Advances in Mathematical Physics

Volume 2013 (2013), Article ID 637375, 20 pages

http://dx.doi.org/10.1155/2013/637375

Research Article

A Quantum Mermin-Wagner Theorem for a Generalized Hubbard Model

Mark Kelbert1,2 and Yurii Suhov2,3,4

¹Swansea University, Singleton Park, Swansea SA2 8PP, UK

²Instituto de Mathematica e Estatistica, USP, Rua de Matão, 1010, Cidada Universitária, 05508-090 São Paulo, SP, Brazil

³Statistical Laboratory, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK

⁴IITP, RAS, Bolshoy Karetny per. 18, Moscow 127994, Russia

Received 20 March 2013; Accepted 14 May 2013

Academic Editor: Christian Maes

Copyright © 2013 Mark Kelbert and Yurii Suhov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 where is a -dimensional unit torus with a flat metric. The phase space of spins is , the subspace of formed by functions symmetric under the permutations of the arguments. The Fock space yields the phase space of a system of a varying (but finite) number of particles. We associate a space with each vertex of a graph satisfying a special bidimensionality property. (Physically, vertex 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) , the minus a half of the Laplace operator on , 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 , , describing a field generated by a heavy atom, (b) two-body potentials , , showing the interaction between pairs of particles belonging to the same atom, and (c) two-body potentials , , scaled along the graph distance between vertices , 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 acts on , represented by a Euclidean space or torus of dimension , preserving the metric and the volume in . Furthermore, we suppose that the potentials , , and are -invariant. The result of the paper is that any (appropriately defined) Gibbs states generated by the above Hamiltonian is -invariant, provided that the thermodynamic variables (the fugacity and the inverse temperature ) 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 has been given, with the set of vertices and the set of edges . The graph has the property that whenever edge , the reversed edge belongs to as well. Furthermore, graph is without multiple edges and has a bounded degree; that is, the number of edges with a fixed initial or terminal vertex is uniformly bounded:

The bi-dimensionality property is expressed in the bound

where stands for the set of vertices in at the graph distance from (a sphere of radius around ):

(The graph distance between is determined as the minimal length of a path on joining and .) This implies that for any the cardinality of the ball

grows at most quadratically with .

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 we associate a copy of a compact manifold which we take in this paper to be a unit -dimensional torus with a flat metric and the volume . We also associate with a bosonic Fock-Hilbert space . Here where is the subspace in formed by functions symmetric under a permutation of the variables. Given a finite set , we set . An element is a complex function:

Here is a collection of finite point sets associated with sites . Following [1], we call a particle configuration at site (which can be empty) and a particle configuration in, or over, . The space of particle configurations in can be represented as the Cartesian product where is the disjoint union and is the collection of (unordered) -point subsets of . (One can consider as the factor of the “off-diagonal” set in the Cartesian power under the equivalence relation induced by the permutation group of order .) The norm and the scalar product in are given by

where measure is the product and is the Poissonian sum measure on :

Here is the volume of torus .

As in [1], we assume that an action

is given, of a group that is a Euclidean space or a torus of dimension . The action is written as

Here vector with components and is the -dimensional vector representing the element , where is a matrix of column rank with rational entries. The action of is lifted to unitary operators in :

where and .

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 of the system in acts as follows:

Here means the Laplacian in variable . Next, stands for the cardinality of the particle configuration (i.e., when ), and the parameter is introduced in (17). (Symbol will be used for denoting the cardinality of a general (finite) set; for example, means the number of vertices in .) Further, denotes the particle configuration with the point removed and point added to .

As in [1], we also consider a Hamiltonian in an external field generated by a configuration where is a (finite or infinite) collection of vertices. More precisely, we only consider with (see (17) below) and set

The novel elements in (11) and (12) compared with [1] are the presence of on-site potentials and and the summand involving transition rates for jumps of a particle from site to .

We will suppose that vanishes if the graph distance . We will also assume uniform boundedness:

in view of (1) it implies that the total exit rate from site 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 is replaced by a fermonic one. Another opening provided by this model is a possibility to consider random potentials , and 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 is assumed to be of class . Consequently, and its first and second derivatives satisfy uniform bounds. Namely,

Here and run through the pairs of variables , . A similar property is assumed for the on-site potential (here we need only a smoothness):

Note that for and the bounds are imposed on their negative parts only.

