Asymptotic expansion of the log-partition function for a

JOURNAL OF MATHEMATICAL PHYSICS 51, 073302 共2010兲

Asymptotic expansion of the log-partition function for a gas of interacting Brownian loops. II. Suren Poghosyana兲 Institute of Mathematics, Armenian National Academy of Sciences, Bagramian 24-B, Yerevan 0019, Armenia 共Received 22 March 2010; accepted 28 May 2010; published online 19 July 2010兲

In an earlier paper 关Poghosyan, S. and Zessin, H., “Asymptotic expansion of the log-partition function for a gas of interacting Brownian loops,” J. Math. Phys. 48, 093301 共2007兲兴 we studied the asymptotic expansion of the log-partition function of a quantum gas in a bounded domain as this domain is dilated to infinity. The volume and the boundary terms of this expansion were found explicitly in terms of functional integrals. Here we obtain the third term of the expansion which in the two-dimensional case has a form of the Euler–Poincaré characteristics of the domain times a potential dependent factor given explicitly in terms of functional integrals. The analysis relies on the Feynman–Kac representation of the logpartition function and on the cluster expansion method. © 2010 American Institute of Physics. 关doi:10.1063/1.3456063兴

I. INTRODUCTION

As in the previous paper 共Ref. 15, hereafter referred to as I兲 we consider a quantum gas with Maxwell–Boltzmann 共MB兲 statistics in the Feynman–Kac representation3,4 共also called Ginibre gas with MB statistics16兲. We assume that the particles interact via bounded, stable pair potential ␾ which is differentiable, has uniformly bounded derivatives and decays fast enough at infinity. The present paper studies the third term of the asymptotics of the logarithm of the grand partition function ln Z共⌳R , z兲 of a quantum gas in a bounded domain ⌳R = 兵Ru 兩 u 苸 ⌳其 as R → +⬁, where z is the activity or fugacity parameter. We consider two-dimensional bounded convex domains ⌳, with convex holes and smooth boundary. The following expansion is the main result of the paper: ln Z共⌳R,z兲 = R2兩⌳兩␤ p共␾,z兲 + R兩⳵ ⌳兩b共␾,z兲 + 2␲␹共⌳兲c共␾,z兲 + o共1兲. Here ␤ is the inverse temperature, 兩⌳兩 is the area, 兩⳵⌳兩 the length of the boundary of ⌳, and ␹共⌳兲 is the Euler–Poincaré characteristic of the domain ⌳. The coefficients p共␾ , z兲, b共␾ , z兲, and c共␾ , z兲 are explicitly expressed as functional integrals and are analytic functions of the activity z in a neighborhood of the origin; p共␾ , z兲 is the pressure and b共␾ , z兲 can be interpreted as the surface tension. The result of this paper can be viewed as a natural generalization of the Kac famous work6 ⬁ where the problem of finding the asymptotics of the partition function Tr exp共␤⌬兲 = 兺n=1 e−␤␭n as ␤ → 0 was considered. Here ␭n are eigenvalues of the Laplacian −⌬ in a bounded domain 共see I, Introduction兲. The important technical tool for our analysis is Main Lemma from Sec. III which describes a decay property of the two-point truncated correlation functions of the system of interacting Brownian loops 共cf. I and Ref. 16兲. a兲

Electronic mail: [email protected].

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The paper is organized as follows. In Sec. II we formulate Theorem 1 which is the main result of the paper. The proof of Theorem 1 is given in Sec. III which starts with the main arguments that were used in I to obtain the first two terms of the asymptotics. Concluding remarks are given in Sec. IV. II. FORMULATION OF THE MAIN RESULT

The state space for n quantum particles in a bounded domain ⌳ 傺 R2 obeying Boltzmann statistics is the Hilbert space L2共⌳n兲 of square-integrable complex functions. The Hamiltonian H⌳,n is given by n

H⌳,n = − 兺 ⌬i + i=1

兺

␾共ui − u j兲,

1ⱕi⬍jⱕn

where ⌬i is the Laplacian for the ith particle with Dirichlet boundary conditions and ␾共u兲 is a multiplication operator. The grand partition function ⬁

Z共⌳,z兲 = 兺 zn Tr e−␤H⌳,n .

