On the construction of point processes in statistical mechanicsBenjamin Nehring, Suren Poghosyan, and Hans Zessin Citation: J. Math. Phys. 54, 063302 (2013); doi: 10.1063/1.4807724 View online: http://dx.doi.org/10.1063/1.4807724 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v54/i6 Published by the American Institute of Physics. Additional information on J. Math. Phys.Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors

JOURNAL OF MATHEMATICAL PHYSICS 54, 063302 (2013)

On the construction of point processes in statisticalmechanics

Benjamin Nehring,1,a) Suren Poghosyan,2,b) and Hans Zessin3,c)

1Institut fur Mathematik der Universitat Potsdam, Am Neuen Palais 10,D-14469 Potsdam, Germany2Institute of Mathematics, Armenian National Academy of Sciences,Marshal Bagramian 24-B, Yerevan 375019, Armenia3Fakultat fur Mathematik der Universitat Bielefeld, Postfach 10 01 31,D-33601 Bielefeld, Germany

(Received 30 January 2013; accepted 10 May 2013; published online 7 June 2013)

We present a new approach to the construction of point processes of classical statisti-cal mechanics as well as processes related to the Ginibre Bose gas of Brownian loopsand to the dissolution inRd of Ginibre’s Fermi-Dirac gas of such loops. This approachis based on the cluster expansion method. We obtain the existence of Gibbs perturba-tions of a large class of point processes. Moreover, it is shown that certain “limitingGibbs processes” are Gibbs in the sense of Dobrushin, Lanford, and Ruelle if the un-derlying potential is positive. Finally, Gibbs modifications of infinitely divisible pointprocesses are shown to solve a new integration by parts formula if the underlying po-tential is positive. C© 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4807724]

I. INTRODUCTION

We reconsider the problem of construction of interacting point processes which are of importancein statistical physics. They include Gibbs processes of classical statistical mechanics and processeswhich are associated with continuous quantum systems in the sense of Ginibre,2 the so-called gasesof winding Brownian loops.

Earlier approaches can be found in the work of Kondratiev et al.5 in the case of Boltzmannstatistics, and in the thesis of Kuna6 as well as in Rebenko.17 But several questions are left openhere. The method we use is a new version of cluster expansions which had been developed inRefs. 12 and 13 and which is summarized in Theorem 1.

By combining this method with the abstract cluster expansions from Ref. 16 we then construct, inTheorem 2, limiting interacting processes in the context of statistical mechanics. As a first applicationwe consider the interacting quantum Bose gas in Feynman-Kac representation. This yields a pointprocess corresponding to the Bose gas of interacting winding loops. This result seems to be new.

One of the main assumptions of Theorem 2 is the positivity of the reference measure. This isnot the case for the Fermi loop gas where appears a signed reference measure. Therefore, assuminghypothetically the existence of a cluster process for the Fermi gas of polygonal loops, and dissolvingclusters into its particles, one would obtain a Gibbs modification of a determinantal point processin Euclidean space. We are able to construct this process by means of our method. In a moregeneral setting such processes are then constructed in Theorem 3. As examples we consider Gibbsmodifications of the Poisson as well as determinantal and permanental processes.

In Theorem 4, we show that under natural regularity conditions the limiting processes areGibbs in the sense of Dobrushin/Lanford/Ruelle (DLR) if the underlying interaction is positive.Finally, it is shown that Gibbs modifications with positive pair potential of infinitely divisible

a)E-mail: benjamin.nehring@gmail.comb)E-mail: suren.poghosyan@unicam.itc)E-mail: zessin@math.uni-bielefeld.de

0022-2488/2013/54(6)/063302/15/$30.00 C©2013 AIP Publishing LLC54, 063302-1

063302-2 Nehring, Poghosyan, and Zessin J. Math. Phys. 54, 063302 (2013)

point processes solve a new integral equation involving the Campbell measure of the process. Thisequation generalizes the integration by parts formula of Nguyen et al.15 which is equivalent to theDLR-equation. Examples of such processes are Gibbs modifications of the ideal Bose gas.

II. RANDOM MEASURES AND POINT PROCESSES

We introduce some basic concepts and standard results from the theory of point processes whichwe take from the monographs.4, 7, 9 The basic underlying phase space is a Polish space (X,B,B0);i.e., a complete separable metric space (X, d). Our main examples of phase spaces are discrete spaces,the Euclidean space E = Rd , the space X of finite configurations in E which may have multiplepoints, and the space of Brownian loops in E. B denotes the corresponding Borel σ -field and B0 thering of all bounded sets in X.

ByM = M(X ), we denote the set of all measures μ onB taking only finite values onB0. We callthem Radon measures here. Denote by F the set of all B-measurable mappings f : X −→ [0,∞],and by Fc the subset of all bounded and continuous f ∈ F with bounded support supp f . We alsoneed the space Fb of bounded f ∈ F with bounded support. Denote by

ζ f (μ) = μ( f ) =∫

Xf (x) μ(d x), μ ∈ M, f ∈ F,

the integral as a function of the underlying measure. The vague topology on M is defined as thetopology generated by all mappings ζ f, f ∈ Fc. And M, provided this topology, is a Polish space; thecorresponding σ -algebra of Borel subsets B(M) is the one generated by all mappings ζB, B ∈ B0.A random measure on the phase space X is a random element in M(X ). But we will also considerother measures on M.

A measure μ ∈ M is called a counting or point measure if it takes only integer values on B0.The set of all point measures is denoted by M·· = M··(X ). M··, considered as a subspace of M, isvaguely closed and thereby a Borel set in M. Moreover, it is a Polish space. X = M··

f (X ) denotesthe subset of finite counting measures on X. If G is a Borel set in X, then we denote by X(G) thecollection of all configurations contained in G. Now a point process in X is a random element inM··(X ).

The Laplace transform of a random measure P is defined by

LP ( f ) =∫M

exp(−ζ f ) d P, f ∈ F.

It determines the process P completely. The first moment measure of P is defined by

νP ( f ) =∫M··

μ( f ) P(d μ), f ∈ F.