As to , we suppose that (a)

and (b) a -function such that whenever . Here stands for the (flat) Riemannian distance between points . As a result of (16), there exists a “hard core” of diameter , and a given atom cannot “hold” more than

particles where is the volume of a -dimensional ball of diameter . We will also use the bound

Formally, (16) means that the operators in (11) and (12) are considered for functions vanishing when in the particle configuration , the cardinality for some .

The function is assumed monotonically nonincreasing with and obeying the relation as , where

Additionally, let be such that

Next, we assume that the functions , , and are -invariant: and ,

In the following we will need to bound the fugacity (or activity, cf. (25)) in terms of the other parameters of the model

1.4. The Gibbs State in a Finite Volume

Define the particle number operator , with the action

Here, for a given , stands for the total number of particles in configuration :

The standard canonical variable associated with is activity .

The Hamiltonians (11) and (12) are self-adjoint (on the natural domains) in . Moreover, they are positive definite and have a discrete spectrum, cf. [14]. Furthermore, , and give rise to positive-definite trace-class operators and :

We would like to stress that the full range of variables is allowed here because of the hard-core condition (16): it does not allow more than particles in where stands for the number of vertices in . However, while passing to the thermodynamic limit, we will need to control and .

Definition 1. We will call and the Gibbs operators in volume , for given values of and (and—in the case of —with the boundary condition ).

The Gibbs operators in turn give rise to the Gibbs states and , at temperature and activity in volume . These are linear positive normalized functionals on the -algebra of bounded operators in space :

where

The hard-core assumption (16) yields that the quantities and are finite; formally, these facts will be verified by virtue of the Feynman-Kac representation.

Definition 2. Whenever , the -algebra is identified with the subalgebra in formed by the operators of the form . Consequently, the restriction of state to -algebra is given by

where

Operators (we again call them RDMs) are positive definite and have . They also satisfy the compatibility property: ,

In a similar fashion one defines functionals and operators , with the same properties.

1.5. Limiting Gibbs States

The concept of a limiting Gibbs state is related to notion of a quasilocal -algebra; see [14]. For the class of systems under consideration, the construction of the quasilocal -algebra is done along the same lines as in [1]: is the norm completion of the algebra . Any family of positive-definite operators in spaces of trace one, where runs through finite subsets of , with the compatibility property

determines a state of , see [12, 13].

Finally, we introduce unitary operators , , in :

where

Theorem 3. Assuming the conditions listed above, for all satisfying (22) and a finite , operators form a compact sequence in the trace-norm topology in as . Furthermore, given any family of (finite or infinite) sets and configurations with , operators also form a compact sequence in the trace-norm topology. Any limit point, , for or as , is a positive-definite operator in of trace one. Moreover, if and and are the limits for and or for and along the same subsequence , then the property (32) holds true.

Consequently, the Gibbs states and form compact sequences as .

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

Definition 5. Any limit point for states and is called a limiting Gibbs state (for given ).

Theorem 6. Under the condition (22), any limiting point, , for or , as , is a positive-definite operator of trace one commuting with operators : ,

Accordingly, any limiting Gibbs state of determined by a family of limiting operators obeying (35) satisfies the corresponding invariance property: finite , any , and ,

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 and of families and . However, they require a proof for the limit points of the families and .

The set of limiting Gibbs states (which is nonempty due to Theorem 3) is denoted by . In Section 3 we describe a class of states of -algebra 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 and of operators and .

Definition 7. Given , denotes the space of path, or trajectories, in , of time-length , with the end-points and . Formally, is defined as follows:

The notation and its alternative, , for the position and the index of trajectory at time will be employed as equal in rights. We use the term the temporal section (or simply the section) of path at time .

Definition 8. Let , and be particle configurations over , with . A matching (or pairing) between and is defined as a collection of pairs , with , , and , with the properties that (i) and there exist unique and such that and form a pair, and (ii) and there exist unique and such that and form a pair. (Owing to the condition , these properties are equivalent.) It is convenient to write .

Next, consider the Cartesian product

and the disjoint union

Accordingly, an element in (38) represents a collection of paths , , , of time-length , starting at and ending up at . We say that is a path configuration in (or over) .

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 , , and .

Definition 9. In what follows, , , stands for the Markov process on , with cádlág trajectories, determined by the generator acting on a function by

In the probabilistic literature, such processes are referred to as Lévy processes; see, for example, [14].