共1兲

n=0

The class of domains ⌳ to be considered in the present paper consists of open convex bounded subsets of R2 with finitely many convex closed holes such that the connected parts of the boundary of ⌳ are one-dimensional closed C3-manifolds. Such ⌳ are called admissible domains. We assume that particles interact via pair interaction ␾, which is an even function on R2 and satisfies the stability condition with a constant B ⱖ 0,

兺

␾共ui − u j兲 ⱖ − Bn.

共2兲

1ⱕi⬍jⱕn

Moreover we assume that ␾ is differentiable and is uniformly bounded together with its derivatives so that 兩␾共u兲兩 ⱕ M,

储 ␾ l储 1 =

冕

R2

du兩␾l共u兲兩 ⬍ + ⬁,

共3兲

where ␾l共u兲 = ␾共u兲共1 + 兩u兩兲l with l ⱖ 16, and 兩ⵜ␾共u兲兩 ⱕ M ⬘,

储ⵜ␾储1 =

冕

R2

du兩ⵜ␾共u兲兩 ⬍ + ⬁.

共4兲

Let X0 = 兵x 苸 C共关0 , ␤兴 , R2兲 兩 x共0兲 = x共␤兲 = 0其 be the space of Brownian loops in R2 which start and end at 0. Let P0 be the non-normalized Brownian bridge measure on X0, P0共X0兲 = 共2␲␤兲−1. We will consider also modified measures P0k , k = 1 , 2 , . . ., given by P0k 共dx0兲 = 共sup兩x0兩兲k P0共dx0兲. Let ␭ = ␭共␤兲 = max兵P01共X0兲 , P02共X0兲其. The main result of this paper is the following. Theorem 1: If the interaction potential ␾ satisfies the above mentioned conditions (2)–(4) and z is from the interval 0 ⬍ z ⬍ 关2l␤e␤B+1␭ max共M,储␾l储1,储ⵜ␾储1兲兴−1 ,

共5兲

then for any admissible domain ⌳ the log-partition function has the following asymptotic expansion: ln Z共⌳R,z兲 = R2兩⌳兩␤ p共␾,z兲 + Rb共⌳, ␾,z兲 + c共⌳, ␾,z兲 + o共1兲.

共6兲

where the coefficients p共␾ , z兲 , b共⌳ , ␾ , z兲 , and c共⌳ , ␾ , z兲 are given explicitly by Eqs. (18), (26),

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and (92), respectively. If ␾ is rotation invariant the coefficients b共⌳ , ␾ , z兲 and c共⌳ , ␾ , z兲 take a simpler form ln Z共⌳R,z兲 = R2兩⌳兩␤ p共␾,z兲 + R兩⳵ ⌳兩b共␾,z兲 + 2␲␹共⌳兲c共␾,z兲 + o共1兲

共7兲

with b共␾ , z兲 and c共␾ , z兲 given by Eqs. (94) and (95). To make this paper self-content we briefly recall the necessary notions and notations from I. Let X = 兵x 苸 C共关0 , ␤兴 , R2兲 兩 x共0兲 = x共␤兲其 be the space of all Brownian loops in R2 provided with the topology of uniform convergence and let B共X兲 be the Borel ␴-algebra in X. We consider a measure ␳ on 共X , B共X兲兲 given by

冕

X

h共x兲␳共dx兲 =

冕 冕 du

X0

R2

P0共dx0兲h共u + x0兲,

共8兲

where h ⱖ 0 is a measurable function on X. By ␳k we denote the measure given by 共8兲, but with P0k instead of P0. For a bounded domain ⌳ in R2 we denote by ␳⌳ the restriction of ␳ to the bounded Borel set X共⌳兲 = 兵x 苸 X 兩 x共t兲 苸 ⌳ for each 0 ⱕ t ⱕ ␤其 of Brownian loops in ⌳. Let M = M共X兲 = 兵␮ 傺 X兩兩␮兩 ⬍ ⬁其 be the space of finite configurations of loops in R2. Without confusing the reader, we use the notation 兩 · 兩 also for the number of elements in a finite set. A configuration ␮ = 兵x1 , . . . , xn其 苸 M can be identified with the finite sum of Dirac measures of its elements: ␮ = ␦x1 + ¯ +␦xn. Thus, M can be identified with the space of all finite simple point measures on R2, provided with its canonical ␴-field. 共See Ref. 16 for precise definitions and details.兲 We define a ␴-finite measure Wz␳ on M by ⬁

W z␳共 ␸ 兲 = 兺

n=0

zn n!

