Asymptotics of Brownian integrals and pressure: Bose-Einstein statistics

Èçâåñòèÿ ÍÀÍ Àðìåíèè. Ìàòåìàòèêà, òîì 42, í. 3, 2007, ñòð. 23-38

ASYMPTOTICS OF BROWNIAN INTEGRALS AND PRESSURE:BOSE-EINSTEIN STATISTICS

S. FRIGIO AND S. K. POGHOSYAN

University of Camerino, ItalyInstitute of Mathematics, Armenian Academy of Sciences

E-mail: [email protected], [email protected]

Àííîòàöèÿ. The paper studies the asymptotics of the Brownian integrals withpaths restricted to a bounded domain of Rν , when the domain is dilated toin�nity. The framework is that of the Bose-Einstein statistics with paths observedwithin random time intervals which are integer multiplies of some �xed β > 0.The three �rst terms of the asymptotics are found explicitly via the functionalintegrals. In the case of a gas of interacting Brownian loops an expression for thevolume term of the asymptotics of the log-partition function is found and thecorrection term is proved to by order be the boundary area of the domain.

1. INTRODUCTIONIn [1] the large volume asymptotics of the Brownian integrals with paths observed

in a �xed time interval β restricted to a bounded domain of Rν was studied.In the present paper we consider similar problem for the Brownian integrals with

random time intervals which are integer multiplies of β. This problem can be conside-red as a natural generalization of the famous Kac problem [2] on the asymptotics ofthe function

∞∑i=0

e−βλi as β goes to zero, where λi are the eigenvalues of the Laplacian−∆ in a bounded domain Λ.

In the special case where the integrand is one, the Brownian integrals are nothingelse but the logarithms of the grand canonical partition functions of the ideal quantumgases in their functional integral representations [3]. The functional integration met-hod allows one to replace the quantum mechanical problem by a correspondingclassical problem for a system of interacting Brownian trajectories. This method withapplication of Feynman-Kac formula was used �rst by Ginibre in [4]. The systems ofinteracting Brownian trajectories we call Ginibre gases (see [5] and [6]).

The case considered in [1] corresponds to the Ginibre gas with Maxwell-Boltzmannstatistics while the present paper considers the case of Ginibre gas with Bose-Einsteinstatistics. The class of admissible domains Λ consists of bounded convex domains withconvex holes possessing smooth boundaries of the class C3.

We obtain the three �rst terms of the asymptotics for the case of small activity(Theorem 1 and 2). The �rst two terms are proportional respectively to the volumeand to the area of the boundary of Λ. We prove that in two dimensional case the

23

24 S. FRIGIO AND S. K. POGHOSYAN

third term is proportional to the Euler-Poincare characteristic of the domain. In thispart our analysis relies on the modi�ed techniques from [1] and involves some speci�cproperties of the Brownian bridge process outlined in the Appendix.

We consider also the Ginibre gas with repulsive two-body interaction at low activi-ty. Applying the previous results together with the results on the decay of correlationsfrom [6] we �nd an explicit expression for the pressure in terms of functional integralsand prove that the correction term is of order of the area of the boundary of Λ(Theorem 3). The proof is based on the cluster expansion method.

Similar result for the case of Maxwell-Boltzmann statistics was obtained in [7].

2. GINIBRE GAS WITH BOSE�EINSTEIN STATISTICSFor β > 0 �xed and j = 1, 2, . . . let Xjβ be the space of Brownian loops of time

interval jβ in Rν , ν ≥ 1, de�ned byXjβ ≡ {X ∈ C ([0, jβ],Rν) |X(0) = X(jβ)}

In the topology of uniform convergence Xjβ is a Polish space with Borel σ-algebra Bjβ .Let X u

jβ be the set of loops X which start and end at the point u ∈ Rν . In (Xjβ , Bjβ)

we consider a non-normalized Brownian bridge measure Pujβ : Pu

jβ (Xjβ) = (πjβ)ν/2.(see the details in [6])

The underlying one particle space X is de�ned as a topological sum of the spacesXjβ :

X =∞⋃

j=1

Xjβ .

The natural σ-algebra in X generated by the σ-algebras Bjβ we denote by B(X ).The elements of X we call composite loops and put |X| = j if X ∈ Xjβ .

Let 0 < z ≤ 1 be a parameter called activity or fugacity. We de�ne a measure Puz

on X u =∞⋃

j=1

X ujβ by the formula

Puz =

∞∑

j=1

zj

jPu

jβ .

Evidently Puz is a �nite measure for all z, 0 < z ≤ 1. Using a natural bijection

τ : X 0 × Rν → X de�ned by τ(X0, u

)= X0 + u, X0 ∈ X 0, u ∈ Rν , we de�ne a

σ-�nite measure ρz on X byρz =

(P 0

z × λ) ◦ τ−1,

where λ is the Lebesgue measure on Rν . The triple (X ,B(X ), ρz) is the one particlespace of our system.