Pictorially, a trajectory of process moves along according to the Brownian motion with the generator and changes the index from time to time in accordance with jumps occurring in a Poisson process of rate In other words, while following a Brownian motion rule on , having index and being at point , the moving particle experiences an urge to jump from to a neighboring vertex and to a point at rate . After a jump, the particle continues the Brownian motion on from and keeps its new index until the next jump, and so on.

For a given pairs , we denote by the nonnormalised measure on induced by . That is, under measure the trajectory at time starts from the point and has the initial index while at time it is at the point and has the index . The value is given by

where denotes the transition probability density for the Brownian motion to pass from to on in time :

In view of (13), the quantity and its derivatives are uniformly bounded:

where 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 and are particle configurations over , with . Let be a pairing between and . Then denotes the product measure on :

Furthermore, stands for the sum measure on :

According to Definition 10, under the measure , the trajectories constituting are independent components. (Here the term independence is used in the measure-theoretical sense.)

As in [1], we will work with functionals on representing integrals along trajectories. The first such functional, , is given by

Here, introducing the notation and for the positions in of paths and at time , we define

Next, with and standing for the indices of and at time ,

Next, consider the functional : for . As before, we assume that . Define

Here is as in (46) and

where, in turn,

The functionals and are interpreted as energies of path configurations. Compare and in [1].

Finally, we introduce the indicator functional :

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 and a finite , the Gibbs operators and act as integral operators in :

Moreover, the integral kernels and vanish if . On the other hand, when , the kernels and admit the following representations:

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 such that and , the number of paths with index is less than or equal to . Likewise, the integral in the RHS of (55) receives a non-zero contribution only from configurations such that, site , the number of paths with index plus the cardinality does not exceed .

2.2. The Representation for the Partition Function

The FK representations of the partition functions in (27) and in (1.4.6) reflect a specific character of the traces and in . The source of a complication here is the jump terms in the Hamiltonians and in (11) and (12), respectively. In particular, we will have to pass from trajectories of fixed time-length to loops of a variable time length. To this end, a given matching 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 . (A similar construction has been performed in [18].)

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

Definition 13. For given , the symbol denotes the disjoint union:

In other words, is the space of paths in , of a variable time-length , where takes values and called the length multiplicity, with the end-points and . The formal definition follows the same line as in (37), and we again use the notation and the notation for the pair of the position and the index of path at time . Next, we call the particle configuration the temporal section (or simply the section) of at time . We also call a path (from to ).

A particular role will be played by closed paths (loops), with coinciding endpoints (where ). Accordingly, we denote by the set . An element of is denoted by or, in short, by and called a loop at vertex . (The upper index indicates that the length multiplicity is unrestricted.) The length multiplicity of a loop is denoted by or . It is instructive to note that, as topological object, a given loop admits a multiple choice of the initial pair : it can be represented by any pair at a time where . As above, we use the term the temporal section at time for the particle configuration and employ the alternative notation addressing the position and the index of at time .

Definition 14. Suppose and are particle configurations over , with . Let be a matching between and . We consider the Cartesian product:

and the disjoint union:

Accordingly, an element in (58) represents a collection of paths , , , of time-length , starting at and ending up at . We say that is a path configuration in (or over) .

Again, loops play a special role and deserve a particular notation. Namely, denotes the Cartesian product:

and stands for the disjoint union (or equivalently, the Cartesian power):

Denote by a collection of loop configurations at vertices starting and ending up at particle configurations (note that some of the ’s may be empty). The temporal section (or, in short, the section), , of at time is defined as the particle configuration formed by the points where , , and .

As before, consider the standard sigma algebras of subsets in the spaces , , , , , and introduced in Definitions 13 and 14. In particular, the sigma algebra of subsets in will be denoted by ; we comment on some of its specific properties in Section 3.1. (An infinite-volume version of is treated in Section 3.2 and after.)

Definition 15. Given points , we denote by the sum measure on :

Further, denotes the similar measure on :

Definition 16. Let and be particle configurations over , with . Let be a matching between and , and we define the product measure :

and the sum measure

Next, symbol stands for the product measure on :

Finally, yields the measure on :

Here, for , we set: . For sites with , 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 and which are modifications of the above functionals and ; confer (46) and (50). Say, for a loop configuration over with an initial and final particle configuration ,

To determine the functionals and , we set, for given and :

A (slightly) shortened notation is used for the index and for the position for , of the section of the loop at time , and similarly with and . (Note that the pairs and may coincide.) Then

Next, the functional takes into account the bosonic character of the model:

The factor in (70) reflects the fact that the starting point of a loop may be selected among points arbitrarily.