冕 冕 X

¯

X

␸共x1, . . . ,xn兲␳共dx1兲 ¯ ␳共dxn兲,

where ␸ ⱖ 0 is a measurable function on M. We remark that the measure ␳ is diffuse, i.e., ␳共兵x其兲 = 0 , ∀ x 苸 X, therefore, Wz␳ is correctly defined on the set of finite sums of Dirac measures ␦x1 + ¯ +␦xn with distinct x1 , . . . , xn, i.e., on M. The restriction of the measure Wz␳ to the set M共X共⌳兲兲 = 兵␮ 傺 X共⌳兲 兩 兩␮兩 ⬍ ⬁其 of finite configurations of loops in ⌳ we denote by Wz␳⌳. For shortness, we write M共⌳兲 for M共X共⌳兲兲 and Mc共⌳兲 for Mc共X共⌳兲兲, where Ac denotes the complement of a set A. Given an interaction ␾ the energy U共␮兲 of a configuration ␮ is defined by U共␮兲 =

1 兺 u共x,y兲, 2 x,y苸␮,

␮ 苸 M,

x⫽y

where u共x,y兲 =

冕

␤

␾共x共s兲 − y共s兲兲ds.

共9兲

0

Then the Feynman–Kac representation of grand partition function 共1兲 of a quantum gas in ⌳R with MB statistics can be written as4,14 Z共⌳R,z兲 =

冕

M共⌳R兲

Wz␳⌳ 共d␮兲e−U共␮兲 . R

共10兲

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III. THE PROOF OF THE THEOREM A. The relevant steps of the proof of theorem from I

Here we briefly describe how the volume, the boundary terms, as well as the first contribution to the constant term were obtained in I. We write ln Z共⌳R , z兲 as an integral of the Ursell function over the finite configurations of loops in ⌳R, we then separate a loop and release all the constraints except that the separated loop starts in ⌳R. This gives the volume term. Then we take away the integral over the configurations where at least one loop leaves ⌳R. Approximating this integral by the integral over the configurations where at least one loop crosses the tangent line, we obtain the boundary term as well as the first contribution Jc1 to the constant term 关see Eq. 共27兲 below兴. We start with the cluster representation of the log-partition function, ln Z共⌳R,z兲 =

冕

M共⌳R兲

Wz␳共d␮兲g共␮兲,

共11兲

where g is the Ursell function defined by g共␮兲 =

兺

兿

␥苸⌫con共␮兲 兵x,y其苸␥

兩␮兩 ⱖ 2,

共e−u共x,y兲 − 1兲,

共12兲

g共␮兲 = 1 for 兩␮兩 = 1 and g共 쏗 兲 = 0. Here ⌫con共␮兲 is the set of all 共unoriented兲 connected graphs with the set of vertices ␮ and the product is over all edges of the graph ␥. The integral on the right-hand side of Eq. 共11兲 is absolutely convergent for all z from the interval,16 0 ⬍ z ⬍ 2␲共e␤B+1储␾储1兲−1 .

共13兲

For any measurable A 傺 M we set WA f共␮兲 =

冕

W␳共d␻兲f共␮, ␻兲,

␮ 苸 M.

共14兲

A

We use the notation f共␮ , ␻兲 instead of f共␮ 艛 ␻兲, and since ␳ is diffuse, we can consider ␮ and ␯ in 共14兲 as disjoint configurations. For brevity we will write W instead of WM. Separating a loop from the configuration ␮ in Eq. 共11兲, we get ln Z共⌳R,z兲 =

冕 冕 du

⌳R

X0

P0共dx0兲W共1M共⌳R兲˜gz兲共u + x0兲 =

冕

⌳R

duIA共u兲 −

冕

⌳R

duIB共u,R兲,

共15兲

where 1A is the indicator function of A, ˜gz共␮兲 = z兩␮兩g共␮兲 / 兩␮兩 , 兩␮兩 ⱖ 1, IA共u兲 =

冕

X0

˜ z兲共u + x0兲 P0共dx0兲W共g

共16兲

and IB共u,R兲 =

冕

X0

P0共dx0兲W共1Mc共⌳R兲˜gz兲共u + x0兲.

共17兲

冕

共18兲

By translation invariance IA共u兲 =

X0

˜ z兲共x0兲. P0共dx0兲W共g

This integral multiplied by ␤−1 is the pressure p共␾ , z兲 which is an analytic function in z for 兩z兩 satisfying Eq. 共13兲.

