The Feynman–Kac Representation and Dobrushin–Lanford–Ruelle States of a Quantum Bose-Gas

This paper focuses on infinite-volume bosonic states for a quantum particle system (a quantum gas) in Rd. The kinetic energy part of the Hamiltonian is the standard Laplacian (with a boundary condition at the border of a ‘box’). The particles interact with each other through a two-body finite-range potential depending on the distance between them and featuring a hard core of diameter a > 0. We introduce a class of so-called FK-DLR functionals containing all limiting Gibbs states of the system. As a justification of this concept, we prove that for d = 2, any FK-DLR functional is shift-invariant, regardless of whether it is unique or not. This yields a quantum analog of results previously achieved by Richthammer.


Introduction. Infinite-Volume Gibbs States and Reduced Density Matrices
The results of the paper and related works. In this paper we focus on bosonic quantum systems and pursue two directions of study: (i) a working definition of an infinite-volume quantum Gibbs state for various types of quantum bosonic systems, and (ii) its justification, which we have chosen to be the shift-invariance property for a 2D Bose-gas. The starting point for this work was the famous definition by Dobrushin-Lanford-Ruelle (DLR) of an infinite-volume Gibbs probability distribution, which is universally accepted in contemporary Mathematical Physics and beyond.
(i) Our approach combines the DLR-equation and the Ginibre representation of density matrix kernel [1] and develops the approach outlined in [2]. Alternative approaches are represented in [3][4][5], based on probability measures on distributions or direct analysis of the spectrum of the Hamiltonian. See also the biblio quoted in the above sources. The first result of the paper in this direction is the proof of existence, by compactness, of a compatible family of infinite-volume reduced density matrices for a given family of local Hamiltonians (1.1.1), under a natural super-stability-type condition (1.1.14) (cf. Theorem 1 in Section 1.3). Next, we establish that the kernels of the infinite-volume reduced density matrices satisfy an analog of the DLR equation, which we call an FK-DLR equation. (FK stands

The Local Hamiltonian
The object of this study is a quantum Bose-gas in a Euclidean space R d , d ≥ 2. The starting point of our analysis is a self-adjoint n-particle Hamiltonian, H n,Λ , in a finite 'box' Λ represented by a cube [−L, L] d , of size 2L > 0, centered at the origin. (Other types of bounded domains in R d can/will also be incorporated.) Operator H n,Λ acts on functions x n 1 = {x(1), . . . , x(n)} ∈ Λ n → φ n (x n 1 ) from L sym,a 2 (Λ n ) by Here, L sym,a 2 (Λ n ) is the subspace in the Hilbert space L 2 (Λ n ) = L 2 (Λ) ⊗n formed by symmetric functions of variables x(j), 1 ≤ j ≤ n, constituting the argument x n 1 , which vanish whenever min x(j) − x(j ) Eu : 1 ≤ j < j ≤ n < a.
(|x| Eu , or briefly |x|, stands for the Euclidean norm of x ∈ R d whereas |x| m denotes the max-norm.) Parameter a > 0 is fixed and represents the diameter of the hard core (see below). It is convenient to denote Λ n a = x n 1 = {x(1), . . . , x(n)} ∈ Λ n : min [|x(j) − x(j )| : 1 ≤ j < j ≤ n] ≥ a (1. 1.2) and identify L sym,a 2 (Λ n ) with L sym 2 (Λ n a ), the Hilbert space of square-integrable symmetric functions φ n x n 1 with support in Λ n a . Operator ∆ j in (1.1.1) acts as a Laplacian in the variable x(j). Further, V : r ∈ [a, +∞) → V(r) ∈ R is a C 2 -function describing a two-body interaction potential depending upon the distance between particles. Pictorially, we set: V(r) = +∞ for 0 ≤ r < a, conforming with the hard-core assumption. In the following condition (1.1.3a), we attempt to control the negative (attracting) part of V(r): we assume that Observe that when V(r) ≥ 0 then W − = 0 (this includes the case of pure hard cores where V(r) ≡ 0 for r ≥ a). In a similar manner, we assure a control over the derivative V : Physically, one can say that the potential V(r), r > a, has a bounded derivative and decays to 0 for large r in a qualified manner. We also set In the case where V(r) = 0 for r ≥ R • , we say that V has a finite range; the smallest value R • ∈ (0, ∞) with this property is called the interaction radius (or the interaction range) and is referred to in the relevant bounds.
For n = 0, we formally set H 0,Λ = 0. In general, the term − 1 2 ∑ 1≤j≤n ∆ j φ n (x n 1 ) represents the kinetic energy part in the Hamiltonian, and the term ∑ 1≤j<j ≤n V (|x(j) − x(j )|) the potential energy (as an operator, it is given as multiplication by this function). Note that if n is large enough (when n disjoint balls of diameter a can't be placed in a box Λ) then the expression for H n,Λ formally becomes infinite; so we will only care about the values of n such that the set Λ n a = ∅. To complete the definition of operator H n,Λ , we need to specify a boundary condition. More precisely, H n,Λ is initially defined by the right hand side (RHS) of Equation (1.1.1) as a symmetric operator on the set of C 2 -functions φ = φ n with the support in the interior of Λ n a , see [16]. A self-adjoint extension of this symmetric operator emerges when we impose the Dirichlet boundary condition: φ(x n 1 ) = 0 for x n 1 = {x(1), . . . , x(n)} ∈ ∂ (a) Λ n a ∪ ∂ out Λ n a . (1.1.5) Here ∂ (a) Λ n a = x n 1 ∈ Λ n a : min [|x(j) − x(j )| : 1 ≤ j < j ≤ n] = a , ∂ out Λ n a = x n 1 ∈ Λ n a : max [|x(j)| m = L : 1 ≤ j ≤ n] = a .
(1. 1.6) Other examples of boundary conditions on ∂ out Λ n a for which the methods of this paper are applicable are Neumann and periodic. (In fact, one can incorporate general elastic boundary conditions. We intend to analyze these in a forthcoming work.) In the Krein-Vishik classification, [17], Dirichlet's boundary condition generates a 'soft' self-adjoint extension whereas Neumann's boundary condition generates a 'rigid' self-adjoint extension. These two self-adjoint extensions are extreme ones (among Dirichlet-form extensions) in the sense of a natural order of the eigenvalues. Moreover, in our scheme the choice of the boundary condition for H n,Λ may vary from one square Λ to another (and even from one value of n to another). This endeavors towards inclusion of a broad class of Hamiltonians, aiming at enhancing possible phase transitions.
To simplify the notation, we omit the indices/arguments z and β whenever it does not lead to a confusion. A straightforward generalization of the above concepts can be done by including an external potential field induced by a particle configuration x(Λ c ) represented by a finite or countable subset in the complement Λ c such that |y − y | ≥ a ∀ pair y, y ∈ x(Λ c ) with y = y . Viz., the Hamiltonian H n,Λ|x(Λ c ) is given by and possesses the properties listed above for H n,Λ . This enables us to introduce the Gibbs operators G n,Λ|x(Λ c ) and G Λ|x(Λ c ) , the partition functions Ξ n (Λ|x(Λ c )) and Ξ(Λ|x(Λ c )), the DM R Λ|x(Λ c ) , the GS ϕ Λ|x(Λ c ) and the RDMs R For an empty exterior particle configuration x(Λ c ) = ∅, the argument x(Λ c ) will be omitted. (Although the Hamiltonian H n,Λ and its derivatives G n,Λ , G Λ and so on, are particular examples of H n,Λ|x(Λ c ) , etc., (with x(Λ c ) being an empty configuration), we will now and again address this specific example individually, for its methodological significance.)

