in the volume Λ⧵Λ⁰ with a boundary condition 𝛀^∗0. Note the presence of the subscript Λ indicating that the loops contributing to Ξ^Λ0_Λ(Λ⧵Λ⁰∣𝛀^∗0) can jump within volume Λ only (owing to the indicator functional 𝛼_Λ). On the other hand, the presence of the indicator functional 𝟏(𝛀^∗_Λ⧵Λ0∈ℱ^Λ0) in the integral (reflected in the upperscript Λ⁰ and the roof sign in the notation Ξ^Λ0_Λ(Λ⧵Λ⁰∣𝛀^∗0)) indicates a particular restriction on the jumps of the loops, forbidding them to visit set Λ⁰ at intermediate times 𝛽𝑚. This is true also for the integral Ξ^Λ0_Λ(Λ⧵Λ⁰∣𝛀^∗0∨𝐱^∗_Γ⧵Λ) in (83): it is a particular example of a partition function in the volume Λ⧵Λ⁰ with a boundary condition 𝛀^∗0∨𝐱^∗_Γ⧵Λ.

Other useful types of partition functions are Ξ_Γ⁰(Λ∣𝛀^∗0∨𝛀^∗_Γ1) andΞ_Γ⁰(Λ∣𝛀^∗0∨𝛀^∗_Γ1∨𝐱^∗_Γ2) where the sets of vertices Λ, Λ⁰, Γ⁰, Γ¹, and Γ² satisfy Λ⊂Γ⁰⊆Γ,Γ¹,Γ²Γ⊂Γ⧵Λ,₁∩Γ²=∅,Λ⁰⊂Γ⧵Λ∪Γ¹∪Γ² (95) and ♯Λ,♯Λ⁰<+∞. Accordingly, 𝛀^∗0 is a (finite) configuration over Λ⁰, 𝛀^∗_Γ1 a (possibly infinite) loop configuration over Γ¹, and 𝐱^∗_Γ2 a (possibly infinite) particle configuration over Γ². The partition functions Ξ_Γ⁰(Λ∣𝛀^∗0∨𝛀^∗_Γ1) and Ξ_Γ⁰(Λ∣𝛀^∗0∨𝛀^∗_Γ1∨𝐱^∗_Γ2) are given by Ξ_Γ⁰Λ∣𝛀^∗0∨𝛀^∗_Γ1=_𝑊∗Λd𝛀^∗Λ𝛼_Γ⁰𝛀^∗Λ𝐵𝛀^∗Λ×exp−𝐡Λ𝛀^∗Λ∣𝛀^∗0∨𝛀^∗_Γ1,Ξ_Γ⁰Λ∣𝛀^∗0∨𝛀^∗_Γ1∨𝐱^∗_Γ2=_𝑊∗Λd𝛀^∗Λ𝛼_Γ⁰𝛀^∗Λ𝐵𝛀^∗Λ×exp−𝐡Λ𝛀^∗Λ∣𝛀^∗0∨𝛀^∗_Γ1∨𝐱^∗_Γ2 (96)(97) with the indicator 𝛼_Γ⁰ as in (74). These partition functions, feature loop configurations 𝛀^∗𝜆 formed by loops Ω^∗_𝑥,𝑖, 𝑖∈Λ, which start and finish in Λ, are confined to Γ⁰ and move in a potential field generated by 𝛀^∗0∨𝛀^∗_Γ1, where 𝛀^∗0={Ω^∗_{(𝑥,𝑖),𝛾0(𝑥,𝑖)}} and 𝛀_Γ1={Ω^∗_𝑥,𝑖,𝑥∈𝐱^∗(𝑖),𝑖∈Γ¹} or 𝛀^∗_Γ1∨𝐱^∗_Γ2, where 𝐱^∗_Γ2={𝐱^∗(𝑖),𝑖∈Γ²}. (The latter can be understood as the concatenation of the loop configuration 𝛀^∗_Γ1 over Γ¹ and the loop configuration over Γ² formed by the constant trajectories sitting at points 𝑥∈𝐱^∗(𝑖), 𝑖∈Γ².) In (96) we assume that, ∀𝜏∈[0,𝛽] and 𝑗∈Γ, the number ♯(𝑥,𝑖,𝑚)∶𝑖∈Λ,𝑙𝜏+𝑚𝛽;Ω^∗_𝑥,𝑖=𝑗,0≤𝑚<𝑘_𝑥,𝑖+♯(𝑥,𝑖)∶𝑖∈Λ⁰,𝑙𝜏+𝑚𝛽;Ω^∗_{(𝑥,𝑖),𝛾0(𝑥,𝑖)}=𝑗,0≤𝑚<𝑘_{(𝑥,𝑖),𝛾0(𝑥,𝑖)}+♯(𝑥,𝑖,𝑚)∶𝑖∈Γ¹,𝑙𝜏+𝑚𝛽;Ω^∗_𝑥,𝑖=𝑗,0≤𝑚<𝑘_𝑥,𝑖 (98) does not exceed 𝜅. Analogously, in (97) it is assumed that the same is true for the above number plus the cardinality ♯𝐱^∗(𝑗).

Such “modified” partition functions will be used in forthcoming sections.

2.4. The FK-DLR Measure 𝝁_Λ in a Finite Volume

The Gibbs states 𝝋_Λ and 𝝋_Λ∣𝐱^∗_Γ⧵Λ give rise to probability measures 𝝁_Λ and 𝝁_Λ∣𝐱^∗_Γ⧵Λ on the sigma algebra 𝖂_Λ of subsets of 𝑊^∗_Λ. The sigma algebra 𝖂_Λ is constructed by following the structure of the space 𝑊^∗_Λ (a disjoint union of Cartesian products); confer Definition 16. The measures 𝝁_Λ and 𝝁_Λ∣𝐱^∗_Γ⧵Λ are determined by their Radon-Nykodym derivative 𝑝_Λ and 𝑝_Λ∣𝐱^∗_Γ⧵Λ relative to the measure d𝛀^∗_Λ: 𝑝_Λ𝛀^∗_Λ𝝁∶=^Λd𝛀^∗_Λd𝛀^∗_Λ=1Ξ(Λ)𝛼_Λ𝛀^∗_Λ𝐵𝛀^∗_Λ×exp−𝐡^Λ𝛀^∗_Λ,𝛀^∗_Λ∈𝑊^∗_Λ,𝑝_Λ∣𝐱^∗_Γ⧵Λ𝛀^∗_Λ𝝁∶=^{Λ∣𝐱∗Γ⧵Λ}d𝛀^∗_Λd𝛀^∗_Λ1ΞΛ∣𝐱^∗_Γ⧵Λ=𝛼_Λ𝛀^∗_Λ𝐵𝛀^∗_Λ×exp−𝐡^Λ𝛀^∗_Λ∣𝐱^∗_Γ⧵Λ,𝛀^∗_Λ∈𝑊^∗_Λ. (99) Given Λ⁰⊂Λ, the sigma algebra 𝖂_Λ⁰ is naturally identified with a sigma subalgebra of 𝖂_Λ. The restrictions of 𝝁_Λ to 𝖂_Λ⁰ and 𝝁_Λ∣𝐱^∗_Γ⧵Λ are denoted by 𝝁^Λ0_Λ and 𝝁^Λ0_Λ∣𝐱^∗_Γ⧵Λ; these measures are determined by their Radon-Nikodym derivatives 𝑝^Λ0_Λ(𝛀^∗_Λ0)∶=𝝁^Λ0_Λ(d𝛀^∗_Λ0)/d𝛀^∗_Λ0 and 𝑝^Λ0_Λ∣𝐱^∗_Γ⧵Λ(𝛀^∗_Λ0)∶=𝝁^Λ0_Λ∣𝐱^∗_Γ⧵Λ(d𝛀^∗_Λ0)/d𝛀^∗_Λ0.