If νP is a Radon measure we say that P is of first order. A more general notion containing this oneis the Campbell measure of P defined by

CP (h) =∫M··

∫X

h(x, μ) μ(d x)P(d μ), h ∈ F.

We will use these notions also when P is replaced by σ -finite measures L.

III. A GENERAL CONSTRUCTION BY MEANS OF THE CLUSTER EXPANSION METHOD

We consider the construction of point processes by means of the cluster expansion method onan abstract level first in the finite case within the setting of Ref. 8. Then we indicate briefly theinfinitely extended case. This construction has been developed in Refs. 12 and 13.

Let E denote the set of finite, signed measures L, M on M··(X ). Each L can be represented in aunique way as the difference L = L+ − L− of measures in E+, the subspace of positive measuresin E, that are purely singular with respect to each others (Jordan decomposition of L). E is a normedspace with respect to ‖L‖ = L+(M··) + L−(M··), the total mass of the variation |L| = L+ + L−

of L. The distance ‖M − L‖ is called variation distance.

063302-3 Nehring, Poghosyan, and Zessin J. Math. Phys. 54, 063302 (2013)

With respect to the convolution operation * and ‖.‖, the vector space E is a commutative realBanach algebra with unit δo, o denoting here the measure zero on M··(X ). Thus in particular ‖L*M‖≤ ‖L‖ · ‖M‖. For all L ∈ E the series

exp L =∞∑

n=0

1

n!L∗n, L∗0 = δo

converges absolutely. All this can be found in Sec. 1.9. of Ref. 8.We are now given a family of positive, symmetric Radon, i.e., locally finite, measures �±

m onXm, m ≥ 1,. These give rise to the measures L± by means of

L±(ϕ) =∞∑

m=1

1

m!

∫Xm

ϕ(δx1 + · · · + δxm ) �±m(d x1 . . . d xm). (1)

Here L± are positive measures on the space X with L± {0} = 0; and we assume the integrabilitycondition

(A) L±(1 − e−ζ f ) < +∞, f ∈ Fb.

It is shown in Ref. 12 that the condition (A) implies that |�m| are Radon measures. Then�m = �+

m − �−m are signed Radon measures. We call them the cumulant measures belonging to the

signed Levy measure L = L+ − L− .

Theorem 1 (Refs. 12 and 13). Let L± be measures on X, given in terms of cumulant measures�±

m by means of (1), satisfying the integrability condition (A). Assume that the corresponding Schurmeasures21

�k(⊗nj=1 f j ) :=

∑J∈π([n])

∏J∈J

�|J |(⊗ j∈J f j ), f1, . . . , fn ∈ Fb

are all non-negative.22 Then there exists a point process P in X with Laplace transform KL ( f )= e−L(1−e−ζ f ), f ∈ Fb. The process P will be denoted P = �L. P is the weak limit of the finiteprocesses

QG(ϕ) = 1

(G)

∞∑n=0

1

n!

∫Gn

ϕ(δx1 + . . . + δxn ) �n(d x1 . . . d xn),

as B0(X ) G ↑ X .

The strategy of the proof: Given G ∈ B0(X ) we localize L± by means of L±G = 1X(G) · L±

and set LG = L+G − L−

G. Here X(G) = M··(G). LG is a finite signed measure on X because of theintegrability assumption (A). By computing Laplace transforms, we obtain that

QG = 1

G· exp LG, G = exp LG(X). (2)

In particular QG = �LG . The second representation (2) of the QG will be needed in the proof toTheorem 3.

Another comment is in order here. The cluster expansion construction of the point process Pis based on two assumptions: The integrability condition (A) of L and the positivity of the Schurmeasures �k. We will see in Sec. IV that in case, where �m are defined by means of the Ursellfunctions for some underlying pair potential, the verification itself of condition (A) is actually anessential part of the cluster expansion method. For this, one has to recall that in this case LG(X)has the meaning of the log-partition function, so that the finiteness of LG is in fact equivalent to theabsolute convergence of the “traditional” cluster expansion of the log-partition function.

Example 1. Let λ ∈ M(X ) and z ∈ (0, 1). Consider the following two families ε = + 1, − 1 ofcumulant measures

�m(ε)(d x1 . . . d xn) = εm−1 zm (m − 1)! δxn (d x1) . . . δxn (d xn−1)λ(d xn).

063302-4 Nehring, Poghosyan, and Zessin J. Math. Phys. 54, 063302 (2013)

The family {�m(+1)}∞m=1 satisfies the conditions of Theorem 1 and the corresponding point processis the so-called Polya sum process as introduced in Ref. 20. In case ε = − 1, we have to additionallyassume that λ ∈ M··(X ) to ensure the non-negativity of the Schur measures. The correspondingpoint process is called the Polya difference process.14

Example 2. Let λ ∈ M(X ), z ∈ (0, ∞), and K : X × X → R a symmetric non-negative definiteand bounded kernel such that supx∈X

∫ |K |(x, y) λ(d y) is finite. Consider the following two familiesε = + 1, − 1 of cumulant measures

�m(ε)(d x1 . . . d xn) = εm−1zm∑ω∈Scy

m

m∏j=1

K (x j , xω( j)) λ(d x1) . . . λ(d xn),

where Scym denotes the set of all permutations of the set {1, . . . , m}, which consist of one cycle. The

associated signed Levy measure can be written as

L(ε)(ϕ) =∞∑

m=1

εm−1 zm

m

∫Xm

ϕ(δx1 + · · · + δxm ) bx1m (d x2 . . . d xm) λ(d x1), (3)

where

bxm(d x2 . . . d xm) = K (x, x2) . . . K (xm, x) λ(d x2) . . . λ(d xm). (4)

The Schur measures are given by

�k(ε)(d x1 . . . d xk) = zk detε({K (xi , x j )}i, j

)λ(d x1) . . . λ(d xk),

where det−1 denotes the determinant and det+1 the permanent. In Ref. 12, it is shown that under thestated conditions and for small z > 0 the assumptions of Theorem 1 are satisfied. In case ε = − 1,we obtain a determinantal and in case ε = + 1 a permanental point process.