The Thermodynamic Limit. The Shift-Invariance Property
The key concept of Statistical Mechanics is the thermodynamic limit; in the context of this work it is lim The quantities and objects established as limiting points in the course of this limit are often referred to as infinite-volume ones (e.g., infinite-volume RDM or GS). The existence and uniqueness of a limiting object is often interpreted as absence of a phase transition, a multitude of such objects (viz., depending on the boundary conditions for the Hamiltonian or the choice of external configuration) is treated as an exhibition of a phase transition, see [3,7,9,[18][19][20][21][22]. However, there exists an elegant alternative where infinite-volume values are identified in terms that, at least formally, do not invoke the thermodynamic limit. For classical systems, this is the DLR equations and for the so-called quantum spin systems-the KMS equations. (The latter involves an infinite-volume dynamics which is not affected by phase transitions in terms of GSs.) Unfortunately, the KMS equations are not directly available for the class of quantum systems under consideration in this paper, since the Hamiltonians H n,Λ and H n,Λ|x(Λ c ) are not bounded.
In this paper we employ a construction generalizing the classical DLR equation and-in dimension d = 2-establish shift-invariance property for the emerging objects (the RDMs). Observe that . . , b d and vector s = s 1 , . . . , s d ∈ R d , the Fock spaces H(Λ 0 ) and H(S(s)Λ 0 ) are related through a pair of mutually inverse shift isomorphisms Here, S(s) stands for the shift isometry R d → R d : and S(s)Λ 0 is for the image of Λ 0 : The isomorphisms U Λ 0 (s) and U Λ 0 (−s) are given by where φ n ∈ L sym 2 ((Λ 0 ) n ), n = 0, 1, . . .. The Fock spaces H(Λ) and H(Λ 0 ) (see (1.1.9)) can be conveniently represented as L 2 (C a (Λ)) and L 2 (C a (Λ 0 )), respectively. Here and below, C(Λ) denotes the collection of finite (unordered) subsets x ⊂ Λ (including the empty set) with the Lebesgue-Poisson measure where is the Lebesgue measure on R d ), and C a (Λ) stands for the subset of C(Λ) formed by x ⊂ Λ with The symbol is used for the cardinality of a given set.) The same meaning is attributed to the notation C a (R d ) and C a (Λ c ) (here, we mean finite or countable sets x ⊂ R d and x ⊂ Λ c , respectively, obeying (1.2.5)). In Theorem 1 below, we speak of a pair of fixed cubes, . On the other hand, a sequence of boxes Λ(k) = [−L(k), L(k)] d R d is present, of sidelengths 2L(k) → ∞ as k → ∞, which may depend on Λ 1 and Λ 0 . We use the term 'box' when referring to a physical volume where a given system is confined and 'cube' while bearing in mind a 'localized' sub-volume as a part of a proof. A box will increase to cover the whole R d whereas a cube will be fixed or vary within a restricted range. Theorem 1. Suppose that z > 0 and β > 0 are given, satisfying condition (1.1.14). For any cube Λ 0 , the family of RDMs {R Λ 0 Λ|x(Λ c ) , Λ R d } is compact in the trace-norm operator topology in H(Λ 0 ), for any choices of particle configurations x(Λ c ) ∈ C a (Λ c ). Any limit-point operator R Λ 0 for {R Λ 0 Λ|x(Λ c ) } is a positive-definite operator in H(Λ 0 ) of trace 1. Furthermore, let Λ 1 ⊂ Λ 0 be a pair of cubes and R Λ 1 , R Λ 0 be a pair of limit-point RDMs such that for a sequence of boxes Λ(k) R d and external configurations x(Λ(k) c ) obeying (1.2.5). Then R Λ 1 and R Λ 0 satisfy the compatibility property The next theorem is established in dimension d = 2.