The con�guration space of our system isM(X ) = {ω ⊂ X ||ω| < ∞}

where | · | stands for the number of elements in a �nite set.An element ω ∈M(X ) is a �nite con�guration of composite loops in Rν of random

time intervals multiple to β. We denote by F(X ) the canonical σ-algebra in M(X ).

ASYMPTOTICS OF BROWNIAN INTEGRALS AND PRESSURE 25

On M(X ) we consider the following σ-�nite measure Wρz given by the formula

(2.1) Wρz =∞∑

n=0

1n!

∫

Xn

ϕ (X1, . . . ,Xn) ρz(dX1) · · · ρz(dXn)

(For the details, see [6]).For any domain Λ ∈ Rν let X (Λ) be the set of all composite loops �living� in Λ:

X (Λ) = {X ∈ X |X(t) ∈ Λ, ∀t ∈ [0, |X|β]} .

In the same way, letM(Λ) = {ω ⊂ X (Λ) | |ω| < ∞}

be the set of �nite con�gurations of composite loops in Λ. The restriction of themeasure ρz on X (Λ) (respectively of Wρz

on M(Λ)) we denote by ρz,Λ (respectivelyby Wz,Λ). The triple (M(Λ),F(Λ),Wz,Λ) we call the ideal Ginibre gas in Λ withBose-Einstein statistics and activity z.

Note that for Λ bounded both measures ρz,Λ and Wz,Λ are �nite. Moreover

(2.2) Wz,Λ(M(Λ)) =∞∑

n=0

1n!

∫

Xu(Λ)

ρz,Λ(dX1) · · · ρz,Λ(dXn) = exp {ρz,Λ(X (Λ))}

is the grand partition function Ξid(Λ, z) of the ideal Ginibre gas in Λ.To de�ne the energy of con�guration ω ∈M(Λ) we consider the space C ([0, β],Rν)

of all continuous trajectories of time intervals β in Rν which we call elementarytrajectories. We will say that an elementary trajectory x is an elementary constituentof a composite loop X ∈ X , and we will write x ∈ X, if for some i, i = 0, 1, . . . , |X|−1,x(t) = X(iβ + t) for all t ∈ [0, β].

Let

(2.3) Φ(x) =

β∫

0

Φ(x(t)) dt, x ∈ C ([0, β],Rν) ,

where Φ : Rν → R is a continuous function (see below for the conditions on Φ). Theenergy U(ω) of a con�guration ω ∈M(X ) is given by

U(ω) =∑

X∈ω

U1(X) +12

∑

X,Y∈ω, X 6=Y

U2(X,Y),

whereU1(X) =

12

∑

x1,x2∈X, x1 6=x2

Φ(x1 − x2),

U2(X,Y) =∑

x∈X, y∈Y

Φ(x− y).

The Boltzmann factor f is de�ned asf(ω) = exp {−U(ω)} , ω ∈M(X ).

The triple (M(Λ),Wz,Λ, Φ) we will call the Ginibre gas in Λ with activity z, interactionΦ and Bose-Einstein statistics.


The main object of our interest is the grand partition function Ξ(Λ, z) of theGinibre gas in a bounded Λ ⊂ Rν which is de�ned by

(2.4) Ξ(Λ, z) = Wz,Λ(f) =∫

M(Λ)

exp {−U(ω)}Wz,Λ(dω),

We want to study the asymptotics of log Ξ(Λ, z) for large bounded Λ. As in thecase of Maxwell-Boltzmann statistics (see [7]) the Brownian integral of a singleloop constrained to a bounded domain Λ,

∫X (Λ)

ρz,Λ(dX) has its own, non-trivial

contribution to the asymptotics of log Ξ(Λ, z). This is purely �quantum� e�ect whichis not the case for the classical analogue of our model. Note that by (2.2),

∫

X (Λ)

ρz,Λ(dX) = log Wz,Λ(M(Λ)) = log Ξid(Λ, z).

Thus we need to study �rst the asymptotics of log Ξid(Λ, z) for large Λ.

3. CONDITIONS ON THE POTENTIAL ANDTHE CLASS OF ADMISSIBLE DOMAINS

We suppose that the function Φ which de�nes the interaction Φ between loops (see(2.3)) satis�es the following conditions:

(a): Φ is an even function: Φ(−u) = Φ(u), u ∈ Rν ;(b): Φ is repulsive: Φ ≥ 0;(c): Φ has the following power decay at in�nity:

∫

Rν

|Φ(u)| (1 + |u|)ldu, l > 0.

The class of potentials Φ satisfying conditions (a)-(c) we denote by P+l .