One of the main results (Theorem 3) of this paper concerns Gibbs modifications of the class ofpoint processes introduced in Theorem 1. That is, if we perturb the finite point processes QG with aclassical Hamiltonian

Qφ

G(ϕ) = 1

φ(G)

∞∑n=0

1

n!

∫Gn

ϕ(δx1 + . . . + δxn ) e−E(δx1 +...+δxn ) �n(d x1 . . . d xn),

where E denotes the energy given by a pair interaction, we give sufficient conditions such that theinfinite volume limit of the Qφ

G as G ↑ X does exist.

IV. POINT PROCESSES OF STATISTICAL MECHANICS

Here we consider the case where the cumulant measures �m are determined by Ursell functionsdefined for some underlying pair potential. It is shown in Ref. 16 that under natural and fairlygeneral conditions on the potential the integrability condition (A) holds true. The non-negativity ofthe Schur measures is satisfied on account of Ruelle’s algebraic method, if the reference measure ispositive. This yields a construction of a large class of processes which includes many examples fromstatistical physics. As main examples, we present the Bose process and some polygonal version ofthe Fermi process of interacting loops.

Given a Polish phase space (X,B,B0) together with some (signed) Radon measure � ∈ M(X )on it. Moreover, a measurable, symmetric function (a pair potential) u : X × X −→ R ∪ {+∞} isgiven. Recall that the corresponding Ursell function is defined by

Uu(x1, . . . , xm) =∑

G∈Cm

∏{i, j}∈G

(e−u(x,y) − 1

), m ≥ 2,

Uu(x1) = 1. Here Cm denotes the set of connected unoriented graphs with m vertices without loops.We consider now cumulant measures of the form

�±m(d x1 . . . d xm) = U±

u (x1, . . . , xm) �(d x1) . . . �(d xm). (5)

063302-5 Nehring, Poghosyan, and Zessin J. Math. Phys. 54, 063302 (2013)

Here U±u denotes the positive, respectively, negative part of the Ursell function. Poghosyan and

Ueltschi work under the following conditions:16

(B1) (weak stability). There exists b ∈ F such that for all n,

∑1≤i< j≤n

u(xi , x j ) ≥ −n∑

j=1

b(x j ) |�|n − a.s.[(x1, . . . , xn)].

(B2) (weak regularity). There exists a ∈ F such that∫X

|�|(d y) |1 − e−u(x,y) | · e(a +2 b)(y) ≤ a(x) |�| − a.s.[x].

We remark that for bounded functions a this implies the regularity of u in the sense of Ruelle.18

The following condition can replace (B2):

(B2′) There exists a ∈ F satisfying∫X

|�|(d y) |u(x, y)| · e(a + b)(y) ≤ a(x) |�| − a.s.[x],

where

u(x, y) ={

u(x, y), u(x, y) < ∞,

1, u(x, y) = ∞.(6)

(B3) (integrability of a, b),

ea +2 b ·|�| ∈ M(X ).

The measure here is |�| having density ea +2 b.

Under condition (B2′), we will always replace condition (B3) by the integrability condition

(B3′),

ea + b ·|�| ∈ M(X ).

The following basic theorem will serve as a main lemma in our reasoning.

Lemma 1 (Poghosyan/Ueltschi16). Assume conditions (B1), (B2) respectively (B1), (B2′). Thenthe following estimate is true: |�| − a.s.[x],∑

m≥1

1

(m − 1)!

∫Xm−1

|Uu(x, x1, .., xm−1)| |�|(d x1)...|�|(d xm−1)

≤ ea(x)+2 b(x) .

(Under condition (B2′) this holds true with eb(x) instead of e2 b(x).)

Let us denote by Eu(μ) the energy of a finite configuration μ defined by the pair potential u bymeans of

Eu(μ) =∑

1≤i< j≤n

u(xi , x j ), if μ =n∑

k=1

δxk . (7)

Theorem 2. If the measure � is positive then, under the above conditions on the pair potentialu, i.e., under (B1), (B2), and (B3) or (B1), (B2′), and (B3′); and for the signed Levy measure Rdefined by means of the Ursell functions the point process �R does exist. Moreover, it is the weak

063302-6 Nehring, Poghosyan, and Zessin J. Math. Phys. 54, 063302 (2013)

limit of the specification with empty boundary condition

QG(ϕ) = 1

(G)

∞∑n=0

1

n!

∫Gn

ϕ(δx1 + . . . + δxn ) e−Eu (δx1 +···+δxk ) �(d x1) . . . �(d xn),

as B0(X ) G ↑ X .

Proof. Lemma 1 implies that R satisfies condition (A). It even implies that | R | is of first order.This means that the intensity measure ν1

| R | of the variation of the signed Levy measure R is locallyfinite. For any f ∈ Fc,

ν1| R |( f ) :=

∫X

ζ f d | R | =

=∑m≥1

1

(m − 1)!

∫Xm

f (x) · |Uu(x, x1, .., xm−1)| |�|(d x)|�|(d x1)..|�|(d xm−1)

< ∞.

Here one uses (B3), respectively, (B3′). This is the main consequence, and we are in the situation ofTheorem 1 above, if the Schur measures are positive. Thus it remains to show the positivity of thesemeasures. Note first that the measures �k can be represented as

�k(d x1 . . . d xk) =∑J

∏J∈J

Uu((x j ) j∈J ) �(d x1) . . . �(d xk).

On the other hand, the density here is given by Ruelle’s algebraic exponential (cf. Sec. 4.4 ofRef. 18), ∑

J∈π([k])

∏J∈J

Uu((x j ) j∈J ) = exp(−Eu(δx1 + · · · + δxk )), (8)

so �k is a positive measure, if � has this property. q.e.d.

Remark 1. If we additionally assume the non-negativity of the pair potential,�R can be identifiedas a Gibbs measure in the Dobrushin-Landford-Ruelle sense, see Theorem 4 in Sec. VII.