2.9)
In the future, the bound (1.1.14) will be assumed without stressing it every time again. Also, referring to external configurations x(Λ c ) and x(Λ(k) c ), we always assume that x(Λ c ) ∈ C a (Λ c ) and x(Λ(k) c ) ∈ C a (Λ c (k)).
A direct corollary of Theorem 1 is the construction of a limit-point Gibbs state. To this end, it suffices to consider a countable collection of cubes and L 0 . By invoking a diagonal process, we can guarantee that, as Λ R d , given any family of external configurations x(Λ c ), one can extract a sequence Λ(k) R d such that (i) ∀ cube Λ 0 from the collection, ∃ the trace-norm limit and (ii) the limiting operators relation (1.2.7) holds true ∀ pair of cubes from the collection, This enables us to define an infinite-volume Gibbs state ϕ by setting (1.2.11) for any cube Λ 0 ⊂ R d . More precisely, ϕ is a state of the quasilocal C * -algebra B(R d ) defined as the norm-closure of the inductive limit B 0 (R d ): Here, S(s)A) stands for the shift of the argument A: if A ∈ B(Λ 0 ) then (1.2.14)

Integral Kernels of Gibbs Operators and RDMs
According to the adopted realization of the Fock space H(Λ) as L 2 (C a (Λ)), its elements are represented by functions φ Λ : The space H(Λ 0 ) is described in a similar manner: here, we will use a short-hand notation x 0 and y 0 instead of x(Λ 0 ), y(Λ 0 ) ∈ C a (Λ 0 ). The first step in the proof of Theorems 1 and 2 is to reduce their assertions to statements about the integral kernels F Λ|x(Λ c ) and their infinite-volume counterpart R Λ 0 ; we call these kernels RDMKs for short. Indeed, Λ|x(Λ c ) and R Λ 0 are integral operators: and The RDMKs F Λ 0 Λ (x 0 , y 0 ) and F Λ 0 Λ|x(Λ c ) (x 0 , y 0 ) -and ultimately F Λ 0 (x 0 , y 0 )-admit a Feynman-Kac (FK) representation providing a basis for future analysis. Here, we state properties of these kernels in Theorems 4 and 5: Theorem 4. Under the conditions of Theorem 1, for any cube Λ 0 and for any choice of particle configurations determines a positive-definite operator R Λ 0 in H(Λ 0 ) of trace 1 (a limit-point RDM). Furthermore, let Λ 1 ⊂ Λ 0 be a pair of squares and F Λ 1 , F Λ 0 a pair of limit-point RDMKs such that ) for a sequence of squares Λ(k) R 2 , boundary conditions on ∂ out Λ(k) * and external configurations x (Λ(k) c ). Then the corresponding limit-point RDMs R Λ 1 and R Λ 0 obey (1.2.7).
Theorem 4 implies Theorem 1 with the help of Lemma 1.5 from [23] (going back to Lemma 1 in [24]). In turn, Theorem 2 is a direct corollary of Theorem 5. Set d = 2 and assume the conditions of Theorem 2. Given a square Λ 0 and a vector s = (s 1 , s 2 ), consider limit-point RDMKs F Λ 0 and F S(s)Λ 0 such that for a sequence of squares Λ(k) R 2 , boundary conditions on ∂ out Λ(k) * and external configurations x(Λ(k) c ). Then, ∀ x 0 , y 0 ∈ C a (Λ 0 ) and s = s 1 , s 2 ∈ R 2 , Therefore, we focus on the proof of Theorems 4 and 5. In fact, we will establish the properties for more general objects-FK-DLR functionals.
An element Ω * from W * (x, y) is called a path collection/configuration, with the initial/terminal particle configurations x, y. (For simplicity, we write x and y instead of x n 1 and y n
Definition 2 (Path measures). The spaces introduced in Definition 1 are equipped with standard sigma-algebras (generated by cylinder subsets and operations on them), see [25]. We consider various measures on these sigma-algebras: x,γ n y and P * x,y = ∑ γ:x↔y P * x,γx -the sum-measures on W * (x, y) and W * (x, y).
). We will use the name Lebesgue-Poisson-Wiener measure (LPWM). Sometimes we will write d Λ x(Λ) and d Λ Ω * (Λ) in order to stress the dependence upon Λ.