Next, we define the functional : for , again assuming that . Set

Here is as in (67) and

where, in turn,

As before, the functionals and have a natural interpretation as energies of loop configurations.

Finally, as before, the functional is the indicator that the collection of loops does not quit :

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 and satisfying (22), the partition functions in (27) and in (27) and (28) admit the representations as converging integrals:

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 such that vertex and , the total number of pairs with , and does not exceed .

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 be finite sets, . The construction developed in Section 2.2 also allows us to write a convenient representation for the integral kernels of the RDMs (see (31)) and . In accordance with Lemma 11 and the definition of in (31), the operator acts as an integral operator in :

where

We employ here and below the notation and for particle configurations and over . Next, ; denotes the concatenated configurations over .

Similarly, the RDM is determined by its integral kernel , again admitting the representation

As in [1], we call and the RDM kernels (in short, RDMKs). The focus of our interest is the numerators and in (78) and (79). To introduce the appropriate representation for these quantities, we need some additional definitions.

Definition 19. Repeating (57)-(58), symbol denotes the disjoint union over matchings between and . Accordingly, element yields a collection of paths lying in . Each path has time lengths , begins at , and ends up at , where , . Like above, we will use for the term a path configuration over . Repeating (63)-(64), we obtain the measures on and on .

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

Lemma 20. The quantity emerging in (78) is set to be when . On the other hand, for ,

where

Similarly, the quantity from (79) vanishes when . For ,

where

These representations hold and finite .

Let us define the functionals , , , , , , and in (79)–(83). (The functionals and are defined as (70) and (74), respectively, replacing with .)

To this end, let be a path configuration represented by a collection of paths ( in short), with end points and , of time-length . The functional is given by

The functional is again an indicator:

Now let us define the indicator function in (80)–(83). The factor equals one if and only if every path from , of time-length , starting at and ending up at remains in at the intermediate times for :

(when  , this is not a restriction).

Furthermore, suppose that is a loop configuration over , with the initial/end configuration , represented by a collection of loops . Then if and only if each loop of time-length , beginning and finishing at , does not enter the set at times for :

(again, if , this is not a restriction).

The functional in (80) gives the energy of the path configuration and is introduced similarly to (67), mutatis mutandis. Next, the functional in (81) represents the energy of the loop configuration in the potential field generated by the path configuration :

Here, the summand yields the energy of the loop configuration ; again confer (67). Further, the term yields the energy of interaction between and : for a path/loop configurations and a we set

Here, for a path , of time-length , and a loop , of time-length ,

Here, in turn, we employ the shortened notation for the positions and indices of the sections and of and at times and , respectively:

Further, the functional in (82) is determined as in (71)–(73), with instead of . Next, for in (83), we set

Here again, the summand is determined as in (67). Next, the term is defined similarly to (72)-(73):

with

As before, the functionals and 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 such that and , the total number of pairs with , and incident to the paths of the configuration and pairs with , and incident to the loops of the configuration does not exceed . Similarly, non-zero contributions to the integral in (82) come only from pairs such that the above inequality holds when we additionally count points .

The integral defined in (81) can be considered as a particular (although important) example of a partition function in the volume with a boundary condition . Note the presence of the subscript indicating that the loops contributing to can jump within volume only (owing to the indicator functional ). On the other hand, the presence of the indicator functional in the integral (reflected in the upperscript and the roof sign in the notation ) indicates a particular restriction on the jumps of the loops, forbidding them to visit set at intermediate times . This is true also for the integral in (83): it is a particular example of a partition function in the volume with a boundary condition .

Other useful types of partition functions are and where the sets of vertices , , , , and satisfy

and . Accordingly, is a (finite) configuration over , a (possibly infinite) loop configuration over , and a (possibly infinite) particle configuration over . The partition functions and are given by

with the indicator as in (74). These partition functions, feature loop configurations formed by loops , , which start and finish in , are confined to and move in a potential field generated by , where and or , where . (The latter can be understood as the concatenation of the loop configuration over and the loop configuration over formed by the constant trajectories sitting at points , .) In (96) we assume that, and , the number

does not exceed . Analogously, in (97) it is assumed that the same is true for the above number plus the cardinality .