A. The Ginibre Bose gas

An important direct application of this theorem is related to the Ginibre’s Bose gas (Ref. 2). Forprecise definitions we refer to Ref. 16. Consider the space X of Brownian loops in E = (Rd , d a).The measure � is defined by means of some nice pair potential φ in E. Given φ, define a self-potentialυ in X and a pair potential u in X as in Refs. 2 and 16. Then for parameters z, β > 0 let

�( f ) =∑m≥1

1

m· zm

∫E

∫X

f (x) e−υ(x) Pamβ (d x) d a, f ∈ F. (9)

Here Pamβ(d x) is the non-normalized Brownian bridge measure of loops of length mβ which start

and end at a ∈ E. This defines a positive measure on the loop space X. It is shown in Sec. V B ofRef. 16 that for a stable and integrable pair potential φ the assumptions (B1), (B2′), and (B3′) holdtrue for all z from the interval

z ≤ exp{−β

[‖φ‖1ζ ( d2 )

(4πβ)d/2+ B

]}. (10)

Hence by Theorem 2 there exists a unique point process P in X with Levy measure R. Thisprocess P is the limiting Bose gas of interacting Brownian loops (in the sense of Ginibre). Here,ζ ( d

2 ) = ∑n≥1 n− d

2 is the Riemann zeta function. When d = 3 and if the potential is repulsive, onecan rewrite (10) in a more transparent way.16 Let a0 = 1

8π‖φ‖1 denote the Born approximation to

063302-7 Nehring, Poghosyan, and Zessin J. Math. Phys. 54, 063302 (2013)

the scattering length. The condition is then

z ≤ exp{− ζ ( 3

2 )√π

a0√β

}. (11)

In this context we will consider below another class of examples with a modified � which is evensigned.

B. The Groeneveld process

As an aside we first mention an interesting class of point processes which are even infinitelydivisible. Consider a positive pair potential u together with the cumulant measures

�m(d x1 . . . d xm) = zm (−1)m−1 Uu(x1, . . . , xm) �(d x1) . . . �(d xm).

Here again � is a positive measure.It is well known (see Goeneveld, Ref. 3) that in case of a positive potential the Ursell functions

have alternating signs, i.e., that the density of the cumulant measures is non negative. Thus theassociated R is positive, so that the process with Levy measure R exists and is given by the clusterdissolution of the Poisson process with intensity measure R and as such infinitely divisible. We callthis process Groeneveld process; we do not know what kind of process this is. Further results in thisdirection can be found in the interesting paper.19

C. Gibbs modifications of determinantal processes

To motivate the main results in Sec. V we now present, in the context of the Ginibre Bose gas, aheuristic argument which leads to some new class of interacting non-classical point processes. Thisargument is based on the hypotheses that a Fermi-Dirac process on the level of clusters exists.

As above for the Bose gas, we consider E = Rd with Lebesgue’s measure d a. We are given apair potential on E, i.e., a measurable symmetric function φ : E × E −→ R ∪ {+∞}, and we nowreplace in the definition of � in (9) the term zm by ( − 1)m − 1zm and, to be more modest, Pa

mβ by themeasure ba

m from example 2. The positive measure � is then replaced by

�( f ) =∑m≥1

1

m(−1)m−1zm ·

∫Em

f (δa1 + · · · + δam ) ·

e−Eφ (δa1 +···+δam ) ·ba1m (d a2 · · · d am) d a1,

(f ∈ F). Recall that Eφ is defined in (7). This is in general a signed measure on the Polish space X offinite configurations in the phase space. Remark that the energy functional Eφ on the space X is theanalog of the self-potential υ on the space X of Brownian loops. The measure � will be the referencemeasure on X.

We finally introduce a pair potential � on X, which resembles the pair potential u betweenBrownian loops. An obvious guess is

�(μ, η) =∫

E

∫E

φ(a, b) μ(d a)η(d b), for all μ, η ∈ X.

Remark that for any μ1, . . . , μn ∈ X the following identity holds true:

E�(δμ1 + . . . + δμn ) + Eφ(μ1) + . . . + Eφ(μn) = Eφ(μ1 + . . . + μn). (12)

The main question is: Does there exist a point process having the Levy measure

R(ϕ) =∞∑

n=1

1

n!

∫Xn

ϕ(δμ1 + . . . + δμn ) U�(μ1, . . . , μn) �(d μ1) . . . �(d μn). (13)

At first we have to check whether the Schur measures �k are positive. But since the first Schurmeasure �1 coincides with �, which is a signed measure, this is certainly not the case. So at least

063302-8 Nehring, Poghosyan, and Zessin J. Math. Phys. 54, 063302 (2013)

our construction does not give a point process corresponding to R. In case such a process wouldexist one would obtain a Fermi process, which is a process realizing configurations of interactingpolygonal loops δa1 + .. + δam ∈ X.

Now in the sequel let us hypothetically assume that such a process exists. How would the localprocesses QG = �RG , G ∈ B0 look like? Consider

τ ( f ) =∑m≥1

(−1)m−1

mzm

∫Em

f (δa1 + · · · + δam ) ba1m (d a2 . . . d am) d a1.

Here f ∈ F. This is a signed measure on X. It is the above � without the density e−Eφ . Recall that τ

is the Levy measure of the determinantal point process with interaction kernel K (see (3) in example2). Using (12), we obtain

�RG (ϕ) = 1

(G)

∞∑n=0

1

n!

∫X(G)n

ϕ(δμ1 + .. + δμn ) e−Eφ (μ1+..+μn ) τ (d μ1)..τ (d μn).

Consider then the so-called cluster dissolution mapping

ξ : M··(X) −→ M··(E), δμ1 + δμ2 + . . . �−→ μ1 + μ2 + . . . .

The image of �RG under ξ , denoted by ξ �RG , becomes an ordinary finite signed measure on thespace X(G). If we recall the definition of the exponential of a finite signed measure from Sec. III weobtain

ξ �RG (ϕ) = 1

(G)exp τG (ϕ e−Eφ ).