As a rule, we will be working with restrictions of the above measures upon the corresponding , and W * a (Λ). The Brownian (or Wiener) bridge on the time interval kβ with the endpoints at 0 is usually defined as a process of the form is a standard Brownian motion; cf., e.g., [26,27]. It is a (non-homogeneous) Markov process, with a strong Markov property that ∀ Markov stopping time T , the behavior of the process before time T ∧ kβ and after are conditionally independent, given W(T ∧ kβ). The Brownian bridge with the initial points at x and the final point at y is constructed as W(t) + x + t(y − x). Definition 3 (Energy-related functionals). Given a path ω * ∈ W * a (x, y), we set: where, for a given finite particle configuration z ∈ C a (R d ), we set: The quantity h(ω) can be interpreted as an energy of path ω. The energy of interaction between two paths, ω * ∈ W * a (x, y) and ω * ∈ W * a (x , y ), is determined by Here, for a given pair of particle configurations z, z ∈ C a (R d ), such that z ∪ z ∈ C a (R d ), z ∩ z = ∅ and at least one of them is finite, we set: The Definitions 1 and 3 hold for loops as well, obviously. Next, for a path collection We will also need the energy for various combined collections of paths, loops and particle configurations. Viz., for Finally, we introduce functionals K, L and α Λ , for path and loop configurations : Here and below, The presence of Dirichlet's boundary conditions is manifested in the indicators

The FK-Representation in a Box
As follows from well-known results about the operators H n,Λ and H n,Λ|x(Λ c ) (see, e.g., [1,20,21]), we have the following properties listed in Lemmas 1 and 2 Lemma 1. For an external particle configuration x(Λ c ) defining the self-adjoint operators H n,Λ|x(Λ c ) , the partition function Ξ Λ|x(Λ c ) (see (1.1.21)) admits the following representation: , Functionals K and L are as in (2.1.11) and (2.1.12). Next, ) stands for the indicator requiring that no path ω * or loop ω * from the whole collection enters the square Λ 0 at 'control' time points lβ with 1 ≤ l < k, where k equals k(ω * ) or k(ω * ).
Mnemonically, the notation Ξ Λ 0 , Ω * 0 means the application of an indicator function χ Λ 0 in the corresponding integral, together with presence of a specific path configuration Ω * 0 in the energy 3) represents a restricted partition function in Λ \ Λ 0 in presence of a path configuration Ω * 0 and in the potential field generated by an external particle configuration x(Λ c ), with the restriction dictated by χ Λ 0 . We would like to note that is only one out of several types of partition functions that we will have to deal with in our analysis.

(2.2.7)
Here, the numerator Ξ represents a partition function in Λ \ Λ 0 in the external field generated by the particle configuration x(Λ c ), in presence of a loop configuration Ω * 0 over Λ 0 , and with Dirichlet boundary condition in box Λ.
The next assertion, Lemma 2, describes compatibility properties of PMs µ Λ|x(Λ c ) relative to the choice of an intermediate cube Λ where Λ 0 ⊂ Λ ⊂ Λ. This property will allow us to use the same formalism in Section 2.3 when box Λ is replaced with the whole space R d . The proof of Lemma 2 is a standard (although tedious) manipulation with the Gibbsian form of PM µ Λ|x(Λ c ) and is omitted. (2.2.10) Furthermore, for a given Ω * . Moreover, the following representation holds: (2.2.12) Here, in analogy with (2.2.11), for a given Ω * Λ\Λ ∈ W * a (Λ \ Λ ), An important property is given in Lemma 3 establishing uniform estimates for quantities q In terms of integrals β 0 dt, we have to lower-bound the classical energy of interaction between a single particle and a particle configuration (possibly infinite) in R d . According to the definition of the value W − , we obtain that ∀ Ω *