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

2.4. The FK-DLR Measure in a Finite Volume

The Gibbs states and give rise to probability measures and on the sigma algebra of subsets of . The sigma algebra is constructed by following the structure of the space (a disjoint union of Cartesian products); confer Definition 16. The measures and are determined by their Radon-Nykodym derivative and relative to the measure :

Given , the sigma algebra is naturally identified with a sigma subalgebra of . The restrictions of to and are denoted by and ; these measures are determined by their Radon-Nikodym derivatives and .

The first key property of the measures and 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 .

Lemma 21. For all satisfying (22), and , the probability density admits the form:

where

and the conditional partition functions and are determined as in (96).

Similarly, for one has:

where

and the conditional partition functions and are determined as in (96).

As in [1], (100) and (102) mean that the conditional densities and relative to -algebra coincide, respectively, with , and , for —and —a.a. and a.a. .

As in [1], we call the expressions and , as well as the expressions and appearing below, the (conditional) RDM functionals (in brief, the RDMFs). The same name will be used for the quantity from (110)-(111) and the quantity from (112)-(113).

The second property is that the RDMKs and are related to the measures and . Again, the proof of this fact is done by inspection.

Lemma 22. The RDMK is expressed as follows: ,

where

Similarly,

where

Here the partition functions and are determined in (81) and (83).

Remark 23. Summarizing the above observations, the measures and are concentrated on the subset in formed by loop configurations such that , the section has ≤κ particles at each vertex .

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

3.1. Definition of Class

In this section we apply the idea from [1] to define the class of states 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 is based on the notion of an FK-DLR probability measure on the space ; the class of these measures will be also denoted by .

Definition 24. Space is the (infinite) Cartesian product (cf. (60)); its elements are loop configurations over . A component is a finite loop configuration (possibly, empty), with an initial/final particle configuration . Formally, is a finite collection of loops , of time-length where , starting and finishing at a point . For reader’s convenience, we repeat (37) for the case under consideration:

By we denote the -algebra in generated by cylindrical events. Given a subset (finite or infinite), we denote by the -subalgebra of generated by cylindrical events localized in . Given a probability measure on , we denote by the restriction of on .

Definition 25. The class under consideration is formed by measures which satisfy the following equation: finite and , the probability density:

is of the form

where

and the conditional partition functions and are determined as in (96).

As in [1], (110) means that the conditional density , relative to -algebra , coincides with , for —a.a. and —a.a. .

Remark 26. The measure inherits the property from Remark 23 and is concentrated on the subset in formed by (infinite) loop configurations such that, for all , the section has ≤κ particles at each vertex .

Given a measure , we associate with it a normalized linear functional on the quasilocal -algebra . First, we set

where

This defines a kernel , , where is a finite set of sites. It is worth reminding the reader of the presence of the indicator functionals in (112) and (113) (in the integral for ). These indicators guarantee the compatibility property: finite ,

Next, we identify the operator (a candidate for the RDM in volume ) as an integral operator acting in by

Equation (114) implies that

Definition 27. The functional is identified with the (compatible) family of operators . If the operators 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 and inverse temperature ). To stress the dependence on and , we sometimes employ the notation .

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

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

Theorem 28. For all satisfying (22), any limiting Gibbs state (see Theorem 3) lies in . Therefore, the class of state s is nonempty.

Theorem 29. Under condition (22), any FK-DLR state is -invariant, in the sense that, finite and , the RDM 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 , we establish compactness of the sequence of the RDMKs and (see (78)–(83)) as functions of variables , with

when . Then we use Lemma 1.1 from [1] to derive that the sequence of the RDMs and is compact in the trace-norm operator topology in .

To verify compactness of the RDMKs and 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 and admit the bounds

where

(Note that under the assumption (22)) Let yields the supremum of the transition function over time for Brownian motion on the torus :

and . Finally, the upper-bound values , , , and have been determined in (14), (15), and (18).

(ii) The gradients of the RDMKs and satisfy and , ,

where

(Again, under the condition (22).) The ingredients and , , , and as in statement (i).

Proof of Lemma 30. First, observe that on a compact manifold. Bound (119) is established in a direct fashion. First, we majorize the energy

contributing to the RHS in (80) and (81) and the energy

contributing to the RHS in (82) and (83). This yields the factor

Next, we majorize the integral in (80) and (82); this gives the factor

The aftermath are the ratios (78) and (79) with ; they do not exceed .