The class of admissible domains Λ consists of open bounded convex subsets of Rν

with n, n ≥ 0, convex closed holes. We assume that the boundary ∂Λ of Λ consists ofn + 1 (ν − 1)-dimensional closed C3 manifolds. At each point r ∈ ∂Λ we de�ne localcoordinates (η, ξ1, . . . , ξν−1) so that η is along the inward drawn unit normal n andξ1, . . . , ξν−1 are along the directions of principal curvatures of ∂Λ at the point r. Inthis local coordinates ∂Λ is given by a C3 function fr:(3.1) η = fr(ξ1, . . . , ξν−1) = fr(ξ), ||ξ|| < δ

for some δ > 0 small enough, ξ = (ξ1, . . . , ξν−1).

4. MAIN RESULTSLet F (X), X ∈ X be a translation invariant function: F (X + u) = F (X), for all

X ∈ X and u ∈ Rν . Hence we can think of F as a function on X 0 and we assumethat F ∈ L2(X 0, P 0

z ) for some z > 0. LetΛR = R · Λ = {R · u|u ∈ Λ} .


Theorem 1. For any admissible domain Λ and for all z from the interval 0 < z ≤ zthe following expansion holds true∫

X (ΛR)

F (X)ρz,ΛR(dX) = Rν |Λ|a0(F, z) + Rν−1a1(Λ, F, z)

+Rν−2a2(Λ, F, z) + o(Rν−2)

as R → ∞, where |Λ| is the volume of Λ and the coe�cient a0, a1 and a2 aregiven explicitly in terms of functional integrals by formulas (5.3), (5.29) and (5.30)respectively.

In the case where the function F is in addition rotation invariant the coe�cientsa1 and a2 have simpler form.Theorem 2. If under the conditions of Theorem 1 the function F is in additionrotation invariant, then∫

X (ΛR)

F (X)ρz,ΛR(dX) = Rν |Λ|a0(F, z) + Rν−1|∂Λ|a1(F, z)

+Rν−2∫

∂Λ

HΛ(r)σ(dr)a2(F, z) + o(Rν−2)

where a1 and a2 are given by (5.31) and (5.32), HΛ(r) is the mean curvature of ∂Λat the point r ∈ ∂Λ and σ is the ν − 1-dimensional surface measure.Remark 1. In dimension two, ν = 2, according to Gauss-Bonnet theorem∫

∂Λ

HΛ(r)σ(dr) = 2πΓ(Λ),

where Γ(Λ) is the Euler-Poincar�e characteristic of Λ, Γ(Λ) = 1−n, if Λ has n holes.Therefore the corresponding term is purely topological.Remark 2. In particular case where F ≡ 1 Theorem 2 gives an asymptotic expansionof the log-partition function log Ξid(ΛR, z) of the ideal Ginibre gas in ΛR, as R →∞.

The next result gives the main term of the asympotic expansion of the log-partitionfunction of the Ginibre gas in ΛR with interaction Φ.Theorem 3. Let Φ ∈ P+

l , l > 1 and z be from the interval

(4.1) 0 < z < exp

−

32

β1−ν/2

πν/2

∞∑

j=1

j−(1+ν/2)

∫

Rν

Φ(u)du

1/2

.

then for any admissible domain Λ ⊂ Rν

log Ξ(ΛR, z) = Rν · p(Φ, z)|Λ|+ O(Rν−1) as R →∞,

where p(Φ, z) is given by

p(Φ, z) =∫

X 0

P 0z (dX)

∫

M(X )

g(ω,X)|ω|+ 1

Wρz (dω),


and β−1p(Φ, z) is called pressure.

Here and below, without confusing the reader, we write (X, ω) for the con�guration{X} ∪ ω. The Ursell function g is de�ned below by formula 6.1.

5. PROOF OF THEOREM 1Let

I(R, z) =∫

X (ΛR)

F (X)ρz,ΛR(dX).

By de�nition of the measure ρz,ΛR

I(R, z) =∫

ΛR

du

∫

X 0

1IX (ΛR)

(X + u)F (X)P 0z (dX)

where 1IAis the indicator function of a set A.

We decompose this integral as follows(5.1) I(R, z) = I0(R, z)− I1(R, z),

whereI0(R, z) =

∫

ΛR

du

∫

X 0

F (X)P 0z (dX),

I1(R, z) =∫

ΛR

du

∫

X 0

(1− 1I

X (ΛR)(X + u)

)F (X)P 0

z (dX).

This gives the volume term:(5.2) I0(R, z) = Rν · |Λ| · a0(F, z)

with

(5.3) a0(F, z) =∫

X 0

F (X)P 0z (dX).