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The contribution to the integral IB共u , R兲 of the loops u + x0 starting far from the boundary ⳵⌳R is small. This is due to the following lemma 共proof in Appendix兲 which describes the decay of correlations. Main Lemma: If ⌽ satisfies conditions (2) and (3), then for k = 1 , 2 , . . . , and z from the interval 0 ⬍ z ⬍ 关2le␤B+1␤储␾l储1 P0k 共X0兲兴−1 ,

共19兲

there exists a constant C = C共␾ , ␤ , z , k , l兲 , such that

冕

X0

P0共dx0兲

冕

X

␳k共dy兲1Mc共BR共u兲兲共y兲W兩gz兩共x0 + u,y兲 ⱕ

C . 共1 + R兲l

Here BR共u兲 is a ball of radius R centered at u 苸 R2 . Let ⌳R,␦ = 兵u 苸 ⌳R 兩 d共u , ⳵⌳R兲 ⬍ ␦R␧其, where d is the Euclidean distance on the plane. Taking from now on ␧ = 81 we find from Main Lemma that 兰⌳R\⌳R,␦IB共u , R兲du = o共1兲 as R → ⬁. Therefore,

冕

⌳R

IB共u,R兲du =

冕

⌳R,␦

IB共u,R兲du + o共1兲.

共20兲

To treat the integral on the right-hand side of Eq. 共20兲, we set up at each point r 苸 ⳵⌳R local coordinates 共␰ , ␩兲, where ␰ is along the tangent vector s = s共r兲 and ␩ is along the inward drawn unit normal n = n共r兲 to ⳵⌳R at r. Then ⳵⌳R is given locally by ␩ = f r,R共␰兲, 兩␰兩 ⬍ ␦R␧, for ␦ ⬎ 0 small enough, where f r,R is a function of class C3. We observe the following evident equalities: f r,R共␰兲 = Rf r共R−1␰兲 and kR共r兲 = f r,R ⬙ 共0兲 = R−1k共r兲, where kR共r兲 is the curvature of ⳵⌳R at the point r 苸 ⳵⌳R , f r ⬅ f r,1 , k ⬅ k1. We choose

␦ = ¯k共⌳兲−1,

with ¯k共⌳兲 = sup 兩k共r兲兩 ⬍ ⬁.

共21兲

r苸⳵⌳

With the help of the local coordinate system we can write11

冕

⌳R,␦

IB共u,R兲du =

冕

⳵⌳R

␴共dr兲

冕

␦R␧

dt共1 − tkR共r兲兲IB共r + tn,R兲,

0

where ␴共dr兲 is the arclength element of ⳵⌳R. Let ⌸r,␦R␧ = 兵共␰, ␩兲兩兩␰兩 ⬍ ␦R␧其,

Fr,␦R␧ = 兵共␰, ␩兲 苸 ⌸r,␦R␧兩␩ ⬎ f r,R共␰兲其. +

From now on we assume that r is a point from the convex part of ⳵⌳R. The concave case can be treated similarly 共see Remark in I兲. Our strategy in I was the following. Using the main lemma we approximated the integral IB over the configurations Mc共⌳R兲 that have at least one loop leaving ⌳R by the integral over the set M共⌸r,␦R␧兲Mc共Fr,+ ␦R␧兲 of configurations that have at least one loop traveling outside ⌳R but none of them visits neither the exterior of the cylinder ⌸r,␦R␧ nor the holes of ⌳R. Then we decomposed this set in two sets as +

+

+

+

M共⌸r,␦R␧兲Mc共Fr,␦R␧兲 = M共⌸r,␦R␧兲Mc共⌸r,␦R␧兲 + M共⌸r,␦R␧兲Mc共Fr,␦R␧兲, with ⌸r,+ ␦R␧ = 兵共␰ , ␩兲 苸 ⌸r,␦R␧ 兩 ␩ ⬎ 0其, and obtained the following equality:

共22兲

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冕

⌳R

IB共u,R兲du =

冕 冕

⳵⌳R

␴共dr兲

冕 冕

␦R␧

dt共1 − tkR共r兲兲J共r + tn,R兲

0

␴共dr兲

+

⳵⌳R

␦R␧

dt共1 − tkR共r兲兲K共r + tn,R兲 + o共1兲

0

= J共R兲 + K共R兲 + o共1兲,

共23兲

with

J共r + tn,R兲 =

冕

P0共dx0兲W共1Mc共⌸+

gz兲共r 兲M共⌸r,␦R␧兲˜

冕

P0共dx0兲W共1Mc共F+

兲M共⌸

X0

r,␦R␧

+ tn + x0兲

and

K共r + tn,R兲 =

X0

r,␦R␧

˜g 兲共r + tn + x0兲.