After Theorem 1 we saw in Eq. (2) that 1(G) exp τG coincides with the local process QG = �τG of

the determinantal point process �τ . This then implies that ξ �RG is a finite point process in G. ByCorollary 6.1.2 in Ref. 13, we conclude ξ �RG = �(ξ R)G . So we have identified not R but ξ R as aLevy measure of a point process in E. What remains to be seen, according to Theorem 1, is thatξ | R | is of first order. This will be established with the help of Lemma 1. Remark that the Schurmeasures of �ξ R are given by

e−Eφ (δa1 +..+δak ) det(K (ai , a j )i, j ) d a1.. d ak .

This is why we call�ξ R a Gibbs modification of the determinantal process with interaction kernel K.From now on we consider a general phase space X.

Definition 1. Let L be a Levy measure as introduced in Sec. III with the corresponding pointprocess �L and the family of local processes {�LG}G∈B0 . Furthermore, let φ : X × X → R ∪ {∞}be a pair potential such that

0 < �LG (e−Eφ ) < ∞, G ∈ B0.

Now introduce another family of finite point processes

�φ

LG(ϕ) := 1

�LG (e−Eφ )�LG (e−Eφ ϕ), ϕ ∈ F, G ∈ B0.

If a weak limit, denoted �φ

L, of �φ

LGas G ↑ X does exist we call it the Gibbs modification of the

process �L.

In Sec. V, we will provide sufficient conditions on the pair potential in order for �φ

L to exist.Remark that the above discussion suggests that the process �φ

L has a Levy measure given by

Rφ

L(ϕ) =∞∑

n=1

1

n!

∫Xn

ϕ(μ1 + . . . + μn) U�(μ1, . . . , μn) Lφ(d μ1) . . . Lφ(d μn), (14)

where Lφ(d μ) = e−Eφ (μ) L(d μ).

063302-9 Nehring, Poghosyan, and Zessin J. Math. Phys. 54, 063302 (2013)

The method of the proof will be to show that Rφ

L satisfies the condition of Theorem 1. Certainly,the first question which arises is what kind of family of cumulant measures corresponds to the aboverepresentation of Rφ

L? But a close look at the proof of Theorem 1 yields that we only need to showthat the local Gibbs modifications �φ

LGhave a Levy measure given by Rφ

L,G the restriction of Rφ

L to

X(G). In addition, we need to establish that | Rφ

L | = Rφ,+L + Rφ,−

L is of first order. The positive Rφ,+L

respective negative Rφ,−L part of Rφ

L are given by the Jordan decomposition of L = L+ − L− so that

| Rφ

L |(ϕ) =∞∑

n=1

1

n!

∫Xn

ϕ(μ1 + . . . + μn) |U�(μ1, . . . , μn)| | Lφ |(d μ1) . . . | Lφ |(d μn),

where | Lφ |(d μ) = e−Eφ (μ) | L |(d μ).

V. CONSTRUCTION OF GIBBS MODIFICATIONS

In the following X is a general phase space. We start with a family of cumulant measures{�m}m ≥ 1 such that the corresponding family of Schur measures {�k}k ≥ 1 is non-negative. Weintroduce a parameter z ∈ (0, ∞) called the activity which will be chosen later small enough. Wedenote by Lz the Levy measure corresponding to the family {zm�m}m ≥ 1 of cumulant measures.Recall Definition 1 here and observe that the Schur measures are now given by {zk�k}k ≥ 1. So theSchur measures are non-negative for any choice of the activity.

Let φ : X × X → R ∪ {∞} be a stable pair potential in the classical sense, that is there isB ≥ 0 such that Eφ(μ) ≥ −B |μ|, μ ∈ X.

Let us denote by φx for x ∈ X the function y �→ φ(x, y).

Theorem 3. Let φ be a stable pair potential. Furthermore, assume that there exists c > 0 andz0 = z0(c) > 0 such that

(1) ν1| Lz0ec+B |( f ) < ∞, f ∈ Fb,

(2) ν1| Lz0ec+B |(|φx |) ≤ c, f or all x ∈ X,

where φ is defined as in (6) and recall that ν1| Lz0ec+B | denotes the first moment measure of | Lz0ec+B |.

Then for z ∈ (0, z0] the process �Lz and its Gibbs modification �φ

Lzdo exist. Furthermore, �φ

Lzis a

process with Levy measure Rφ

Lz, that is �φ

Lz= �Rφ

Lz, where Rφ

Lzis defined as in (14).

Proof. Let z ∈ (0, z0]. Due to (1) | Lz | is of first order, which implies condition (A) of Theorem 1.Since the non-negativity of the Schur measures was already established we obtain the existence ofthe process to the Levy measure Lz . Let us now deduce the existence of the process with Levymeasure Rφ

Lz.

1. Let us start by showing that | Rφ

Lz|, z ∈ (0, z0] is of first order. For any f ∈ Fb,

ν1| Rφ

Lz|( f ) =

∫X

| Lφz |(d μ) ζ f (μ) ·

∞∑n=1

1

(n − 1)!

∫Xn−1

| Lφz |(d μ2) . . . | Lφ

z |(d μn) |U�(μ,μ2, . . . , μn)|.

Certainly in order to obtain the finiteness of ν1| Rφ

Lz|( f ), we want to apply the bound as formulated by

Lemma 1. So here the underlying space X is now given by the set X of finite point configurations inX, the signed reference measure � is given by Lφ

z and the pair potential u by �. The aim is now toshow that this triple (X, Lφ

z ,�) does satisfy the conditions (B1), (B2′), and (B3′). Observe that onaccount of (12) the pair potential � on X is stable in the weak sense of Ref. 16, that is it satisfiesassumption (B1) with b(μ) = B|μ| + Eφ(μ), μ ∈ X.

063302-10 Nehring, Poghosyan, and Zessin J. Math. Phys. 54, 063302 (2013)

We now establish (B2′), that is we have to find a ∈ F+(X) such that∫| Lφ

z |(d μ) |�(η,μ)| e(a+b)(μ) ≤ a(η), η ∈ X, (15)

where � is defined with the help of φ as in (12). Let a(μ) = c |μ|, where c > 0 is given as in Theorem3. Introduce I (x) := φ−1

x ({∞}) ∈ B(X ) and I (η) = ⋃x∈η

I (x), η ∈ X. Then the integral in (15) can

be split up into a hardcore and non-hardcore part∫{ζI (η)>0}

| Lz |(d μ) e(c+B)|μ| +∫

X(E\I (η))

| Lz |(d μ) |�(η,μ)| e(c+B)|μ| .