To study I1 from (5.1) we put

ΛR,δ ={

u ∈ ΛR|d(u, ∂Λ) < δ√

R}

where d is the Euclidean distance in Rν . Then(5.4) I1(R, z) = I2(R, z) + I ′2(R, z),

whereI2(R, z) =

∫

ΛR,δ

du

∫

X 0

(1− 1I

X (ΛR)(X + u)

)F (X)P 0

z (dX),

I ′2(R, z) =∫

ΛR\ΛR,δ

du

∫

X 0

(1− 1I

X (ΛR)(X + u)

)F (X)P 0

z (dX).


Evidently

|I ′2(R, z)| ≤∫

ΛR\ΛR,δ

du

∫

X 0

1I{sup ||X||≥δ

√R}

(X) |F (X)|P 0z (dX)

By Schwarz inequality and Lemma 1 from [6]∫X 0

1I{sup ||X||≥δ

√R}

(X) |F (X)|P 0z (dX)(5.5)

≤ ‖F‖L2

[P 0

z

(sup ‖X‖ ≥ δ

√R

)]1/2

≤ C (ν)β−ν/4‖F‖L2 exp[−C(β, z)δR]

for all z, 0 < z < 1, where C(β, z) =(| ln z|32β

)1/2

and

‖F‖L2=

∫

X 0

F 2(X)P 0+,z(dX)

1/2

.

Hence (for simplicity we denote all the constants by the same letter C indicating onlythe dependence on the parameters)

(5.6) |I ′2(R, z)| ≤ |Λ|C (ν, β, z) ‖F‖L2 exp[−C(β, z)δR].

Now consider I2(R, z). We have

(5.7) I2(R, z) = I3(R, z) + I ′3(R, z),

where

I3(R, z) =∫

ΛR,δ

du

∫

X 0

(1− 1I

X (ΛR)(X + u)

)1I{sup ||X||<δ

√R}

(X)F (X)P 0z (dX),

I ′3(R, z) =∫

ΛR,δ

du

∫

X 0

(1− 1I

X (ΛR)(X + u)

)1I{sup ||X||≥δ

√R}

(X)F (X)P 0z (dX).

According to (5.5)

(5.8) |I ′3(R, z)| ≤ |Λ|C (ν, β, z) ‖F‖L2 exp[−C(β, z)δR].

To estimate I3(R, z) we use the local coordinates. Similarly to (3.1) ∂ΛR is givenlocally by

η = fr,R(ξ), ‖ξ‖ < δ√

R.

We have the following relations between the functions fr,R and fr ≡ fr,1:

(5.9) fr,R(ξ) = Rfr,1(R−1ξ).

Let ki(r|R), i = 1, . . . , ν−1, be the principal curvatures of ∂ΛR at the point r ∈ ∂ΛR.From (5.9) it follows that

(5.10) ki(r|R) = R−1ki(r|1), i = 1, . . . , ν − 1.


Then (see, for example, [8])

I3(R, z) =∫

∂ΛR

σR(dr)

δ√

R∫

0

ν−1∏

i=1

(1− τki(r|R)) dτ

·∫

X 0

(1− 1I

X (ΛR)(X + r + τn)

)1Isup ‖X‖<δ

√R(X)F (X)P 0

z (dX).

For each X ∈ X 0 such that sup ‖X‖ < δ√

R we putγ(X) ≡ γr,R(X) = inf

t[Xn(t)− fr,R(XT (t)] .

Here(5.11) Xn(t) = 〈X(t),n〉, XT (t) = X(t)− 〈X(t),n〉nwhere 〈·, ·〉 stands for the scalar product in Rν . It is easy to check that, for anyX ∈ X 0 with sup ‖X‖ < δ

√R, 1I

X (ΛR)(X + r + τn) = 0 i� τ + γ(X) < 0. Therefore

I3(R, z) =∫

∂ΛR

σR(dr)δ√

R∫0

ν−1∏i=1

(1− τki(r|R)) dτ ·

· ∫X 0

1Iτ+γ(X)<0

(X)1Isup ‖X‖<δ

√R(X)F (X)P 0

z (dX).(5.12)

Using the equality1Isup ‖Xn‖<δ

√R

= 1Isup ‖X‖<δ

√R

+ 1Isup ‖X‖≥δ

√R1Isup ‖Xn‖<δ

√R,

we can rewrite (5.12) as

I3(R, z) =∫

∂ΛR

σR(dr)δ√

R∫0

ν−1∏i=1

(1− τki(r|R)) dτ ·

· ∫X 0

1Iτ+γ(X)<0

(X)1Isup ‖Xn‖<δ

√R(X)F (X)P 0

z (dX)−(5.13)

− ∫∂ΛR

σR(dr)δ√

R∫0

ν−1∏i=1

(1− τki(r|R)) dτ ·

· ∫X 0

1Iτ+γ(X)<0

(X)1Isup ‖X‖≥δ

√R(X)1I

sup ‖Xn‖<δ√

R(X)F (X)P 0

z (dX) ≡

≡ IA(R, z) + IA(R, z).