+ 兲 z r,␦R␧

共24兲

It was shown 关see I, Eqs. 共28兲 and 共30兲–共32兲兴 that J共R兲 = − Rb共⌳, ␾,z兲 − Jc1 + o共1兲.

共25兲

Here b共⌳, ␾,z兲 = −

冕

⳵⌳

␴共dr兲

冕 冕 ⬁

dt

X0

0

P0共dx0兲W共1Mc共⌸+兲˜gz兲共r + tn + x0兲 r

共26兲

and

Jc1 =

冕

⳵⌳

␴共dr兲k共r兲

冕 冕 ⬁

dtt

X0

0

P0共dx0兲W共1Mc共⌸+兲˜gz兲共r + tn + x0兲, r

共27兲

where ⌸r+ = 兵共␰ , ␩兲 兩 ␩ ⬎ 0其. The integrals on the right-hand side of Eqs. 共26兲 and 共27兲 are absolutely convergent for z satisfying 0 ⬍ z ⬍ ␲共2l−1e␤B+1储␾l储1兲−1 .

共28兲

We observe that Rb共⌳ , ␾ , z兲 is the boundary term and Jc1 is the first contributions to the constant term of asymptotic expansion 共6兲. Now we consider the term K共R兲 from 共23兲. In I we have just shown that K共R兲 = o共R兲. Below, analyzing this term in more details, we will separate from K共R兲 the second and the third contributions to the constant term and will show that the rest is o共1兲. Note that, due to the factor 1Mc共F+ ␧兲M共⌸+ ␧兲, the integration on the right-hand side of Eq. r,␦R

r,␦R

共24兲 is over those configurations 兵r + tn + x0 , ␻其 in ⌸r,+ ␦R␧, where at least one loop leaves ⌳R. There are two possibilities: either the loop r + tn + x0 stays in ⌳R then at least one loop from the configuration ␻ leaves ⌳R or the loop r + tn + x0 itself leaves ⌳R then ␻ is any configuration in ⌸r,+ ␦R␧. We will treat these cases separately. Therefore we write K共r + tn , R兲 as

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K共r + tn,R兲 =

冕

X0

P0共dx0兲1X共F+

r,␦R␧

⫻共r + tn + x0兲 + ⫻W共1Mc共F+

r,␦R␧

兲共r

冕

X0

兲M共⌸

+ tn + x0兲W共1Mc共F+

r,␦R␧

P0共dx0兲1Xc共F+

r,␦R␧

兲共r

兲M共⌸

˜g 兲

+ 兲 z r,␦R␧

+ tn + x0兲

˜g 兲共r + tn + x0兲

+ 兲 z r,␦R␧

= KA共r + tn,R兲 + KB共r + tn,R兲,

共29兲

respectively, K共R兲 =

冕 冕

⳵⌳R

␴共dr兲

冕 冕

␦R␧

dt共1 − tkR共r兲兲KA共r + tn,R兲

0

+

⳵⌳R

␴共dr兲

␦R␧

dt共1 − tkR共r兲兲KB共r + tn,R兲

0

= K 共R兲 + K 共R兲. A

共30兲

B

B. Analysis of KA„R…: The second contribution to the constant term

In this subsection we will separate from KA共R兲 the second contribution Jc2 to the constant term 关see Eq. 共71兲 below兴. Note that integration in KA共r + tn , R兲 is over the configurations 共r + tn + x0 , ␻兲, where the loop r + tn + x0 stays in ⌳R and at least one loop from the configuration ␻ leaves ⌳R. To get rid of the dependence on ⌳R, we decompose KA共r + tn , R兲 as KA共r + tn,R兲 =

冕

X0

P0共dx0兲1X共⌸+

r,␦R␧

˜ z兲共r + tn + x0兲 − ⫻共g

兲共r

冕

+ tn + x0兲WMc共F+

X0

⫻共r + tn + x0兲WMc共F+

r,␦R␧

P0共dx0兲1Xc共F+

r,␦R␧

r,␦R␧

兲M共⌸

+ 兲 r,␦R␧

兲M共⌸

兲X共⌸

+ 兲 r,␦R␧

+ 兲 r,␦R␧

˜ z兲共r + tn + x0兲 共g

= KA1共r + tn,R兲 − KA2共r + tn,R兲.