The Infinite-Volume FK-DLR Equations and RDMKs
The infinite-volume versions of the RDMK arise when we mimic properties listed in Lemmas 1 and 2 by getting rid of the reference to the enveloping box Λ (including the external particle configuration x(Λ c ) and the functional α Λ indicating Dirichlet's boundary condition). The first place to do so is the PM µ Λ|x(Λ c ) ; to this end we need to consider its infinite-volume analog µ R d representing an To simplify technical aspects of the presentation, we will omit the reference to the initial configuration x(R d ) and write Ω R d ∈ W * (R d ) or Ω * ∈ W * (R d ) (given a loop configuration Ω * , the initial particle configuration is uniquely determined and is denoted by x(Ω * )).
Furthermore, we will use the notation W * (Λ c ) for the subset in W * (R d ) formed by loop configurations Ω * Λ c with x(Ω * Λ c ) ∈ C a (Λ c ). (We call such Ω * Λ c a loop configuration over Λ c .) Here, the functional q Λ 0 (Ω * 0 ) admits the following representation: ∀ pair of cubes Λ 0 ⊂ Λ ⊂ R d , Observe similarities with Equation (2.2.9). At the same time, note the absence the indicator α Λ in the RHS of (2.3.2). Here, for a given (infinite) loop configuration Ω * yields a partition function in Λ \ Λ 0 , in the external field generated by Ω * Λ c and in presence of a loop configuration Ω * 0 ∈ W * a (Λ 0 ) (but without a boundary conditions): Here, Λ stands for the cube [− L, L] d of side-length 2 L centered at the origin in R d , and Ω * Λ\Λ denotes the restriction of Ω * Λ c to Λ \ Λ. . The class of FK-DLR PMs (for a given pair of values z ∈ (0, 1), β ∈ (0, +∞)) is denoted by K(z, β), or, briefly, K. It is straightforward that any PM µ ∈ K is supported by the set W * a (R d ): µ(W * a (R 2 )) = 1.

3.15)
Then, according to (2.3.13)-(2.3.14), for Λ 1 ⊂ Λ 0 , The family of operators R Λ 0 defines a linear normalized functional on the quasilocal C * -algebra We call the functional A ∈ B(R d ) → ϕ(A) an FK-DLR functional generated by µ; to stress this fact, we sometimes use the notation ϕ µ . If in addition ϕ is a state (that is, the operators R Λ 0 are positive-definite), then we say that ϕ is an FK-DLR state. In this case, we call the operator R Λ 0 an infinite-volume FK-DLR RDM.
The class of FK-DLR functionals is denoted by F = F(z, β) and its subset consisting of the FK-DLR states by Before we move further, we would like to introduce a property conventionally called a 'Ruelle superstability bound'. It is closely related to the so-called Campbell formula assessing integrals of summatory functions Σ g :  and will follow from the representation

Results on Infinite-Volume FK-DLR PMs and Gibbs States
Our results about classes K, F and F + are summarized in the following theorems.
Theorem 6. The class K of FK-DLR PMs is non-empty. Moreover, the family of FK-DLR PMs µ Λ is compact in the weak topology, and every limiting point µ for this family lies in K. Furthermore, the family of the Gibbs states ϕ Λ is compact in the w * -topology, and every limiting point for this family gives an element from F + . The same is true for any family of the PMs µ Λ|x(Λ c ) and states ϕ Λ|x(Λ c ) with configurations x(Λ c ) ∈ C a (Λ c ). Consequently, the set F + (z, β) is non-empty.

4.1)
In terms of the corresponding infinite-volume RDMs R Λ 0 : Remark 2. The statement of Theorem 7 is straightforward for the limit points R Λ 0 of the family {R Λ 0 Λ , Λ R 2 }, but requires a proof for the family {R Λ 0 Λ, x c (N) }.
The argument for equi-continuity of RDMKs is based on uniform bounds upon the gradients ∇ x F Λ 0 Λ|x(Λ c ) (x 0 , y 0 ) and ∇ y F Λ 0 Λ|x(Λ c ) (x 0 , y 0 ), for x ∈ x 0 , y ∈ y 0 . Both cases are treated in a similar fashion; for definiteness, we consider gradients ∇ y F Λ 0 Λ|x(Λ c ) (x 0 , y 0 ), x 0 , y 0 ∈ C a (Λ 0 ). It can be seen from representations (2.2.2)-(2.2.4) and (2.2.16) that there are two contributions into the gradient. The first contribution comes from varying the measure P * x 0 ,y 0 . The second one emerges from varying the functional q Λ 0 Λ|x(Λ c ) (Ω * 0 ), more precisely, the numerator Ξ In fact, it is clear that the second contribution will come out only when we vary the term Of course, we are interested in varying a chosen point y ∈ y 0 .

(3.3)
Given j = 1, . . . , n, write k( j) for k(ω * ( j)) and ω * j and ζ * j for ω * ( j) and ζ * ( j). Then the first gradient in For the last three contributions we have: The quantity W (1) is given in (1.1.3b). For the first contribution: Thus, The integral of the gradient ∇ y(j) |x(j) − y(j)| 2 in (3.3) does not exceed a constant C. Hence, .