Let us estimate the second term IA(R, z). It is clear that for each admissible domainΛ

k = max1≤i≤ν−1

supr∈∂Λ

|ki(r|1)| < ∞.

Assuming δ < k−1, we have that

∣∣∣∣∣ν−1∏

i=1

(1− τki(r|R)

∣∣∣∣∣ < 2ν−1, 0 < τ < δR.


Hence using (5.5) we see that

(5.14)∣∣∣IA(R, z)

∣∣∣ ≤ |Λ|C (ν, β, z) ‖F‖L2 exp[−C(β, z)δR].

The �rst term IA(R, z) in (5.13) we decompose as

IA(R, z) =∫

∂ΛR

σR(dr)∞∫0

ν−1∏i=1

(1− τki(r|R)) dτ ·

· ∫X 0

1Iτ+γ(X)<0


√R(X)F (X)P 0

z (dX)−

− ∫∂ΛR

σR(dr)∞∫

δ√

R

ν−1∏i=1

(1− τki(r|R)) dτ ·(5.15)

· ∫X 0

1Iτ+γ(X)<0


√R(X)F (X)P 0

z (dX) ≡

≡ IA1 (R, z) + IA(R, z)

Let us show that(5.16)

∣∣∣IA(R, z)∣∣∣ ≤ C exp

(−C(β, z)δ

√R

)

where C = C(ν, β, z, Λ, F, δ) does not depend on R. >From (5.10) it follows that

(5.17)ν−1∏

i=1

(1− τki(r|R)) =ν−1∑s=0

τsas(r|R) =ν−1∑s=0

R−sτsas(R−1r|1),

where a0(r|R) = 1,

as(r|R) = (−1)s∑

1≤i1<···<is≤ν−1

ki1(r|R) · · · kis(r|R), s = 1, · · · , ν − 1.

Hence∣∣∣IA(R, z)

∣∣∣ ≤ν−1∑s=0

∫

∂ΛR

|as(r|R)|σR(dr)

∞∫

δ√

R

τsdτ

·∫

X 0

1Iτ+γ(X)<0


√R(X)|F (X)|P 0

z (dX).

Now with the help of (5.9) and the condition that fr,R is of class C3 one can easilyobtain that

(5.18) fr,R(ξ) = R−1 12

ν−1∑s=0

ki(R−1r|1)ξ2i + R−2εr,R(ξ), ‖ξ‖ < δ

√R

where(5.19) |εr,R(ξ)| ≤ C(ν)C(Λ)‖ξ‖3

uniformly in r ∈ ∂ΛR and R ≥ 1. This implies that for all ξ, ‖ξ‖ < δR and R largeenough

|fr,R(ξ)| ≤ kδ2.


Using the fact that supt ‖X(t)‖ > τ −kδ2 for any loop X starting at the point r + τnwith τ > δ

√R and such that τ + γ(X) < 0, we can write:

∫

X 0

1Iτ+γ(X)<0

(X)1Isup ‖Xn‖<σ

√R(X)|F (X)|P 0

z (dX)

≤∫

X 0

1Isup ‖Xn‖>τ−kδ2

(X)|F (X)|P 0z (dX)

≤ ‖F‖L2

[P 0

+,z

(sup ‖Xn‖ > τ − kδ2

)]1/2

≤ C(ν)β−ν/4‖F‖L2 exp[C(β, z)kδ2] exp[−C(β, z)τ ].

Hence

|IA(R, z)| ≤ C(ν, β, z, k, δ)‖F‖L2

ν−1∑s=0

∫

∂ΛR

as(r|R)σR(dr)

∞∫

σ√

R

τ s exp[−C(β, z)τ ]dτ

(5.20) ≤ C(ν, β, z, Λ, δ)‖F‖L2 exp[−C(β, z)δ√

R],

which proves the formula (5.16).Hence combining the formulas (5.1), (5.4), (5.6)-(5.8), (5.13) and (5.16) we �nd

that

I1(R, z) =∫

∂ΛR

σR(dr)∞∫0

ν−1∏i=1

(1− τki(r|R)) dτ ·

· ∫X 0

1Iτ+γ(X)<0


√R(X)F (X)P 0

z (dX) + O(e−C√

R).

Applying Fubini's theorem and formula (5.17) to the last integral we have

I1(R, z) =ν−1∑s=0

∫∂ΛR

as(r|R)σR(dr)∫X 0

1Isup ‖Xn‖<δ

√R(X)F (X)P 0

z (dX) ·

·−γ(X)∫

0

τsdτ + O(e−C√

R)

or

(5.21) I1(R, z) =ν−1∑s=0

Ls(z, R) + O(e−C√

R).

withLs(z, R) = 1

s+1

∫∂ΛR

as(r|R)σR(dr) ·(5.22)

· ∫X 0

1Isup ‖Xn‖<δ

√R(X)F (X) (−γ(X))s+1

P 0z (dX).