The first key property of the measures 𝝁_Λ and 𝝁_Λ∣𝐱^∗_Γ⧵Λ is expressed in the so-called FK-DLR equation. We state it as Lemma 21 below; its proof repeats a standard argument used in the classical case for establishing the DLR equation in a finite volume Λ⊂Γ.

Lemma 21. For all 𝑧,𝛽>0 satisfying (22), and Λ⁰⊂Λ^′⊂Λ, the probability density 𝑝^Λ0_Λ admits the form: 𝑝^Λ0_Λ𝛀^∗_Λ0=_{𝑊∗′Λ⧵Λ}𝑞^Λ0_Λ⧵Λ′𝛀^∗_Λ0∣𝛀^∗_Λ⧵Λ′𝝁^Λ⧵Λ′_Λd𝛀^∗_Λ⧵Λ′, (100) where 𝑞^Λ0_Λ⧵Λ′𝛀^∗_Λ0∣𝛀^∗_Λ⧵Λ′=exp−𝐡^Λ0𝛀^∗_Λ0∣𝛀^∗_Λ⧵Λ′×Ξ_ΛΛ^′⧵Λ⁰∣𝛀^∗_Λ0∨𝛀^∗_Λ⧵Λ′Ξ_ΛΛ^′∣𝛀^∗_Λ⧵Λ′, (101) and the conditional partition functions Ξ_Λ(Λ^′⧵Λ⁰∣𝛀^∗∨𝛀^∗_Λ⧵Λ′) andΞ_Λ(Λ^′∣𝛀^∗_Λ⧵Λ′) are determined as in (96).
Similarly, for 𝑝^Λ0_Λ∣𝐱^∗_Γ⧵Λ one has: 𝑝^Λ0_Λ∣𝐱^∗_Γ⧵Λ𝛀^∗_Λ0=_{𝑊∗′Λ⧵Λ}𝑞^Λ0_Λ⧵Λ′𝛀^∗_Λ0∣𝛀^∗_Λ⧵Λ′∨𝐱^∗_Γ⧵Λ×𝝁^Λ⧵Λ′_{Λ∣𝐱∗Γ⧵Λ}d𝛀^∗_Λ⧵Λ′, (102) where 𝑞^Λ0_Λ⧵Λ′𝛀^∗_Λ0∣𝛀^∗_Λ⧵Λ′∨𝐱^∗_Γ⧵Λ=exp−𝐡^Λ0𝛀^∗_Λ0∣𝛀^∗_Λ⧵Λ′∨𝐱^∗_Γ⧵Λ×Ξ_ΛΛ^′⧵Λ⁰∣𝛀^∗_Λ0∨𝛀^∗_Λ⧵Λ′∨𝐱^∗_Γ⧵ΛΞ_ΛΛ^′∣𝛀^∗_Λ⧵Λ′∨𝐱^∗_Γ⧵Λ (103) and the conditional partition functions Ξ_Λ(Λ^′⧵Λ⁰∣𝛀^∗_Λ0∨𝛀^∗_Λ⧵Λ′∨𝐱^∗_Γ⧵Λ) andΞ_Λ(Λ^′∣𝛀^∗_Λ⧵Λ′∨𝐱^∗_Γ⧵Λ) are determined as in (96).

As in [1], (100) and (102) mean that the conditional densities 𝑝^Λ0_Λ(𝛀^∗_Λ0∣𝛀^∗_Λ⧵Λ′) and 𝑝^Λ0_Λ∣𝐱^∗_Γ⧵Λ(𝛀^∗_Λ0∣𝛀^∗_Λ⧵Λ′) relative to 𝜎-algebra 𝖂^Λ⧵Λ′ coincide, respectively, with 𝑞^Λ0_Λ⧵Λ′(𝛀^∗0∣𝛀^∗_Λ⧵Λ′), and 𝑞^Λ0_Λ⧵Λ′(𝛀^∗0∣𝛀^∗_Λ⧵Λ′∨𝐱^∗_Γ⧵Λ), for 𝝁^Λ⧵Λ′_Λ—and 𝝁^Λ⧵Λ′_{Λ∣𝐱∗Γ⧵Λ}—a.a. 𝛀^∗_Λ⧵Λ′∈𝑊^∗_Λ⧵Λ′ and a.a. 𝛀^∗_Λ0∈𝑊^∗_Λ0.

As in [1], we call the expressions 𝑞^Λ0_Λ⧵Λ′(𝛀^∗_Λ0∣𝛀^∗_Λ⧵Λ′) and 𝑞^Λ0_Λ⧵Λ′(𝛀^∗_Λ0∣𝛀^∗_Λ⧵Λ′∨𝐱^∗_Γ⧵Λ), as well as the expressions 𝑞^Λ0_Λ⧵Λ′(𝛀^∗0∣𝛀^∗_Λ⧵Λ′) and 𝑞^Λ0_Λ⧵Λ′(𝛀^∗0∣𝛀^∗_Λ⧵Λ′∨𝐱^∗_Γ⧵Λ) appearing below, the (conditional) RDM functionals (in brief, the RDMFs). The same name will be used for the quantity 𝑞^Λ0_Λ⧵Λ′(𝛀^∗_Λ0∣𝛀^∗_Λ⧵Λ′) from (110)-(111) and the quantity 𝑞^Λ0_Γ⧵Λ(𝛀^∗0∣𝛀^∗_Γ⧵Λ) from (112)-(113).

The second property is that the RDMKs 𝐅^Λ0_Λ(𝐱^∗0,𝐲^∗0) and 𝐅^Λ0_Λ∣𝐱^∗_Γ⧵Λ(𝐱^∗0,𝐲^∗0) are related to the measures 𝝁_Λ and 𝝁_Λ∣𝐱^∗_′Γ⧵Λ. Again, the proof of this fact is done by inspection.