Let us call these summands T1 and T2. Then

T1 = | Lzec+B |(1{ζI (η)>0}) ≤ | Lzec+B |(ζI (η)) ≤∑x∈η

ν1| Lzec+B |(1I (x)).

Introduce the non-hard core part of the potential by defining

φ′′(x, y) ={

0, y ∈ I (x),φ(x, y), else.

Certainly, we have

|�(η,μ)| ≤∑x∈η

μ(|φ′′x |), μ ∈ X(X \ I (η)), (16)

whence T2 ≤ ∑x∈η

ν1| Lzec+B |(|φ′′

x |). This finally yields

∫| Lφ

z |(d μ) |�(η,μ)| e(a+b)(μ) ≤∑x∈η

ν1| Lz ec+B |(1I (x) + |φ′′

x |).

So due to the uniform bound (2) and |φx | = 1I (x) + |φ′′x |, we have obtained condition (B2′). Now by

Lemma 1 combined with condition (1),

ν1| Rφ

Lz|( f ) ≤

∫X

| Lφz |(d μ) ζ f (μ) e(a+b)(μ) = ν1

| Lz ec+B |( f ) < ∞.

2. It remains to show that for any G ∈ B0 and z ∈ (0, z0] the Gibbs modification �φ

Lz,Gof �Lz,G

has a signed Levy measure given by Rφ

Lz ,G, the restriction of Rφ

Lzto X(G). Since | Rφ

Lz| is of first

order, Rφ

Lz ,Gis a finite signed measure. As for the proof of Theorem 1, the following combinatorial

result (see, e.g., Ref. 1) will be needed now:

exp(∞∑

k=1

hk

k!) =

∞∑n=0

1

n!

∑J∈π([n])

∏J∈J

h|J |, (17)

where∞∑

k=1

hkk! is an absolutely convergent series. Using (17), we obtain

KRφ

Lz ,G( f ) = exp(− Rφ

Lz ,G(1 − e−ζ f )) =

1

φ(G)

∞∑n=0

1

n!

∫X(G)n

e−(μ1+..+μn )( f )∑

J∈π([n])

∏J∈J

U�((μ j )J ) Lφz (d μ1).. Lφ

z (d μn),

063302-11 Nehring, Poghosyan, and Zessin J. Math. Phys. 54, 063302 (2013)

where φ(G) = exp(Rφ

Lz ,G(1)) and, using Ruelle’s algebraic approach (8), the above expression

equals

1

φ(G)

∞∑n=0

1

n!

∫X(G)n

e−(μ1+...+μn )( f ) e−E�(δμ1 +...+δμn ) Lφz (d μ1) . . . Lφ

z (d μn).

Using (12) this can be written as

1

φ(G)

∞∑n=0

1

n!

∫X(G)n

e−(μ1+...+μn )( f ) e−Eφ (μ1+...+μn ) Lz(d μ1) . . . Lz(d μn).

Now Eq. (2) after Theorem 1 implies that

KRφ

Lz ,G( f ) = 1

φ(G)exp(Lz,G)(e−Eφ e−ζ f ) = �Lz,G (e−Eφ e−ζ f )

�Lz,G (e−Eφ ).

Thus the local Gibbs modification �φ

Lz,Ghas a Laplace transform given by KRφ

Lz ,G. q.e.d.

Corollary 1. Let L be a signed Levy measure defined by a family of cumulant measures {�n}n ≥ 1.Assume there exist α, β ≥ 0, and αf ≥ 0 for f ∈ Fb such that

(1) |�n|( f ⊗ X⊗(n−1)) ≤ (n − 1)! α f βn−1, f ∈ Fb, n ≥ 1,

(2) |�n|(|φx | ⊗ X⊗(n−1)) ≤ (n − 1)! α βn−1, x ∈ E, n ≥ 1,

where 00 := 1. Then Theorem 3 holds with c = 1 and z0 = e−B−1

α+β.

Proof. We verify condition (2) of Theorem 3,

ν1| Lz0 e1+B |(|φx |) =

∞∑n=1

(z0 e1+B)n

(n − 1)!|�n|(|φx | ⊗ X⊗(n−1))

≤ α

α + β

∞∑n=1

( β

α + β

)n−1 = 1.

Condition (1) in Theorem 3 is established in the same way. q.e.d.

VI. EXAMPLES OF GIBBS MODIFICATIONS

The underlying general phase space should be thought as a discrete space of the Euclideanspace E, or the collection X of finite configurations of particles in E or the space of Brownian loopsin E. Let λ be a non-negative reference measure on X and h : X × X → R ∪ {∞} some measurablefunction. Then define ϒ(h) = sup

y∈X

∫λ(d x) |h(x, y)|.

A. Poisson and Gibbs processes

Let λ ∈ M(X ). Consider the point process whose cumulant measures vanish besides the firstone �1 = λ. It is called the Poisson point process with intensity measure λ and we denote it byPλ. Let φ be a stable pair potential such that ϒ(φ) < ∞. As one can easily verify, condition (1)of corollary 1 holds with αf = λ( f ), β = 0; and (2) with α = ϒ(φ). So Corollary 1, respectively,Theorem 3 yields that for

z ∈(

0,e−B−1

ϒ(φ)

], (18)

the Gibbs modification of Pzλ does exist. It is called classical Gibbs process and its ex-istence is well known (see, e.g., Ref. 10). But instead of the classical regularity condition

063302-12 Nehring, Poghosyan, and Zessin J. Math. Phys. 54, 063302 (2013)

supy∈X

∫ |1 − e−φ(x,y)| λ(d x) < ∞ we require ϒ(φ) < ∞. Remark that if φ consists only of ahard core part we obtain the so-called Poisson exclusion processes which have been studied inRef. 11.