Let e1, . . . , eν−1 be unit vectors drawn along the directions of the principal curvaturesof ∂ΛR at the point r ∈ ∂ΛR. For each X ∈ X 0, with sup ‖X‖ < δR, we choose


tn = tn(X) and tR = tR(X) from the interval [0, |X|β] so that Xn(tn) = inft

Xn(t)and

Xn(tR)− fr,R(XT (tR)) = inft

(Xn(t)− fr,R(XT (t))).

By Proposition 1 from Appendix 1 tn is P 0z -almost surely unique and by Proposition

2 from Appendix 2 tn → tR, as R →∞, P 0z -almost surely for all z, 0 < z ≤ 1.

Let us show that the following representation of γ(X) is valid:

(5.23) −γ(X) = −Xn(tn) +R−1

2

ν−1∑

i=1

ki(R−1r|1) 〈XT (tn), ei〉2 + R−1εr,R(X),

where

(5.24) |εr,R(X)| ≤ C(ν, β)

{ν−1∑

i=1

(〈XT (tR), ei〉2 − 〈XT (tn), ei〉2

)+ R−1‖X‖3

}.

Note that from Proposition 2 and the Lebesgue dominant convergence theorem itfollows that

(5.25)∫

∂ΛR

σR(dr)∫

X 0

εr,R(X)P 0z (dX) = o

(Rν−1

), as R →∞.

Let us prove (5.23). We have that

−γ(X) = fr,R(XT (tn))−Xn(tn) + ∆ (X|r,R) ,

where

0 ≤ ∆(X|r,R) = fr,R(XT (tR))−Xn(tR)− fr,R(XT (tn)) + Xn(tn).

Using 5.18 we �nd that

∆ (X|r,R) ≤ fr,R(XT (tR))− fr,R(XT (tn)) =R−1

2

ν−1∑

i=1

ki

(R−1r|1) ·

·[〈XT (tR), ei〉2 − 〈XT (tn), ei〉2

]+ R−2 [εr,R(XT (tR))− εr,R(XT (tn))] .

This according to 5.19 and Proposition 2 implies (5.23) and (5.24).With the help of (5.23) we can treat the terms Ls(z, R) from (5.22). Consider

L0(z, R). We have that

L0(z,R) = Rν−1∫

∂Λ

σ(dr)∫X 0

F (X)(−Xn(tn) + R−1

2

ν−1∑i=1

ki(r|1) ·

· 〈XT (tn), ei〉2 + R−1εr,R(X))

P 0z (dX) =

= −Rν−1∫

∂Λ

σ(dr)∫X 0

F (X) inf XnP 0z (dX) +(5.26)

+Rν−2

2

∫∂Λ

σ(dr)∫X 0

ν−1∑i=1

ki(r|1)F (X) 〈XT (tn), ei〉2 P 0z (dX) + o(Rν−2)


In a similar way, according to (5.17),

L1(z, R) = − 12Rν−2

∫∂Λ

ν−1∑i=1

ki(r|1)σ(dr) ·(5.27)

· ∫X 0

F (X)X2n(tn)P 0

z (dX) + O(Rν−3).

It is easy to check that

(5.28)ν−1∑s=2

Ls(z, R) = O(Rν−3).

Indeed for R large enough |γ(X)|s ≤ C sup ‖X‖s.By Lemma 1 from [6] it is easy to check that sup ‖X‖s ∈ L1(X 0, P 0

z ). Thereforeν−1∑s=2

|Ls(z,R)| ≤ν−1∑s=2

Rν−1

s+1

∫∂Λ

R−sas(r|1)σ(dr) ·

· ∫X 0

F (X) sup ‖X‖s+1P 0z (dX) = O(Rν−3).

Now from (5.21), (5.26)-(5.28) it follows thatI1(R, z) = Rν−1a1(Λ, F, z) + Rν−2a2(Λ, F, z) + o(Rν−2)

where

(5.29) a1 = −∫

∂Λ

σ(dr)∫

X 0

F (X) inf XnP 0z (dX),

(5.30) a2 =12

∫

∂Λ

σ(dr)∫

X 0

F (X)ν−1∑

i=1

ki(r|1)[〈XT (tn), ei〉2 −X2

n(tn)]P 0

z (dX)