Lemma 22. The RDMK 𝐅^Λ0_Λ(𝐱^∗0,𝐲^∗0) is expressed as follows: ∀Λ⁰⊂Λ^′⊂Λ, 𝐅^Λ0_Λ𝐱^∗0,𝐲^∗0=_{𝑊∗𝐱00,𝐲}ℙ^∗_𝐱0,𝐲0d𝛀^∗0𝛼_Λ𝛀^∗0×𝐵𝛀^∗0𝟏𝛀^∗0∈ℱ^Λ0×_{𝑊∗′Λ⧵Λ}𝝁^Λ⧵Λ′d𝛀^∗_Λ⧵Λ′𝟏𝛀^∗_Λ⧵Λ′∈ℱ^Λ0×𝑞^Λ0_Λ⧵Λ′𝛀^∗0∣𝛀^∗_Λ⧵Λ′, (104) where 𝑞^Λ0_Λ⧵Λ′𝛀^∗0∣𝛀^∗_Λ⧵Λ′=exp−𝐡^Λ0𝛀^∗0∣𝛀^∗_Λ⧵Λ′×Ξ^Λ0_ΛΛ^′⧵Λ⁰∣𝛀^∗0∨𝛀^∗_Λ⧵Λ′Ξ_ΛΛ^′∣𝛀^∗_Λ⧵Λ′. (105)
Similarly, 𝐅^Λ0_Λ∣𝐱^∗_Γ⧵Λ𝐱^∗0,𝐲^∗0=_{𝑊∗𝐱00,𝐲}ℙ^∗_𝐱0,𝐲0d𝛀^∗0𝛼_Λ𝛀^∗0×𝐵𝛀^∗0𝟏𝛀^∗0∈ℱ^Λ0×_{𝑊∗′Λ⧵Λ}𝝁^Λ⧵Λ′_{Λ∣𝐱∗Γ⧵Λ}d𝛀^∗_Λ⧵Λ′𝟏𝛀^∗_Λ⧵Λ′∈ℱ^Λ0×𝑞^Λ0_Λ⧵Λ′𝛀^∗0∣𝛀^∗_Λ⧵Λ′∨𝐱^∗_Γ⧵Λ, (106) where 𝑞^Λ0_Λ⧵Λ′𝛀^∗0∣𝛀^∗_Λ⧵Λ′𝐱^∗_Γ⧵Λ=exp−𝐡^Λ0𝛀^∗0∣𝛀^∗_Λ⧵Λ′∨𝐱^∗_Γ⧵Λ×Ξ^Λ0_ΛΛ^′⧵Λ⁰∣𝛀^∗0∨𝛀^∗_Λ⧵Λ′∨𝐱^∗_Γ⧵ΛΞ_ΛΛ^′∣𝛀^∗_Λ⧵Λ′∨𝐱^∗_Γ⧵Λ. (107) Here the partition functions Ξ^Λ0_Λ(Λ^′⧵Λ⁰∣𝛀^∗0∨𝛀^∗_Λ⧵Λ′) and Ξ^Λ0_Λ(Λ^′⧵Λ⁰∣𝛀^∗0∨𝛀^∗_Λ⧵Λ′∨𝐱^∗_Γ⧵Λ) are determined in (81) and (83).

Remark 23. Summarizing the above observations, the measures 𝝁_Λ and 𝝁_Λ∣𝐱^∗_Γ⧵Λ are concentrated on the subset in 𝑊^∗_Λ formed by loop configurations 𝛀^∗_Λ such that ∀𝜏∈[0,𝛽], the section 𝛀^∗_Λ(𝜏) has ≤κ particles at each vertex 𝑖∈Λ.

3. The Class of Gibbs States 𝔊 for the Fock Space Model

3.1. Definition of Class 𝔊

In this section we apply the idea from [1] to define the class of states 𝕲=𝕲_𝑧,𝛽 for the model introduced in Section 2 and state a number of results. These results will hold under condition (34) which is assumed from now on. As in [1], the definition of a state 𝝋∈𝕲 is based on the notion of an FK-DLR probability measure 𝝁 on the space 𝑊^∗_Γ; the class of these measures will be also denoted by 𝕲.

Definition 24. Space 𝑊^∗_Γ is the (infinite) Cartesian product ×_𝑖∈Γ𝑊^∗_{𝑖} (cf. (60)); its elements are loop configurations 𝛀^∗_Γ={𝛀^∗(𝑖),𝑖∈Γ} over Γ. A component 𝛀^∗(𝑖) is a finite loop configuration (possibly, empty), with an initial/final particle configuration 𝐱^∗(𝑖)⊂𝑀. Formally, 𝛀^∗(𝑖) is a finite collection of loops Ω^∗_𝑥,𝑖, of time-length 𝛽𝑘_𝑥,𝑖 where 𝑘_𝑥,𝑖=1,2,…, starting and finishing at a point (𝑥,𝑖)∈𝑀×Γ. For reader’s convenience, we repeat (37) for the case under consideration: Ω^∗_𝑥,𝑖∶𝜏∈0,𝛽𝑘_𝑥,𝑖⟼𝑥Ω^∗_𝑥,𝑖,𝜏,𝑖̃Ω^∗_𝑥,𝑖Ω,𝜏∈𝑀×Γ,_∗𝑥,𝑖iscad́laǵ;Ω^∗_𝑥,𝑖(0)=Ω^∗_𝑥,𝑖𝛽𝑘_𝑥,𝑖−=(),Ω𝑥,𝑖_∗𝑥,𝑖hasfinitelymanyjumpson0,𝛽𝑘_𝑥,𝑖;𝚍𝑖̃Ωifajumpoccursattime𝜏,then_∗𝑥,𝑖,𝜏−,𝑖̃Ω^∗_𝑥,𝑖,𝜏=1. (108)
By 𝖂=𝖂_Γ we denote the 𝜎-algebra in 𝑊^∗_Γ generated by cylindrical events. Given a subset Γ⊂Γ (finite or infinite), we denote by 𝖂^Γ=𝖂^Γ_Γ the 𝜎-subalgebra of 𝖂 generated by cylindrical events localized in Γ. Given a probability measure 𝝁=𝝁_Γ on (𝑊^∗_Γ,𝖂_Γ), we denote by 𝝁^Γ=𝝁^Γ_Γ the restriction of 𝝁 on 𝖂^Γ.

Definition 25. The class 𝕲 under consideration is formed by measures 𝝁 which satisfy the following equation: ∀ finite Λ⊂Γ and Λ⁰⊆Λ, the probability density: 𝑝^Λ0𝛀^∗0=𝑝^Λ0_𝝁𝛀^∗0𝝁∶=^Λ0Γd𝛀^∗0𝝂d𝛀^∗0,𝛀^∗0∈𝑊^∗_Λ0, (109) is of the form 𝑝^Λ0𝛀^∗_Λ0=_{𝑊∗Γ⧵Λ}𝑞^Λ0_Γ⧵Λ𝛀^∗0∣𝛀^∗_Γ⧵Λ𝝁^Γ⧵Λd𝛀^∗_Γ⧵Λ, (110) where 𝑞^Λ0_Γ⧵Λ𝛀^∗0∣𝛀^∗_Γ⧵Λ=exp−𝐡^Λ0𝛀^∗0∣𝛀^∗_Γ⧵Λ×Ξ_ΓΛ⧵Λ⁰∣𝛀^∗0∨𝛀^∗_Γ⧵ΛΞ_ΓΛ∣𝛀^∗_Γ⧵Λ, (111) and the conditional partition functions Ξ_Γ(Λ⧵Λ⁰∣𝛀^∗0∨𝛀^∗_Γ⧵Λ) andΞ_Γ(Λ∣𝛀^∗_Γ⧵Λ) are determined as in (96).

As in [1], (110) means that the conditional density 𝑝^{Λ0∣Γ⧵Λ}(𝛀^∗0∣𝛀^∗_Γ⧵Λ), relative to 𝜎-algebra 𝖂^Γ⧵Λ, coincides with 𝑞^Λ0_Γ⧵Λ(𝛀^∗0∣𝛀^∗_Γ⧵Λ), for 𝝁_Γ^Γ⧵Λ—a.a.𝛀^∗_Γ⧵Λ∈𝑊^∗𝛽_Γ⧵Λ and 𝝂_Λ⁰—a.a. 𝛀^∗0∈𝑊^∗_Λ0.