Passing to (121), let us discuss the gradients only. (The gradients in the entries of are included by symmetry.) The gradient in (121), of course, affects only the numerators and in (78) and (79). The bounds (121) are done essentially as in [1]. For definiteness, we discuss the case of the RDMK ; the RDMK is treated similarly. There are two contributions into the gradient: one comes from varying the measure , and the other from varying the functional .

The first contribution can again be uniformly bounded in terms of the constant . 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 yields a uniform bound in (43)).

The second contribution yields, again as in [1], an expression of the form:

where the functional 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 and converge to a limiting RDM along a subsequence in . The diagonal process yields convergence for every finite . A parallel argument leads to compactness of the measures for any given as . We only give here a sketch of the corresponding argument, stressing differences with its counterpart in [1].

In the probabilistic terminology, measures represent random marked point fields on with marks from the space where and is the space of loops of time-length starting and finishing at and exhibiting jumps, that is, changes of the index. (The space introduced in Definitions 7 and 13 can be considered as a copy of placed at site and point .) The measure describes the restriction of to volume (i.e., to the sigma algebra ) and is given by its Radon-Nikodym derivative relative to the reference measure on (cf. (66), (100)). The reference measure is sigma-finite. Moreover, under the condition (22), the value is uniformly bounded (in both and ). This enables us to verify tightness of the family of measures and apply the Prokhorov theorem. Next, we use the compatibility property of the limit-point measures and apply the Kolmogorov theorem. This establishes the existence of the limit-point measure .

By construction, and owing to Lemmas 21 and 22, the limiting family yields a state belonging to the class . 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 and introduced in (111) and (112). For definiteness, we assume that vertex , so that lies in the ball for large enough. As in [1], the problem is reduced to checking that satisfying (22), and finite ,

here we need to establish this convergence (128) uniformly in the argument with and in outside a set of the measure tending to as . The latter is formed by path configurations that contain trajectories visiting sites where grows with ; see Lemma 31 below. The action of upon a path configuration is defined by

We want to establish that , for any large enough, the conditional RDMFs satisfy

As in [1], we deduce (130) with the help of a special construction of “tuned” actions on loop configurations (over which there is integration performed in the numerators

in the expression for and ). The tuning in is chosen so that it approaches (or the -dimensional zero vector in the additive form of writing), the neutral element of , while we move from towards .

Formally, (130) follows from the estimate (132) below: finite ,, and , for any large enough, and with ,

In (132), the loop configuration is determined by specifying its temporal section . That is, we need to specify the sections:

for loops constituting . To this end we set

In other words, we apply the action to the temporal sections of all loops located at vertex at a given time, regardless of position of their initial points in .

Observe that (130) is deduced from (132) by integrating in and normalizing by ; see (80) with . (The Jacobian of the map equals .)

Thus, our aim becomes proving (132). The tuned family is composed of individual actions :

Elements are powers (multiples, in the additive parlance) of element figuring in (128)–(132) (resp., of the corresponding vector ; cf. (9)) and defined as follows. Let denote the vector from corresponding to , and we select positive integer values and set

where

In turn, the function is chosen to satisfy

with

Lemma 31. Given satisfying (22) and a finite set , there exists a constant such that , the set of path configurations with which include trajectories visiting points in has the measure that does not exceed .

Proof of Lemma 31. The condition that implies that the total number of sub trajectories of length in does not exceed which is a fixed value in the context of the lemma. Each such trajectory has a Poisson number of jumps; this produces the factor .

Back to the proof of Theorem 29: let be the collection of the inverse elements:

The vectors corresponding to are . We will use this specification for and for , or even for , as it agrees with the requirement that when and for . Accordingly, we will use the notation .

Observe that the tuned family does not change the contribution into the energy functional coming from potentials and : it affects only contributions from potential .

The Taylor formula for function , together with the above identification of vectors , gives

Here is a constant, stands for the norm of the vector representing the element , and the value is taken from (14).

Next, the square can be specified as

By using convexity of the function exp and (141), ,

where

The next observation is that

where, owing to the triangle inequality, for all

This yields

where function is determined in (139).

Therefore, it remains to estimate the sum . To this end, observe that when . The next remark is that the number of sites in the sphere grows linearly with . Consequently,

Therefore, given for large enough, the term in the RHS of (145) becomes >1. Hence,

Equation (149) implies that the quantity

obeys

uniformly in boundary condition . Integrating (151) 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.

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