This together with (5.1) completes the proof of Theorem 1.Now suppose that the function F (X) is in addition rotation invariant. Then the

integral ∫

X0

F (X) inf XnP 0z (dX)

does not depend on the orientation of the unit normal n in Rν , because the measureP 0

z also is rotation invariant. Hence a1 takes a simple form:a1 = |∂Λ|a1(F, z)

with

(5.31) a1(F, z) = −∫

X0

F (X) inf〈X,d1〉P 0z (dX)

where d1 is any �xed unit vector in Rν . In the same way

a2 =12

∫

∂Λ

ν−1∑

i=1

ki(r|1)σ(dr)∫

X 0

F (X)[⟨

XT (t),d2

⟩2 − ⟨Xn(t),d1

⟩2]P 0

z (dX),


ora2 =

∫

∂Λ

HΛ(r)σ(dr)a2(F, z),

where

HΛ(r) =1

ν − 1

ν−1∑

i=1

ki(r|1)

is the mean curvature of ∂Λ at the point r and

(5.32) a2(F, z) =ν − 1

2

∫

X 0

F (X)[⟨

XT (t),d2

⟩2 − ⟨X(t),d1

⟩2]P 0

z (dX).

Here d1,d2 is an arbitrary �xed pair of orthogonal unit vectors in Rν and t is de�nedby

⟨Xn(t),d1

⟩= inf 〈Xn(t),d1〉. Theorem 2 is proved.

6. PROOF OF THEOREM 3Let g be the Ursell function given by the formula:

(6.1) g(ω) =∏

X∈ω

e−U1(X)∑

γ∈Γcon(ω)

∏

[X,X]∈E(γ)

(e−U2(X,X) − 1

),

where γ ∈ Γcon(ω) is the set of all connected graphs consructed on ω, E(γ) is the setof edges of the graph γ.

To develop the large volume asymptotics of the log-partition function log Ξ(ΛR, z)of the Ginibre gas with interaction we use the cluster representation log Ξ(ΛR, z) interms of the Ursell function:

log Ξ(ΛR, z) =∫

M(ΛR)

g(ω)Wz,ΛR(dω)

(See for details [6]). It follows from Corollary 3 and formulas (12) and (32) in [6] thatthe Ursell function g ∈ L1(M(ΛR),Wz,Λ), R ≥ 1, for all z from the intervall (4.1).

An appplication of formula (4) from [6] gives

log Ξ(ΛR, z) =∫

X (ΛR)

ρz(dX)∫

M(ΛR)

g1(X, ω)Wz,ΛR(dω)

where g1(ω) = g(ω)|ω| , ω ∈M\{∅}. This implies

(6.2) log Ξ(ΛR, z) = A0(R, z)−A1(R, z),

whereA0(R, z) =

∫

X (ΛR)

Gz(X)ρz(dX),

A1(R, z) =∫

X (ΛR)

ρz(dX)∫

Mc(ΛR)

g1(X, ω)Wρz (dω),

Gz(X) =∫

M

g1(X, ω)Wρz (dω).


Note that Gz is translation invariant function: Gz(X + u) = Gz(X), for any u ∈ Rν

and X ∈ M. This follows from the translation invariance of the Ursell function andthe measure Wρz

. By [6], Lemma 4, Gz ∈ L2(X 0, P 0z ) for all z from the intervall (4.1).

According to Theorem 1

A0(R, z) = Rν |Λ|a0(Gz) + Rν−1a1(Λ, Gz) + Rν−2a2(Λ, Gz) + o(Rν−2).

Now consider A1(R, z). We will show below that A1(R, z) = O(Rν−1). Similarly to(5.4) we decompose A1 as:

(6.3) A1(R, z) = A2(R, z) + A′2(R, z),

where

A2(R, z) =∫

ΛR,δ

du

∫

Xu

1IX (ΛR)

(X)Puz d(X)

∫

Mc(ΛR)

g1(X, ω)Wρz (dω),

A′2(R, z) =∫

ΛR\ΛR,δ

du

∫

Xu

1IX (ΛR)

(X)Puz d(X)

∫

Mc(ΛR)

g1(X, ω)Wρz(dω).

Applying Corollary 1 from [6] we �nd that

|A′2(R, z)| ≤ ∫ΛR\ΛR,δ

du∫Xu

Puz d(X)

∫Mc(Bu(δR)

|g1(X, ω)|Wρz (dω) ≤

≤ C (1 + δR)−lRν |Λ| = O(Rν−1),(6.4)

where C = C (Φ, β, ν, z, l) > 0 and Bu(R) is a ball in Rν of radius R centered atu ∈ Rν .

Consider A2(R, z). Using the local coordinate system we can write

A2(R, z) =∫

∂ΛR

σR(dr)δR∫0

ν−1∏i=1

(1− tki(r|R)) dt ·(6.5)

· ∫X 0

1IX (ΛR)

(X0 + r + tn)P 0z (dX0)

∫Mc(ΛR)

g1(X0 + r + tn, ω)Wρz (dω).