B. Permanental and determinantal processes and its modifications

We here continue the study of determinantal and permanental processes taken up in example 2.Let K be a kernel with the properties as given there. In addition let φ be a stable pair potential suchthat ϒ(φ) < ∞. As one can straightforwardly check condition (1) of Corollary 1 is satisfied withαf = ‖K‖∞ λ(f) and β = ϒ(K) and condition (2) with α = ‖K‖∞ ϒ(φ). Corollary 1, respectively,Theorem 3 now says that for a small activity

z ∈(

0,e−B−1

‖K‖∞ ϒ(φ) + ϒ(K )

],

the corresponding determinantal �Lz (−1), respectively, permanental process �Lz (+1) does exist andalso their Gibbs modifications do exist. If φ consists only of a hard core part we obtain the existenceof determinantal, respectively, permanental exclusion processes.

VII. SOME INTEGRAL EQUATIONS FOR POINT PROCESSES OF STATISTICALMECHANICS

It is shown that the processes �L of Sec. IV are Gibbs in the DLR-sense, if the potential uis non-negative and satisfies (B2′) and (B3′). As a consequence we obtain a new integration byparts formula for Gibbs modifications of infinitely divisible point processes, which seem to be a farreaching generalization of the equation

(�′

�,φ

).

A. Point processes of statistical mechanics revisited

Let us go back to the general setting given in Sec. IV. So (X,B,B0) denotes a Polish phase spaceand � ∈ M(X ) is a given positive Radon measure on it. Let u: X × X → [0, ∞] be a non-negativepair potential. In Theorem 2, we have shown that under the conditions (B1), (B2′), and (B3′) thelimiting Gibbs point process �R with Levy measure

R(ϕ) =∞∑

n=1

1

n!

∫Xn

ϕ(δx1 + . . . + δxn ) Uu(x1, . . . , xn) �(d x1) . . . �(d xn)

does exist.A more delicate question is whether �R is a Gibbs point process in the DLR sense, that is

whether it is a solution to the equation

(�′) CP (h) =

∫h(x, μ + δx ) e−Eu (x,μ) �(d x)P(d μ), h ∈ F ;

here the conditional energy Eu(x, μ) is given by μ(ux) for any x ∈ X and μ ∈ M··(X ), since uis non-negative. CP denotes the Campbell measure of P. The equivalence of this equation to theDLR-equations in the context of classical statistical mechanics had been shown in Ref. 15.

In Ref. 13 theorem 5.2.4 we saw that, if one assumes classical stability and regularity of u asin Ref. 18 and with a reference measure given by � = zλ, where z ∈ (0, ∞) and λ ∈ M(X ), then�L is a solution to

(�′) for small z. Here we strengthen the stability condition, that is we consider

purely repulsive pair potentials, and weaken the regularity condition, that is we only require (B2′)and (B3′).

Theorem 4. Let u be a non-negative pair potential and � ∈ M(X ). Then under the conditions(B2′) and (B3′) from Sec. IV, �R solves

(�′).

063302-13 Nehring, Poghosyan, and Zessin J. Math. Phys. 54, 063302 (2013)

Proof. We follow the proof in Ref. 13 theorem 5.2.4. Due to Ref. 20 the finite processesQG = �RG , G ∈ B0 satisfy

CQG (h) =∫X

∫G

h(x, μ + δx ) e−Eu (x,μ) �(d x)QG(d μ) , h ∈ F.

Let in the sequel h = f ⊗ e−ζg , f, g ∈ Fb. In Ref. 13 corollary 3.3.2, it was shown that CQG (h) →C�R (h) as G ↑ X if |R| is of first order. The right hand side of the above equation can be written as∫

X

f (x) e−g(x) LQG (g + ux ) �(d x). (19)

The lemma, which replaces the main lemma 5.2.5 in Ref. 13 is now given by the following.

Lemma 2. Let ϒ = ζg+ux . Then 1 − e−ζg+ux ≤ ϒ on X and there is c ≥ 0 such that | R |(ϒ)≤ a(x) + c, where a is given as in (B2′).

Proof. The inequality 1 − e−ζg+ux ≤ ϒ is clear. Now we have

| R |(ϒ) ≤∫X

(g(y) + u(x, y)) ea(y) �(d y) ≤∫X

g(y) ea(y) �(d y) + a(x).

The first inequality follows by Lemma 1 and the second by definition of (B2′). So we can choosec := �(g ea), which is finite due to (B3′). q.e.d.

To finish the proof, one can show as in Ref. 13 lemma 5.2.7 that L�R (g + ux )= KR(g + ux ), LQG (g + ux ) = KRG (g + ux ), and so LQG (g + ux ) → L�R (g + ux ) as G ↑ X forany x ∈ X. Moreover, the bound as given by Lemma 2 yields LQG (g + ux ) ≤ ea(x)+c, G ∈ B0, x ∈ X.If we replace LQG (g + ux ) by ea(x)+c we obtain the finiteness of the integral (19) due to condition(B3′). So by Lebesgue’s dominated convergence theorem we are allowed to take the limit G ↑ Xinside the integral of (19) and obtain the assertion as in Ref. 13 theorem 5.2.4. q.e.d.

B. The Ginibre Bose gas revisited

Let us again consider the Bose process of Ginibre. If the underlying pair potential φ is stableand integrable, then the corresponding pair potential u on the loop space satisfies the conditions(B1), (B2′), and (B3′) for a small value of activity, see (10), and thereby we obtain the existenceof the limiting Bose gas by means of Theorem 2. Now if we additionally impose that φ is a purelyrepulsive potential we are able to describe Ginibre’s Bose process as a Gibbs process by means ofTheorem 4. Formulated a bit more generally we have the following.