Remark 26. The measure 𝝁_Γ inherits the property from Remark 23 and is concentrated on the subset in 𝑊^∗_Γ formed by (infinite) loop configurations 𝛀^∗_Γ such that, for all 𝜏∈[0,𝛽], the section 𝛀^∗_Λ(𝜏) has ≤κ particles at each vertex 𝑖∈Λ.

Given a measure 𝝁∈𝕲, we associate with it a normalized linear functional 𝝋=𝝋_𝝁 on the quasilocal 𝐶^∗-algebra 𝕭. First, we set 𝐅^Λ0𝐱^∗0,𝐲^∗0=_{𝑊∗𝐱00,𝐲}ℙ^∗_𝐱0,𝐲0d𝛀^∗0𝐵𝛀^∗0𝟏𝛀^∗0∈ℱ^Λ0×_{𝑊∗Γ⧵Λ}𝝁^Γ⧵Λd𝛀^∗_Γ⧵Λ𝟏𝛀^∗_Γ⧵Λ∈ℱ^Λ0×𝑞^Λ0_Γ⧵Λ𝛀^∗0∣𝛀^∗_Γ⧵Λ, (112) where 𝑞^Λ0_Γ⧵Λ𝛀^∗0∣𝛀^∗_Γ⧵Λ=exp−𝐡^Λ0𝛀^∗0∣𝛀^∗_Γ⧵Λ×Ξ^Λ0_ΛΛ⧵Λ⁰∣𝛀^∗0∨𝛀^∗_Γ⧵ΛΞ_ΓΛ∣𝛀^∗_Γ⧵Λ. (113)

This defines a kernel 𝐅^Λ0(𝐱^∗0,𝐲^∗0), 𝐱^∗0,𝐲^∗0∈𝑀^∗Λ0, where Λ⁰⊂Γ is a finite set of sites. It is worth reminding the reader of the presence of the indicator functionals 𝟏(⋅∈ℱ^Λ0) in (112) and (113) (in the integral for Ξ^Λ0_Λ(Λ⧵Λ⁰∣𝛀^∗0∨𝛀^∗_Γ⧵Λ)). These indicators guarantee the compatibility property: ∀ finite Λ⁰⊂Λ¹, 𝐅^Λ0𝐱^∗0,𝐲^∗0=_{𝑀10∗Λ⧵Λ𝑗∈Λ1⧵Λ0𝑧∈𝐳∗(𝑗)}𝑣(d𝑧)×𝐅^Λ1𝐱^∗0∨𝐳^∗_Λ1⧵Λ0,𝐲^∗0∨𝐳^∗_Λ1⧵Λ0. (114)

Next, we identify the operator 𝐑^Λ0 (a candidate for the RDM in volume Λ⁰) as an integral operator acting in 𝓗_Λ⁰ by 𝐑^Λ0𝝓𝐱^∗0=_𝑀0∗Λ𝐅^Λ0𝐱^∗0,𝐲^∗0𝝓𝐲^∗0d𝐲^∗0. (115)

Equation (114) implies that tr_{𝓗Λ10⧵Λ}𝐑^Λ1=𝐑^Λ0. (116)

Definition 27. The functional 𝝋∈𝕲 is identified with the (compatible) family of operators 𝐑^Λ0. If the operators 𝐑^Λ0 are positive definite (a property that is not claimed to be automatically fulfilled), we again call it an FK-DLR state in the infinite volume (for given values of activity 𝑧 and inverse temperature 𝛽). To stress the dependence on 𝑧 and 𝛽, we sometimes employ the notation 𝕲(𝑧,𝛽).

3.2. Theorems on Existence and Properties of FK-DLR States

We are now in position to state results about class 𝕲. We assume the conditions on the potentials 𝑈⁽¹⁾ and 𝑈⁽²⁾ from the previous section, including the hard-core condition for 𝑈⁽¹⁾.

Theorem 28. For all 𝑧,𝛽∈(0,+∞) satisfying (22), any limiting Gibbs state 𝝋∈𝕲⁰ (see Theorem 3) lies in 𝕲. Therefore, the class of state 𝕲 s is nonempty.

Theorem 29. Under condition (22), any FK-DLR state 𝝋∈𝕲 is 𝙶-invariant, in the sense that, ∀ finite Λ⁰⊂Γ and ∀𝚐∈𝙶, the RDM 𝐑^Λ0 satisfies (35). Consequently, (36) holds true.

4. Proof of Theorems 3, 6, 28, and 29

4.1. Proof of Theorems 3 and 28

The proof is based on the same approach as that used in [1]. First, given Λ⁰⊂Γ, we establish compactness of the sequence of the RDMKs 𝐅^Λ0_Λ(𝐱^∗0,𝐲^∗0) and 𝐅^Λ0_Λ∣𝐱^∗_Γ⧵Λ(𝐱^∗0,𝐲^∗0) (see (78)–(83)) as functions of variables 𝐱^∗0={𝐱^∗0(𝑖)},𝐲^∗0={𝐲^∗0(𝑖)}∈𝑀^∗Λ0, with ♯𝐱_Λ^∗0=♯𝐲_Λ^∗0,♯𝐱^∗0(𝑖),♯𝐲^∗0(𝑖)<𝜅,𝑖∈Λ, (117) when Λ↗Γ. Then we use Lemma 1.1 from [1] to derive that the sequence of the RDMs 𝐑^Λ0_Λ and 𝐑^Λ0_Λ∣𝐱^∗_Γ⧵Λ is compact in the trace-norm operator topology in ℋ_Λ⁰.

To verify compactness of the RDMKs 𝐅^Λ0_Λ(𝐱^∗0,𝐲^∗0) and 𝐅^Λ0_Λ∣𝐱^∗_Γ⧵Λ(𝐱^∗0,𝐲^∗0) we, again as in [1], use the Ascoli-Arzela theorem, which requires the properties of uniform boundedness and equicontinuity. These properties follow from the following.