Again applying Corollary 1 from [6] we have that

|A2(R, z)| ≤ 2ν−1∫

∂ΛR

σR(dr)δR∫0

dt∫X 0

P 0z (dX0) ·(6.6)

· ∫Mc(Br+tn(t))

∣∣g1(X0 + r + tn, ω)∣∣ Wρz (dω) ≤

≤ C (Φ, β, ν, z, l)∫

∂ΛR

σR(dr)∞∫0

(1 + t)−ldt = O(Rν−1).

This completes the proof of Theorem 3.

Acknowledgments: We thank Sylvie Roelly for useful discussions.


7. APPENDIXProposition 1. The time t at which one dimensional composite Brownian loopattains its in�mum is P 0

z -almost surely unique for all z : 0 < z ≤ 1.

Proof: Let X 0 be the space of all one dimensional composite Brownian loops. Let

T (X) ={

t ∈ [0, |X|β]|X(t) = infs

X(s)}

.

We need to show thatP 0

z {X ∈ X |cardT (X) > 1} = 0.

Let h(X) = sup T (X) and h(X) = inf T (X), X ∈ X 0. For each X ∈ X 0 let X ∈ X 0

be de�ned by X(t) = X(jβ − t) if X ∈ X 0jβ . Evidently · : X 0 → X 0 is one to one

mapping which preserves the measure P 0jβ , j = 1, 2, . . ., on each Xjβ . Therefore X

preserves the measure P 0z . Taking into account that h(X) = h(X) we have that

∫

X 0

h(X)P 0z (dX) =

∫

X 0

h(X)P 0z (dX) =

∫

X 0

h(X)P 0z (dX)

Thus h− h ≥ 0 with ∫

X 0

(h(X)− h(X))P 0z (dX) = 0

which implies that P 0z {X ∈ X 0|cardT (X) > 1} = 0.

Proposition 2. For each X ∈ X 0, and all z, 0 < z ≤ 1, tR(X) → tn(X), as R →∞,P 0

z -almost surely.

Proof: It is su�cient to show that(7.1) |Xn(tR)−Xn(tn)| → 0, as R →∞,

for each X ∈ X 0. Indeed, if τ(X) is a limiting point for the set {tR(X), R ≥ 1}then (7.1) implies that 〈X · n〉(τ) = 〈X · n〉(tn) = inf〈X · n〉 and by Proposition 1τ(X) = tn(X) P 0

z -almost surely.Let us prove (7.1). By de�nitions of tR and tn

inft

(X(tn)− fr,R(XT (t)))− inft

Xn(t) ≤ Xn(tn)− fr,R(XT (tn))−Xn(tn)

which impliesXn(tR)− fr,R(XT (tR))−Xn(tn) ≤ −fr,R(XT (tn)),

which together with the bound|fr,R(ξ)| ≤ CR−1‖ξ‖2,

(see (5.18)) gives0 ≤ Xn(tR)−Xn(tn) ≤ 2CR−1‖X‖2.

Formula (7.1) is proved.


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Orbital Magnetism,� Ann. Inst. Henry Poincare 66 (2), 147-183 (1997).[2] M. Kac, �Can One Hear the Shape of a Drum,� Am. Math. Monthly 73, 1-23 (1966).[3] J. Ginibre, �Some Applications of Functional Integration in Statistical Mechanics,� in Statistical

Mechanics and Quantum Field Theory Ed. by C. De Witt and R. Stora (1971), pp. 327-427.[4] J. Ginibre, �Reduced Density Matrices of Quantum Gases. I. Limit in in�nite volume,� J. Math.

Phys. 6, 238-251 (1965). �II. Cluster property,� J. Math. Phys. 6, 252-263 (1965).[5] S. Poghosyan and H. Zessin, �Decay of Correlations for the Ginibre Gas Obeying Maxwell-

Boltzmann Statistics,� Markov Processes and Related Fields 7 (4), 561-580 (2001).[6] S. K. Poghosyan, �Strong Cluster Properties of Ginibre Gas. Quantum Statistics,� Izv. NAN

Armenii. Matematika 40 (4), 58-81 (2005) [Journal of Contemporary Mathematical Analysis40 (4), 57-80 (2005)].

[7] S. Poghosyan and H. Zessin, �Existence of the Pressure for Ginibre Gas in n-connectedDomains,� Izv. NAN Armenii. Matematika 38 (2), 63-72 (2003) [Journal of ContemporaryMathematical Analysis 38 (2), 61-71 (2003)].

[8] B. O'Neil, Elementary Du�erential Geometry (Academic Press, NY, 1966).

Ïîñòóïèëà 5 ìàðòà 2006

Asymptotics of Brownian integrals and pressure: Bose-Einstein statistics

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