Theorem 5. Let Lz be a non-negative Levy measure and φ be a non-negative potential. Thenunder the conditions of Theorem 3, the Gibbs modification�φ

Lzof the infinitely divisible point process

�Lz is a solution to

(�

φ

Lz

)CP (h) =

∫h(x, ν + μ) e−Eφ (ν | μ) CLz (d x d ν)P(d μ),

where the conditional energy Eφ(ν | μ) of the finite configuration δx1 + . . . + δxn = ν ∈ X given theboundary μ ∈ M··(X ) is defined by

Eφ(x1, μ) + Eφ(x2, μ + δx1 ) + . . . + Eφ(xn, μ + δx1 + . . . + δxn−1).

Proof. In the proof to Theorem 3, we have shown that Lφz and � satisfy the condition (B2′).

The stability condition is satisfied since � ≥ 0. In the first step, we will show that there exists aGibbs point process in the space X to the pair potential � with reference measure Lφ

z . According toTheorem 2 it remains to show that condition (B3′) is valid, but it is well known that the boundedsets in X are given by {ζG > 0}, G ∈ B0 and all measurable subsets of those sets.

063302-14 Nehring, Poghosyan, and Zessin J. Math. Phys. 54, 063302 (2013)

Recall that a(μ) = c |μ|, μ ∈ X, so that∫{ζG>0}

ea(μ) Lφz (d μ) ≤ Lφ

z ec (ζG) ≤ ν1Lz ec (G) < ∞.

Whence we conclude that ea Lφz is a locally finite measure on X. So we obtain that the Gibbs point

process �R in X corresponding to the signed Levy measure

R(ϕ) =∞∑

n=1

1

n!

∫Xn

ϕ(δμ1 + . . . + δμn ) U�(μ1, . . . , μn) Lφz (d μ1) . . . Lφ

z (d μn)

does exist and satisfies(�′) by Theorem 4, that is,

C�R (h) =∫

h(ν, μ + δν) e−μ(�ν ) Lφz (d ν)�R(d μ), h ∈ F(X × M··(X)).

Corollary 6.1.2 in Ref. 13 now gives ξ�R = �φ

Lz, that is if we dissolve the clusters realized by the

Gibbs process�R we obtain the Gibbs modification of the infinitely divisible process�Lz . Therefore,

C�φ

Lz(h) =

∫h(x, ξ (μ)) ξ (μ)(d x)�R(d μ)

=∫

h(x, ξ (μ)) ν(d x)μ(d ν)�R(d μ)

=∫

h(x, ξ (μ + δν)) ν(d x) e−μ(�ν ) Lφz (d ν)�R(d μ)

=∫

h(x, ξ (μ) + ν) e−(μ(�ν )+Eφ (ν)) CLz (d x d ν)�R(d μ).

Now observe that for μ = δη1 + δη2 + . . . ∈ M··(X) and ν ∈ X we have

μ(�ν) =∞∑j=1

�(ν, η j ) =∞∑j=1

∑x∈ν

∑y∈η j

φ(x, y) = �(ν, ξ (μ)).

Whence we obtain the assertion by noting Eφ(ν | η) = �(ν, η) + Eφ(ν) for ν ∈ X, η ∈ M··(X ).q.e.d.

Equation(�

φ

Lz

)contains equation

(�′) for the Levy measure of the Poisson process. But it

contains also, for the case φ ≡ 0, the equation which characterizes infinitely divisible processes.(c.f. Theorems 5.2.5. and 5.2.6. of Ref. 8).

C. The ideal Bose gas

Here we consider a particular permanental point process the ideal Bose gas. Let E = Rd , whered ≥ 1, λ the Lebesgue measure on E and let

g(x) = 1

(2πβ)d/2exp

(− ‖ x ‖2

2β

), x ∈ E,

be the Gaussian density where β > 0 is a parameter called the inverse temperature. Consider thefollowing interaction kernel G(x, y) = g(x − y), x, y ∈ E. Certainly, G is bounded ‖G‖∞ = (2πβ)− d/2,non-negative definite and satisfies ϒ(G) = 1. Moreover, let φ be a non-negative potential such thatϒ(φ) < ∞. So if we recall Sec. VI B on permanental processes we obtain for

z ∈(

0,(2πβ)d/2

(2πβ)d/2 + ϒ(φ)e−1

]the existence of the permanental process �Lz (+1), which is called the ideal Bose gas and its Gibbsmodification �φ

Lz (+1), which is a solution to(�

φ

Lz

)due to Theorem 5.

063302-15 Nehring, Poghosyan, and Zessin J. Math. Phys. 54, 063302 (2013)

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quantum continuous systems with Boltzmann statistics via cluster expansion,” Methods Funct. Anal. Topol. 3, 62–81(1997).

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Berlin, 1988).8 Matthes, K., Kerstan, J., and Mecke, J., Infinitely Divisible Point Processes (Wiley, 1978).9 Mecke, J., Random Measures, Classical lectures (Walter Warmuth Verlag, 2011).

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Warmuth Verlag (Potsdam University, 2012).14 Nehring, B. and Zessin, H., “The Papangelou process. A concept for Gibbs, Fermi, and Bose processes,” Izv. Nats. Akad.

Nauk Armenii Mat. 46, 49–66 (2011).15 Nguyen, X. X. and Zessin, H., “Integral and differential characterizations of the Gibbs process,” Math. Nachr. 88, 105–115

(1979).16 Poghosyan, S. and Ueltschi, D., “Abstract cluster expansion with applications to statistical mechanical systems,” J. Math.

Phys. 50, 053509 (2009).17 Rebenko, A., “Euclidean Gibbs states for quantum continuous systems via cluster expansion, II. Bose and Fermi statistics,”

Methods Funct. Anal. Topol. 5, 86–100 (1999).18 Ruelle, D., Statistical Mechanics (Benjamin, 1969).19 Waymire, E., “Infinitely divisible Gibbs states,” Rocky Mt J. Math. 14, 665–678 (1984).20 Zessin, H., “Der Papangelou Prozeß,” Izv. Nats. Akad. Nauk Armenii Mat. 44, 61–72 (2009).21 We propose this terminology here because the measure �k is intimately related to the generalized cycle index, respectively,

generalized Schur function of representation theory. But note, the term Schur measure is used also in a completely differentcontext.

22 Here π [n] denotes the set of all partitions of the set [n] = {1, . . . , n}.