Lemma 30. (i) Under condition (22) the RDMKs 𝐅^Λ0_Λ(𝐱^∗0,𝐲^∗0) and 𝐅^Λ0_Λ∣𝐱^∗_Γ⧵Λ(𝐱^∗0,𝐲^∗0) admit the bounds 𝐅^Λ0_Λ𝐱^∗0,𝐲^∗0,𝐅^Λ0_Λ∣𝐱^∗_Γ⧵Λ𝐱^∗0,𝐲^∗0≤𝜅♯Λ⁰!𝑝_𝑀^𝜅♯Λ0Φ^♯Λ0, (118) where Φ=_𝑘≥1𝑧^𝑘exp(𝑘Θ),withΘ=𝜅𝛽𝑈⁽¹⁾+𝜅𝑈⁽²⁾+𝜅𝐽(1)𝑉. (119) (Note that Φ<∞ under the assumption (22)) Let 𝑝_𝑀^(𝑘) yields the supremum of the transition function over time 𝑘𝛽 for Brownian motion on the torus 𝑀: 𝑝_𝑀^(𝑘)=sup_{𝑥,𝑦∈𝑀}𝑝^𝑘𝛽𝑥,𝑦=𝑝^𝑘𝛽(0,0), (120) and 𝑝_𝑀=sup_𝑘≥1𝑝_𝑀^(𝑘). Finally, the upper-bound values 𝑈⁽¹⁾, 𝑈⁽²⁾, 𝐽(1), and 𝑉 have been determined in (14), (15), and (18).
(ii) The gradients of the RDMKs 𝐅^Λ0_Λ(𝐱^∗0,𝐲^∗0) and 𝐅^Λ0_Λ∣𝐱^∗_Γ⧵Λ(𝐱^∗0,𝐲^∗0) satisfy ∀𝑖∈Λ and 𝑥∈𝐱^∗0(𝑖), 𝑦∈𝐲^∗0(𝑖), ∇_𝑥𝐅^Λ0_Λ𝐱^∗0,𝐲^∗0,∇_𝑥𝐅^Λ0_Λ∣𝐱^∗_Γ⧵Λ𝐱⁰,𝐲⁰∇_𝑦𝐅^Λ0_Λ𝐱^∗0,𝐲^∗0,∇_𝑦𝐅^Λ0_Λ∣𝐱^∗_Γ⧵Λ𝐱⁰,𝐲⁰≤♯Λ⁰ΘΦ^′𝑝_𝑀𝜅♯Λ⁰!𝑝_𝑀^𝜅♯Λ0Φ^♯Λ0, (121) where Φ^′=_𝑘≥1𝑘𝑧^𝑘exp(𝑘Θ). (122) (Again, Φ^′<∞ under the condition (22).) The ingredients 𝑝_𝑀 and 𝑈⁽¹⁾, 𝑈⁽²⁾, 𝐽(1), and 𝑉 as in statement (i).

Proof of Lemma 30. First, observe that 𝑝_𝑀<∞ on a compact manifold. Bound (119) is established in a direct fashion. First, we majorize the energy 𝐡^Λ0𝛀^∗0∣𝛀^∗_Λ⧵Λ0=𝐡^Λ0𝛀^∗0+𝐡𝛀^∗_Λ⧵Λ0∣∣𝛀^∗0 (123) contributing to the RHS in (80) and (81) and the energy 𝐡^Λ0𝛀^∗0∣𝛀^∗_Λ⧵Λ0∨𝐱^∗_Γ⧵Λ=𝐡^Λ0𝛀^∗0∣𝐱^∗_Γ⧵Λ+𝐡𝛀^∗_Λ⧵Λ0∣∣𝛀^∗0 (124) contributing to the RHS in (82) and (83). This yields the factor _{𝜔∈𝛀∗0}𝑧^{𝑘(𝜔∗)}exp𝑘𝜔^∗Θ. (125) Next, we majorize the integral =∫_𝑊^∗_{𝐱∗0,𝐲∗0}ℙ^∗_{𝐱∗0,𝐲∗0}(d𝛀^∗0) in (80) and (82); this gives the factor ♯Λ⁰𝑝_𝑀𝜅♯Λ⁰!𝑝_𝑀^𝜅♯Λ0Φ^♯Λ0. (126) The aftermath are the ratios (78) and (79) with 𝐱^∗0=𝐲^∗0=∅; they do not exceed 1.
Passing to (121), let us discuss the gradients ∇_𝑥 only. (The gradients in the entries of ∇_𝑦 are included by symmetry.) The gradient in (121), of course, affects only the numerators Ξ^Λ0_Λ(𝐱^∗0,𝐲^∗0;Λ⧵Λ⁰) and Ξ^Λ0_Λ(𝐱^∗0,𝐲^∗0;Λ⧵Λ⁰∣𝐱^∗_Γ⧵Λ) in (78) and (79). The bounds (121) are done essentially as in [1]. For definiteness, we discuss the case of the RDMK 𝐅^Λ0_Λ(𝐱^∗0,𝐲^∗0); the RDMK 𝐅^Λ0_Λ∣𝐱^∗_Γ⧵Λ(𝐱⁰,𝐲⁰) is treated similarly. There are two contributions into the gradient: one comes from varying the measure ℙ^∗_{𝐱∗0,𝐲∗0}(d𝛀^∗0), and the other from varying the functional exp[−𝐡^Λ0(𝛀^∗0)−𝐡^Λ⧵Λ0(𝛀^∗_Λ⧵Λ0∣𝛀^∗0)].
The first contribution can again be uniformly bounded in terms of the constant 𝑝_𝑀. The detailed argument, as in [1], includes a deformation of a trajectory and is done similarly to [1] (the presence of jumps does not change the argument because 𝑝_𝑀 yields a uniform bound in (43)).
The second contribution yields, again as in [1], an expression of the form: _{𝑊∗𝐱∗0,𝐲∗0}ℙ^∗_{𝐱∗0,𝐲∗0}d𝛀^∗0×_{𝑖∈Λ0𝑥∈𝐱∗0(𝑖)}𝐡_𝑥,𝑖Ω^∗_{(𝑥,𝑖),𝛾0(𝑥,𝑖)},𝛀^∗_Λ⧵Λ0×exp−𝐡^Λ0𝛀^∗0−𝐡^Λ⧵Λ0𝛀^∗_Λ⧵Λ0∣𝛀^∗0, (127) where the functional 𝐡_𝑥,𝑖(Ω^∗_{(𝑥,𝑖),𝛾0(𝑥,𝑖)},𝛀^∗_Λ⧵Λ0) is uniformly bounded. Combining this an upper bound similar to (119) yields the desired estimate for the gradients in (121).

Hence, we can guarantee that the RDMs 𝐑^Λ0_Λ and 𝐑^Λ0_Λ∣𝐱^∗_Γ⧵Λ converge to a limiting RDM 𝐑^Λ0 along a subsequence in Λ↗Γ. The diagonal process yields convergence for every finite Λ⁰⊂Γ. A parallel argument leads to compactness of the measures 𝜇^Λ0_Λ for any given Λ⁰ as Λ↗Γ. We only give here a sketch of the corresponding argument, stressing differences with its counterpart in [1].

In the probabilistic terminology, measures 𝜇_Λ represent random marked point fields on 𝑀×Γ with marks from the space 𝑊^∗=𝑊^∗₀ where 𝑊^∗₀=⋃_𝑘≥1𝑊₀^𝑘𝛽 and 𝑊₀^𝑘𝛽 is the space of loops of time-length 𝑘𝛽 starting and finishing at 0∈𝑀 and exhibiting jumps, that is, changes of the index. (The space 𝑊^∗_𝑥,𝑖 introduced in Definitions 7 and 13 can be considered as a copy of 𝑊^∗ placed at site 𝑖∈Γ and point 𝑥∈𝑀.) The measure 𝜇^Λ0_Λ describes the restriction of 𝜇_Λ to volume Λ⁰ (i.e., to the sigma algebra 𝔚^∗_Λ0) and is given by its Radon-Nikodym derivative 𝑝^Λ0_Λ relative to the reference measure d𝛀^∗_Λ0 on 𝑊^∗_Λ0 (cf. (66), (100)). The reference measure is sigma-finite. Moreover, under the condition (22), the value 𝑝^Λ0_Λ(𝛀^∗_Λ0) is uniformly bounded (in both Λ↗Γ and 𝛀^∗_Λ0∈𝑊^∗_Λ0). This enables us to verify tightness of the family of measures {𝜇^Λ0_Λ,Λ↗Γ} and apply the Prokhorov theorem. Next, we use the compatibility property of the limit-point measures 𝜇^Λ0_Γ and apply the Kolmogorov theorem. This establishes the existence of the limit-point measure 𝜇_Γ.