(3.4)
This shows equicontinuity of functions F Λ 0 Λ|x(Λ c ) (x 0 , y 0 ). Hence, the family of RDMKs {F Λ 0 Λ|x(Λ c ) } is compact in space C 0 (C a (Λ 0 ) × C a (Λ 0 )). Let F Λ 0 be a limit-point as Λ R d . Then we have the Hilbert-Schmidt convergence Consequently, the RDM R Λ 0 Λ|x(Λ c ) in H(Λ 0 ) converges to the infinite-volume RDM R Λ 0 determined by the kernel F Λ 0 , in the Hilbert-Schmidt norm: As was mentioned, applying Lemma 1 from [24] (see also Lemma 1.5 from [23]), we obtain the trace-norm convergence: Invoking a standard diagonal process implies that the sequence of states ϕ Λ|x(Λ c ) is w * -compact. Alongside with the above argument, one can establish that the PMs µ Λ|x(Λ c ) form a compact family as Λ R 2 . More precisely, we would like to show that for all given cube Λ 0 , the family of PMs µ Λ 0 Λ|x(Λ c ) on (W * a (Λ 0 ), W(Λ 0 )) is compact. To this end, it suffices to check that the family {µ Λ 0 Λ|x(Λ c ) } is tight as the Prokhorov theorem will then guarantee compactness.
Following an argument from [23], tightness is a consequence of two facts. (a) The reference measure d Λ 0 Ω * 0 on W Λ 0 (see Definition 1 (vii)) is supported by loop configurations with the standard continuity modulus 2 ln (1/ ).
(2.2.6)) is bounded from above by a constant similar to the RHS of (3.1). As a result, the family of limit-point PMs {µ Λ 0 : Λ 0 ⊂ R 2 } has the compatibility property and therefore satisfies the assumptions of the Kolmogorov theorem. This implies that there exists a unique PM µ on (W * a (R d ), W(R d )) such that the restriction of µ on the sigma-algebra W(Λ 0 ) coincides with µ Λ 0 .
The fact that µ is an FK-DLR PM follows from the above construction. Hence, each limit-point state ϕ falls in class F + (z, β). This completes the proof of Theorems 1 and 6.

µ(S(s)D) = µ(D).
The proof of Theorem 8 is based on a modification of an argument developed in [11][12][13]. We want to stress that the paper [13] treating some classes of (Gibbsian) RMPPs does not cover our situation because a number of the assumptions used in [13] are (unfortunately) not fulfilled here. Specifically, the condition (2.2) from [13] does not hold in our situation, as well as conditions specifying what is called a bpsi-function on P. 704 of [13]. (In short, the paper [13] employs an approach based on sup-norm conditions whereas the situation under consideration in this paper requires the use of integral-type norms.) The aforementioned modification requires that we use (and inspect) the construction from [12] for particle configurations arising as t-sections of loop and path configurations at a given time point t ∈ [0, β].
Because the argument in the proof does not depend on the direction of the vector s, we will assume that s = (s, 0) lies along the horizontal axis. Also, due to the group property, we can assume that s ∈ (0, 1/2). By using constructions developed in [12,13,19], the assertion of Theorem 8 can be deduced from Theorem 9. Let µ be an FK-DLR PM, Λ 0 be a square [−L 0 , L 0 ] ×2 and an event D ⊂ W * a (R 2 ) be given, localized in Λ 0 : D ∈ W(Λ 0 ). Then For the proof of Theorem 9, we employ a strategy essentially mimicking the one from [11][12][13], particularly [12]. Consequently, we will follow the scheme from [12] rather closely, although, as was said earlier, we introduce considerable alterations. For a given (large) L, we work with the squares Λ = Λ(L) and Λ 0 where We write the terms µ(S(±s)D) and µ(D) as integrals of conditional expectations relative to the sigma-algebra W(Λ c ): (the case of µ(D) is recovered at s = 0, with S(0) = Id.) Furthermore, again as in [11,13], we employ maps T ± L = T ± L,L 0 (s) : W * (R 2 ) → W * a (R 2 ). (The symbol used in [11,13] is T instead of T. The idea of using maps T ± L goes back to [9,10].) These are applied to the concatenated loop configuration Ω * Λ ∨ Ω * Λ c in the expressions from Equation (4.3), in the corresponding case of shift S(±s). Important properties of maps T ± L are: are one-to-one, and a number of 'nice' properties hold true when the loop configuration Ω * Λ ∨ Ω * Λ c lies in a 'good' set G L ⊂ W * a (R 2 ). (Viz., for Ω * Λ ∨ Ω * Λ c ∈ G L the loops from Ω * Λ ∩ W * a (Λ 0 ) will not interact with loops from Ω * Λ c .) The set G L carries asymptotically a full measure as L → ∞. See below.
(ii) For a 'good' loop configuration Ω * = Ω * Λ ∨ Ω * Λ c ∈ G L over R 2 , the 'external' part Ω * Λ c is preserved under T ± n . In other words, the maps are non-trivial only on the part Ω * Λ (although the way Ω * Λ is transformed depends upon Ω * Λ c (and on Ω * Λ , of course)). For that reason, we will often address T ± L as a 'tuned' shift Ω * → Ω , With this agreement: (iii) The transformation (4.4) preserves the cardinality: Ω * Λ = Ω * Λ and transforms a loop ω * ∈ Ω * Λ as ω * → ω * where k( ω * ) = k(ω * ). Consequently, functionals K and L are preserved: see below. We stress that the argument of function R ± L consists of a loop ω * ∈ W * a , a time point (iv) For brevity, let us omit henceforth the symbols ± whenever possible. The value Consequently, in accordance with (4.5), for ω * ∈ W * a (x) with x ∈ Λ 0 and t ∈ [0, k(ω * )β] the point ω * (t) = ω * (t) + s. Therefore, the loops ω * from Ω * 0 = Ω * Λ ∩ W * a (Λ 0 ) are shifted intact by the amount s under the map (4.4). Consequently, the integral energy h(Ω * 0 ) is not changed under tuned shifts. (v) The set S(s)(D ∩ G L ) will have a µ-measure close to that of S(s)D; moreover, the probability µ(S(s)(D ∩ G L )) will be written in the form where function J ± L = J ± L,s gives the Jacobian of transformation T ± L (s). By virtue of the properties above (cf. (i) and (iv)), the impact of T L upon the energy h(T L Ω * Λ |Ω * Λ c ) will be felt through the loop configuration Ω * Essentially, the same remains true about the Jacobian J L (Ω * Λ ∨ Ω * Λ c ).
(vi) In fact, a detailed analysis shows that second-order incremental expressions and are close to 1. It turns out that this fact suffices for the assertion of Theorem 9.
Formally, Theorem 9 is derived from (4.9) (II) The probabilities µ(S(±s)(D ∩ G L )) are represented in the form (4.6) with the following properties: The proof of Theorem 10 is carried on in Sections 5-7.

Remark 3.
It is the pair of inequalities (IIIa), (IIIb) (together with the definition of the 'good' set G L ) where one crucially uses the fact that the physical dimension of the system equals 2.
We now show how to deduce the statement of Theorem 9 from that of Theorem 10. Owing to Theorem 10 (I), (II), we can write: the LHS of (4.1) (4.10) Next, by the AM/GM inequality, the RHS of (4.10) is no less than

(4.11)
Now, by virtue of Theorem 10 (I)-(III), the RHS of (4.11) is greater than or equal to (4.12) Since δ can be made arbitrarily small, we obtain the inequality (4.1).

L
As was said earlier, the maps Ω * → T ± L Ω * Λ ∨ Ω * Λ c are determined by transforming the t-sections {T ± L Ω * Λ }(t) of the loop configuration Ω * Λ , for each t ∈ [0, β]. Denoting by T ± L = T ± L (±s) the map acting on particle configurations from C a (Λ), we can write: Again, we would like to stress that the way the t-section {Ω * is not moving when Ω * ∈ G L . More precisely, set:
In other words, a loop ω * ∈ Ω * is affected only at points ω * (t) lying in Λ.
Observe that t L (y; t) = 0 for y ∈ Λ c and t L (y; t) = 1 for y ∈ Λ R(L) .

(5.22)
The Jacobian J ± L (Ω * Λ ∨ Ω * Λ c ) of the transform T ± L turns out to be of the form: where (∂ 1 t L )(y) stands for the partial derivative ∂ t L ∂y 1 (y; t), y = (y 1 , y 2 ). (The fact that the functions t L are non-differentiable on sets of positive co-dimension is not an obstacle here because of involvement of Wiener's integration.) The crucial quantity J +

(5.24)
We see that the quantity (5.24) is close to 1 when we are able to check that the sum is close to 0. We conclude this section with a straightforward assertion justifying the definition (5.3) that introduces the intermediate square Λ.
In other words, (a) for Ω * ∈ L L , every loop ω * from Ω * which starts at a point x(ω * ) outside square Λ does not reach square Λ, while (b) for Ω * ∈ L By virtue of the Campbell theorem, the last integral equals which by the Ruelle superstability bound (2.3.19) does not exceed with ρ := ze βW − ; cf. (1.1.14).
Next, we observe that the loop ω * with the endpoint x = (x 1 , x 2 ) ∈ Λ c (i.e., with max |x j | m ≥ L) can reach Λ only if at least one of its one-dimensional components (i.e., a scalar Brownian bridge with the endpoint x j , j = 1 or 2) deviates from its origin by at least (|x j | − L) + L 3/4 . Therefore, the last displayed expression is upper-bounded by plus bounds for the tail of the normal distribution (see [30], Formula (3)): It is not hard to see that the RHS of (5.27) tends to 0 as L → ∞. This completes the proof.
In what follows, we will assume that a loop configuration Ω * lies in L L . Together with (5.22), this will imply that the loops ω * ∈ Ω * with x(ω * ) ∈ Λ c remains unaffected by transformations T ± (s).