By construction, and owing to Lemmas 21 and 22, the limiting family {𝐑^Λ0} yields a state belonging to the class 𝕲. This completes the proof of Theorems 3 and 28.

4.2. Proof of Theorems 6 and 29

The assertion of Theorem 6 is included in Theorem 29. Therefore, we will focus on the proof of the latter. The proof based on the analysis of the conditional RDMFs 𝑞^Λ0_Γ⧵Λ(𝛀^∗0∣𝛀^∗_Γ⧵Λ) and 𝑞^Λ0_Γ⧵Λ(𝛀^∗0∣𝛀^∗_Γ⧵Λ) introduced in (111) and (112). For definiteness, we assume that vertex 𝑜∈Λ⁰, so that Λ⁰ lies in the ball Λ_𝑛 for 𝑛 large enough. As in [1], the problem is reduced to checking that ∀𝑧,𝛽∈(0,∞) satisfying (22), 𝚐∈𝙶 and finite Λ⁰⊂Γ, lim_𝑛→∞𝑞^Λ0_{Γ⧵Λ(𝑛)}𝚐𝛀^∗0∣𝛀^∗_{Γ⧵Λ(𝑛)}𝑞^Λ0_{Γ⧵Λ(𝑛)}𝛀^∗0∣𝛀^∗_{Γ⧵Λ(𝑛)}=1; (128) here we need to establish this convergence (128) uniformly in the argument 𝛀^∗_{Γ⧵Λ(𝑛)}={𝛀^∗(𝑖),𝑖∈Γ⧵Λ(𝑛)} with ♯𝛀^∗(𝑖)≤𝜅 and in 𝛀^∗0 outside a set of the ℙ_𝐱^∗𝛽_0,𝐲0 measure tending to 0 as 𝑛→∞. The latter is formed by path configurations 𝛀^∗0 that contain trajectories visiting sites 𝑖∈Γ⧵Λ(𝑟(𝑛)) where 𝑟(𝑛) grows with 𝑛; see Lemma 31 below. The action of 𝚐 upon a path configuration 𝛀^∗0={Ω^∗_{(𝑥,𝑖),𝛾0(𝑥,𝑖)},𝑖∈Λ⁰,𝑥∈𝐱^∗0} is defined by 𝚐𝛀^∗0=𝚐Ω^∗_{(𝑥,𝑖),𝛾0(𝑥,𝑖)},where𝚐Ω^∗_{(𝑥,𝑖),𝛾0(𝑥,𝑖)}(𝜏)=𝚐Ω^∗_{(𝑥,𝑖),𝛾0(𝑥,𝑖)}(𝜏). (129)

We want to establish that ∀𝑎∈(1,∞), for any 𝑛 large enough, the conditional RDMFs satisfy 𝑎𝑞^Λ0_{Γ⧵Λ(𝑛)}𝚐𝛀^∗0∣𝛀^∗_{Γ⧵Λ(𝑛)}+𝑎𝑞^Λ0_{Γ⧵Λ(𝑛)}𝚐⁻¹𝛀^∗0∣𝛀^∗_{Γ⧵Λ(𝑛)}≥2𝑞^Λ0_{Γ⧵Λ(𝑛)}𝛀^∗0∣𝛀^∗_{Γ⧵Λ(𝑛)}. (130)

As in [1], we deduce (130) with the help of a special construction of “tuned” actions 𝚐_{Λ(𝑛)⧵Λ}⁰ on loop configurations 𝝎_{Λ(𝑛)⧵Λ}⁰ (over which there is integration performed in the numerators Ξ^Λ0_ΓΛ(𝑛)⧵Λ⁰∣𝚐𝛀^∗0∨𝛀^∗_{Γ⧵Λ(𝑛)},Ξ^Λ0_ΓΛ(𝑛)⧵Λ⁰∣𝚐⁻¹𝛀^∗0∨𝛀^∗_{Γ⧵Λ(𝑛)} (131) in the expression for 𝑞^Λ0_{Γ⧵Λ(𝑛)}(𝚐𝛀^∗0∣𝛀^∗_{Γ⧵Λ(𝑛)}) and 𝑞^Λ0_{Γ⧵Λ(𝑛)}(𝚐⁻¹𝛀^∗0∣𝛀^∗_{Γ⧵Λ(𝑛)})). The tuning in 𝚐_{Λ(𝑛)⧵Λ}⁰ is chosen so that it approaches 𝚎 (or the 𝑑-dimensional zero vector in the additive form of writing), the neutral element of 𝙶, while we move from Λ⁰ towards Γ⧵Λ(𝑛).

Formally, (130) follows from the estimate (132) below: ∀ finite Λ⁰⊂Γ,𝛀^∗0∈𝑊_Λ(𝑛), 𝚐∈𝙶 and 𝑎∈(1,∞), for any 𝑛 large enough, ∀𝛀^∗_{Λ(𝑛)⧵Λ0}={𝛀^∗(𝑖),𝑖∈Λ(𝑛)⧵Λ⁰} and 𝛀^∗_{Γ⧵Λ(𝑛)}={𝛀^∗(𝑖),𝑖∈Γ⧵Λ(𝑛)} with ♯𝛀^∗(𝑖)≤𝜅, 𝑎2exp−𝐡^Λ(𝑛)𝚐𝛀^∗0∨𝚐_{Λ(𝑛)⧵Λ}⁰𝛀^∗_{Λ(𝑛)⧵Λ0}∣𝛀^∗_{Γ⧵Λ(𝑛)}+𝑎2exp−𝐡^Λ(𝑛)×𝚐⁻¹𝛀^∗0∨𝚐⁻¹_{Λ(𝑛)⧵Λ0}𝛀^∗_{Λ(𝑛)⧵Λ0}∣𝛀^∗_{Γ⧵Λ(𝑛)}≥exp−𝐡^Λ(𝑛)𝛀^∗0∨𝛀^∗_{Λ(𝑛)⧵Λ0}∣𝛀^∗_{(𝑛)Γ⧵Λ}. (132)

In (132), the loop configuration 𝚐_{Λ(𝑛)⧵Λ}⁰𝛀^∗_{Λ(𝑛)⧵Λ0} is determined by specifying its temporal section {𝚐_{Λ(𝑛)⧵Λ}⁰Ω^∗_𝑥,𝑖(𝜏+𝛽𝑚),𝑖∈Λ(𝑛)⧵Λ⁰,𝑥∈𝐱^∗(𝑖),0≤𝑚<𝑘_𝑥,𝑖}. That is, we need to specify the sections: 𝑢𝜏+𝛽𝑚;𝚐_{Λ(𝑛)⧵Λ}⁰Ω^∗_𝑥,𝑖,𝑙𝜏+𝛽𝑚;𝚐_{Λ(𝑛)⧵Λ}⁰Ω^∗_𝑥,𝑖 (133) for loops Ω^∗_𝑥,𝑖 constituting 𝚐_{Λ(𝑛)⧵Λ}⁰𝛀^∗_{Λ(𝑛)⧵Λ0}. To this end we set 𝑢𝜏+𝛽𝑚;𝚐_{Λ(𝑛)⧵Λ}⁰Ω^∗_𝑥,𝑖=𝚐_𝑗^(𝑛)𝑢𝜏+𝛽𝑚;Ω^∗_𝑥,𝑖if𝑙𝜏+𝛽𝑚;𝚐_{Λ(𝑛)⧵Λ}⁰Ω^∗_𝑥,𝑖=𝑗. (134) In other words, we apply the action 𝚐_𝑗^(𝑛) to the temporal sections of all loops Ω^∗_𝑥,𝑖 located at vertex 𝑗 at a given time, regardless of position of their initial points (𝑥,𝑖) in Λ(𝑛)⧵Λ⁰.

Observe that (130) is deduced from (132) by integrating in d𝛀^∗_{Λ(𝑛)⧵Λ0} and normalizing by Ξ_{Λ(𝑛)⧵Λ}⁰(𝛀^∗_{Γ⧵Λ(𝑛)}); see (80) with Λ^′=Λ(𝑛). (The Jacobian of the map 𝛀^∗_{Λ(𝑛)⧵Λ0}↦𝚐_{Λ(𝑛)⧵Λ}⁰𝛀^∗_{Λ(𝑛)⧵Λ0} equals 1.)

Thus, our aim becomes proving (132). The tuned family 𝚐_{Λ(𝑛)⧵Λ}⁰ is composed of individual actions 𝚐_𝑗^(𝑛)∈𝙶: 𝚐_{Λ(𝑛)⧵Λ}⁰=𝚐_𝑗^(𝑛),𝑗∈Λ(𝑛)⧵Λ⁰. (135) Elements 𝚐_𝑗^(𝑛) are powers (multiples, in the additive parlance) of element 𝚐∈𝙶 figuring in (128)–(132) (resp., of the corresponding vector 𝜃∈𝑀; cf. (9)) and defined as follows. Let 𝜃_𝑗^(𝑛) denote the vector from 𝑀 corresponding to 𝚐_𝑗^(𝑛), and we select positive integer values 𝑟(𝑛)=⌈log(1+𝑛)⌉ and set 𝜃_𝑗^(𝑛)=𝜃𝜐𝑛,𝑗, (136) where 𝜐=≤𝑛,𝑗1,𝚍𝑜,𝑗𝑟(𝑛),𝜗𝚍𝑗,𝑜−𝑟(𝑛),𝑛−𝑟(𝑛),𝚍𝑜,𝑗>𝑟(𝑛). (137) In turn, the function 𝜗 is chosen to satisfy 𝜗()()+𝟏()𝑎,𝑏=𝟏𝑎≤00<𝑎<𝑏𝑄(𝑏)×^𝑏_𝑎𝑧(𝑢)d𝑢,𝑎,𝑏∈ℝ, (138) with 𝑄(𝑏)=^𝑏₀𝜁(𝑢)(𝑢)()()1d𝑢∼loglog𝑏,where𝜁=𝟏𝑢≤2+𝟏𝑢>2𝑢ln𝑢,𝑏>0. (139)

Lemma 31. Given 𝑧,𝛽∈(0,∞) satisfying (22) and a finite set Λ⁰, there exists a constant 𝐶∈(0,∞) such that ∀𝐱⁰,𝐲⁰∈𝑀^∗Λ0, the set of path configurations 𝛀^∗0∈𝑊^∗_{𝐱∗0,𝐲∗0} with 𝐡^Λ0(𝛀^∗0)<+∞ which include trajectories visiting points in Γ⧵Λ(𝑟(𝑛)) has the ℙ^∗_𝐱0,𝐲0 measure that does not exceed 𝐶/(⌈(1+𝑟(𝑛))⌉)!.

Proof of Lemma 31. The condition that 𝐡^Λ0(𝛀^∗0)<+∞ implies that the total number of sub trajectories of length 𝛽 in 𝛀^∗0 does not exceed 𝜅×♯Λ⁰ which is a fixed value in the context of the lemma. Each such trajectory has a Poisson number of jumps; this produces the factor 1/(⌈(1+𝑟(𝑛))⌉)!.

Back to the proof of Theorem 29: let 𝚐⁻¹_{Λ(𝑛)⧵Λ0} be the collection of the inverse elements: 𝚐⁻¹_{Λ(𝑛)⧵Λ0}=𝚐_𝑗^(𝑛)−1,𝑗∈Λ(𝑛)⧵Λ⁰. (140) The vectors corresponding to 𝚐_𝑗^(𝑛)−1 are −𝜃_𝑗^(𝑛)∈𝑀. We will use this specification for 𝚐_𝑗^(𝑛) and 𝚐_𝑗^(𝑛)−1 for 𝑗∈Λ(𝑛), or even for 𝑗∈Γ, as it agrees with the requirement that 𝚐_𝑗^(𝑛)≡𝚐 when 𝑗∈Λ⁰ and 𝚐_𝑗^(𝑛)≡𝚎 for 𝑗∈Γ⧵Λ(𝑛). Accordingly, we will use the notation 𝚐_Λ(𝑛)={𝚐_𝑗^(𝑛),𝑗∈Λ(𝑛)}.

Observe that the tuned family 𝚐_{Λ(𝑛)⧵Λ}⁰ does not change the contribution into the energy functional 𝐡^{Λ(𝑛)∣Γ⧵Λ(𝑛)} coming from potentials 𝑈⁽¹⁾ and 𝑈⁽²⁾: it affects only contributions from potential 𝑉.

The Taylor formula for function 𝑉, together with the above identification of vectors 𝜃_𝑗^(𝑛), gives 𝑉𝚐_𝑗^(𝑛)𝑥,𝚐_𝑗^(𝑛)_′𝑥^′+𝑉𝚐_𝑗^(𝑛)−1𝑥,𝚐_𝑗^(𝑛)_′⁻¹𝑥^′−2𝑉𝑥,𝑥^′≤𝐶𝜃²𝜐𝑛,𝑗−𝜐𝑛,𝑗^′2𝑉,𝑥,𝑥^′∈𝑀. (141) Here 𝐶∈(0,∞) is a constant, |𝜃| stands for the norm of the vector 𝜃 representing the element 𝚐, and the value 𝑉 is taken from (14).

Next, the square |𝜐(𝑛,𝑗)−𝜐(𝑛,𝑗^′)|² can be specified as 𝜐(𝑛,𝑗)−𝜐(𝑛,𝑗^′)²=𝑗0,if𝚍𝑗,𝑜,𝚍^′,𝑜≤𝑟(𝑛),𝑗0,if𝚍𝑗,𝑜,𝚍^′𝜗𝚍−,𝑜≥𝑛𝑗,𝑜𝑟(𝑛),𝑛−𝑟(𝑛)−𝜗𝚍(𝑗^′,𝑜)−𝑟(𝑛),𝑛−𝑟(𝑛)²,if𝑟(𝑛)<𝚍𝑗,𝑜,𝚍𝑗^′𝜗,𝑜≤𝑛,𝚍(𝑗,𝑜)−𝑟(𝑛),𝑛−𝑟(𝑛)²,if𝑟(𝑛)<𝚍𝑗,𝑜≤𝑛,𝚍𝑗^′,𝑜∈]𝑟(𝑛)[,𝜗,𝑛𝚍(𝑗^′,𝑜)−𝑟(𝑛),𝑛−𝑟(𝑛)²,if𝑟(𝑛)<𝚍𝑗^′,𝑜≤𝑛,𝚍𝑗,𝑜∈]𝑟(𝑛),𝑛[. (142)