Estimates for the Jacobians
To guarantee properties (I) and (IIIa) of Theorem 9, we need to secure that the good set G L carries a large measure and contains only those loop configurations Ω * ∈ W * (R 2 ) for which the expression can be appropriately controlled. To this end, consider a random variable Σ J (Ω * ) = Σ J L (Ω * ) given by the RHS of (5.25): The formal definition of the set G L will require that the quantity Σ J (Ω * ) is small (more precisely that some majorants for Σ J (Ω * ) are small); see below. Formally, the property that J + is close to 1 follows from Lemma 5. If > 0 is chosen small enough then the mean-value of Σ J (Ω * ) vanishes as L → ∞: Proof. Let us start with technical definitions. Given t ∈ [0, β] and x, x ∈ {Ω * }(t), we write: x ↔ x whenever a < |x − x | < a + and Recall, the values z, β > 0 are such that the bound (1.1.14) is satisfied. Referring below to a small > 0, we mean conditions like this: Constants C j ∈ (0, ∞) appearing in the argument vary with β and z (through ρ) but are independent of L.

(6.16)
Consequently, the contribution of this sum to R 2 dx also does not exceed a constant.
More generally, for a given r > R(L) we consider the contribution into (6.14) from loops ω * with x(ω * ) ∈ Λ r such that |ω * (t)| m = r for some t ∈ [0, k(ω * )β]. Repeating the above argument, we conclude that this contribution again is less than or equal to a constant times J L (r). Note that all constants can be made uniform; this implies that (6.14) (6.17) As in [12], the quantity in the RHS of (6.17) (which is = C 0 × Γ(L)) goes to 0 as L → ∞. This finishes the proof.
It is instructive to note that the relation (6.9) does not require a smallness for .
We now pass to random variable Σ Proof. In the beginning, we again use the Campbell theorem (in conjunction with an argument similar to Equation (6.25) from [12]). Then the integral in (6.18) is less than or equal to a constant (say, C 4 ) times the sum I 2,1 + I 2,2 . Here the term I 2,1 = I 2,1 L is specified as follows: where the loop ω * 0 has been identified as ω * and value l 0 as l. Likewise, where again the loop ω * 0 has been identified as ω * and value l 0 as l. So, it suffices to verify that lim L→∞ I 2,1 = lim L→∞ I 2,2 = 0.

(6.20b)
Employing in addition the Ruelle superstability bound (2.3.19), we conclude that (6.19b) does not exceed (6.21) Expanding the sum of squares in the parentheses, we obtain three expressions; in view of similarity of the argument used for analysing each of them, we focus on the one with the term |ω * (lβ + t)| 2 m : ×1(ω * m (l m β + t) ↔ ω * (l β + t)) .

This argument can be iterated for the integrals dω
, where we have to take into account the double sum ∑ 0≤l i ,l i <k(ω * i ) . However, it only affects the constant in front of .
(The presence of the sum ∑ 0≤l<k(ω * ) in (6.26) does not affect the core of the argument.) This completes the proof of Proposition 2 and Lemma 5.

Estimates for the Change in the Energy
In this section, we assess the expression (4.8) and complete the proof of Theorem 10. The argument is based on the same idea as in Section 8.6 of [12] (again, we partially borrow the system of notation from there). In the course of the argument, we will produce a further (and final) specification of the set G L ⊂ W * a (R 2 ) of good loop configurations. Namely, given Ω * ∈ W * a (R 2 ), we set, as before, Here, E [{T ± (s)Ω * } Λ (t)|{Ω * Λ c }(t)] is defined as the sum is obtained by omitting the terms containing s (cf. (2.3.11) and (2.3.12)). Recall, our aim is to guarantee that on the good set G L , the absolute value of the variable Σ h L (Ω * ) is small. Two straightforward bounds turn out to be helpful:
A formal summary of properties of transformations T ± (s) is given in the following Theorem: Theorem 11. Given Ω * ∈ G L , the transformations T ± L (s) : Ω * → Ω * ∈ W * a (R 2 ) possess the following properties: (i) The maps T ± (s) are measurable and 1 − 1.
The assertions of Theorems 8 and 10 then follow.

Concluding Remarks and Future Research
The series of publications involving the present authors, [23,28,29] and [24,31], have been motivated, on the one hand, by a spectacular success on Mermin-Wagner type theorems, [6], achieved in the past for a broad class of two-dimensional classical and quantum systems and, on the other hand, by a recognised progress in experimental quantum physics creating and working with thin materials like graphene. There has been increasing interest in graphene since its discovery. Much research has been done on this linear dispersion and, in particular, on the transport properties of graphene. This may be a topic of future research. We intend also to elaborate the similar technique for the Hubbard model, which is a highly oversimplified model for strongly interacting electrons in a solid, in line with [28]. 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.
Author Contributions: All the authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.
Funding: This work was partially funded by the Russian Academic Excellence Program '5-100'.