By using convexity of the function exp and (141), ∀𝑎>1, 𝑎2exp−𝐡^Λ(𝑛)𝚐_Λ(𝑛)𝛀^∗0∨𝛀^∗_{Λ(𝑛)⧵Λ0}∣𝛀^∗_{Γ⧵Λ(𝑛)}+𝑎2exp−𝐡^Λ(𝑛)𝚐⁻¹_Λ(𝑛)𝛀^∗0∨𝛀^∗_{Λ(𝑛)⧵Λ0}∣𝛀^∗_{Γ⧵Λ(𝑛)}−1≥𝑎exp2𝐡^{Λ(𝑛)∣Γ⧵Λ(𝑛)}×𝚐_Λ(𝑛)𝛀^∗0∨𝛀^∗_{Λ(𝑛)⧵Λ0},𝛀^∗_{Γ⧵Λ(𝑛)}−12𝐡^Λ(𝑛)𝚐_Λ(𝑛)⁻¹𝛀^∗0∨𝛀^∗_{Λ(𝑛)⧵Λ0}∣𝛀^∗_{(𝑛)Γ⧵Λ}≥𝑎exp−𝐡^Λ(𝑛)𝛀^∗0∨𝛀^∗_{Λ(𝑛)⧵Λ0}∣𝛀^∗_{Γ⧵Λ(𝑛)}𝑒^−𝐶Υ/2, (143) where Υ=Υ(𝑛,𝚐)=𝛽𝜅²×_{𝑗,𝑗′(𝑛)∈Λ×Γ}𝐽𝚍𝑗,𝑗^′𝜐(𝑛,𝑗)−𝜐(𝑛,𝑗^′)². (144) The next observation is that Υ≤3𝛽𝜅²𝜃²×_{𝑗,𝑗′(𝑛)∈Λ×Γ}𝟏𝚍𝑗,𝑜≤𝚍𝑗^′,0𝐽𝚍𝑗,𝑗^′×𝜗𝚍𝑗,𝑜−𝑟(𝑛),𝑛−𝑟(𝑛)−𝜗𝚍𝑗^′,𝑜−𝑟(𝑛),𝑛−𝑟(𝑛)², (145) where, owing to the triangle inequality, for all 𝑗,𝑗^′∶𝚍(𝑗,𝑜)≤𝚍(𝑗^′,𝑜)0≤𝜗𝚍𝑗,𝑜−𝑟(𝑛),𝑛−𝑟(𝑛)−𝜗𝚍𝑗^′,𝑜−𝑟(𝑛),𝑛−𝑟(𝑛)≤𝚍𝑗,𝑗^′𝜁𝚍𝑗,𝑜−𝑟𝑄(𝑛−𝑟(𝑛.)) (146) This yields Υ≤𝛽𝜅²3𝜃²𝑄(𝑛−𝑟(𝑛))²×_{𝑗,𝑗′(𝑛)∈Λ×Γ}𝐽𝚍𝑗,𝑗^′𝚍𝑗,𝑗^′2𝜁𝚍(𝑗,0)−𝑟(𝑛)²≤3𝜃²𝑄(𝑛−𝑟(𝑛))²sup_{𝑗∈Γ𝑗′∈Γ}𝐽_𝑗,𝑗′𝚍𝑗,𝑗^′2×_{𝑗∈Λ0𝑛+𝑟}𝜁𝚍𝑗,0−𝑟(𝑛)², (147) where function 𝜁 is determined in (139).

Therefore, it remains to estimate the sum ∑_{𝑗∈Λ0𝑛+𝑟}𝜁(𝚍(𝑗,0)−𝑟(𝑛))². To this end, observe that 𝑢𝜁(𝑢)<1 when 𝑢∈(3,∞). The next remark is that the number of sites in the sphere Σ_𝑛 grows linearly with 𝑛. Consequently, _{𝑗∈Λ0𝑛+𝑟}𝜁𝚍(𝑗,𝑜)−𝑟(𝑛)²=_{1≤𝑘≤𝑛+𝑟0}𝜁(𝑘−𝑟(𝑛))×_{𝑗∈Σ𝑘}𝜁(𝑘−𝑟(𝑛))≤𝐶_{01≤𝑘≤𝑛+𝑟0}𝜁(𝑘−𝑟(𝑛))≤𝐶₁𝑄𝑛+𝑟₀−𝑟(𝑛),𝐶Υ≤𝑄(𝑛−𝑟(𝑛))⟶∞,as𝑛⟶∞. (148)

Therefore, given 𝑎>1 for 𝑛 large enough, the term 𝑎𝑒^−𝐶Υ/2 in the RHS of (145) becomes >1. Hence, 𝑎2exp−𝐡^Λ(𝑛)𝚐_Λ(𝑛)𝛀^∗0∨𝛀^∗_{Λ(𝑛)⧵Λ0}∣𝛀^∗_{Γ⧵Λ(𝑛)}+𝑎2exp−𝐡^Λ(𝑛)𝚐⁻¹_Λ(𝑛)𝛀^∗0∨𝛀^∗_{Λ(𝑛)⧵Λ0}∣𝛀^∗_{Γ⧵Λ(𝑛)}≥exp−𝐡^Λ(𝑛)𝛀^∗0∨𝛀^∗_{Λ(𝑛)⧵Λ0}∣𝛀^∗_{(𝑛)Γ⧵Λ}. (149) Equation (149) implies that the quantity 𝑞^{Λ0∣Γ⧵Λ(𝑛)}𝛀^∗0∣𝛀_{(𝑛)Γ⧵Λ}=_{𝑊0Λ(𝑛)⧵Λ}d𝛀^∗_{Λ(𝑛)⧵Λ0}×exp−𝐡^Λ0𝛀^∗0∨𝛀^∗_{Λ(𝑛)⧵Λ0}∣𝛀^∗_Γ⧵Λ0Ξ_Λ(𝑛)𝛀^∗_{Γ⧵Λ(𝑛)}, (150) obeys lim_𝑛→∞𝑞^Λ0_𝛽^{∣Γ⧵Λ(𝑛)}𝚐𝛀^∗0∣𝛀^∗_{Γ⧵Λ(𝑛)}+𝑞^Λ0_𝛽^{∣Γ⧵Λ(𝑛)}𝚐⁻¹𝛀^∗0∣𝛀^∗_{Γ⧵Λ(𝑛)}≥2lim_𝑛→∞𝑞^Λ0_𝛽^{∣Γ⧵Λ(𝑛)}𝛀^∗0∣𝛀^∗_{Γ⧵Λ(𝑛)} (151) uniformly in boundary condition 𝝎_{Γ⧵Λ(𝑛)}. Integrating (151) d𝜇_Γ^{Γ⧵Λ(𝑛)}(𝝎_{Γ⧵Λ(𝑛)}) yields (132).

Acknowledgments

This work has been conducted under Grant 2011/20133-0 provided by the FAPESP, Grant 2011.5.764.45.0 provided by The Reitoria of the Universidade de São Paulo, and Grant 2012/04372-7 provided by the FAPESP. The authors express their gratitude to NUMEC and IME, Universidade de São Paulo, Brazil, for the warm hospitality.