共31兲

Now the integration in K 共r + tn , R兲 is over the configurations 兵r + tn + x , ␻其 that have two or more loops visiting the domain ⌸r,+ ␦R␧ \ Fr,+ ␦R␧. The contribution of such configurations is small 关due to the bound 共39兲兴. By similar arguments we approximate KA1共r + tn , R兲 by the integral over the configurations 兵r + tn + x0 , ␻其, where the loop r + tn + x0 stays in ⌸r,+ ␦R␧ and exactly one loop from ␻ visits the domain ⌸r,+ ␦R␧ \ Fr,+ ␦R␧. Then we approximate the last integral by the integral where this loop crosses the parabola tangent to ⳵⌳R, but not the tangent. Thus, we get the quantity Jc2. Decomposition 共31兲 implies A2

0

KA共R兲 = KA1共R兲 − KA2共R兲,

共32兲

where KA1共R兲 = and

冕

⳵⌳R

␴共dr兲

冕

␦R␧

0

dt共1 − tkR共r兲兲KA1共r + tn,R兲

共33兲

073302-8

J. Math. Phys. 51, 073302 共2010兲

Suren Poghosyan

KA2共R兲 =

冕

␴共dr兲

⳵⌳R

冕

␦R␧

dt共1 − tkR共r兲兲KA2共r + tn,R兲.

共34兲

0

With the help of the equality

冕

Mc共X共⌳兲兲

h共␻兲W␳共d␻兲 =

冕

W␳共d␻兲共− 1兲兩␻兩+1Wh共␻兲,

M+共Xc共⌳兲兲

共35兲

which holds true for any absolutely integrable h, we have WMc共F+

r,␦R␧

=

兲M共⌸

冕

+ 兲 r,␦R␧

+ + M+共Xc共Fr,␦R␧兲兲M共⌸r,␦R␧兲

⫻WM共⌸+ =

˜ z兲共r + tn + x0兲 共g

r,␦R␧

冕

+

˜ z兲共r 兲共g

W␳共d␻兲共− 1兲兩␻兩+1

+ tn + x0, ␻兲

␳共dy兲WM共⌸+

r,␦R␧

+

Xc共Fr,␦R␧兲X共⌸r,␦R␧兲

⬁

兺 m=2

⫻共r + tn + x0,y兲 + ⫻

冕

+

+

关Xc共Fr,␦R␧兲X共⌸r,␦R␧兲兴m

⫻WM共⌸+

r,␦R␧

˜ z兲共r 兲共g

˜ z兲 兲共g

共− 1兲m m!

␳共dy 1兲 ¯ ␳共dy m兲

+ tn + x0,y 1, . . . ,y m兲

= S1共r + tn + x0兲 + Sˆ2共r + tn + x0兲.

共36兲

According to this we split KA1 into two parts

KA1共r + tn,R兲 =

冕 冕 X0

P0共dx0兲1X共⌸+

+

X0

r,␦R␧

兲共r

P0共dx0兲1X共⌸+

r,␦R␧

+ tn + x0兲S1共r + tn + x0兲

兲共r

+ tn + x0兲Sˆ2共r + tn + x0兲

ˆ A1共r + tn,R兲. = KA1 1共r + tn,R兲 + K

共37兲

We estimate the series Sˆ2共r + tn + x0兲 with the help of the following lemma 共see Appendix兲. Lemma 2: Let ¯u = ␤ max兵M , 共2␲␤兲−1储␾储1其 , then for all z from the interval ¯ e␤B+1兲−1 0 ⬍ z ⬍ 共u

共38兲

and all ␻ 苸 M the following inequality holds true: ˜ z兩兲共␻兲 ⱕ W共兩g

冋

共兩␻兩 − 1兲! ze␤B+1¯u ¯ eu 1 − ze␤B+1¯u

Due to the inequality just after formula 共40兲 in I

册

兩␻兩

.

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J. Math. Phys. 51, 073302 共2010兲

Asymptotic expansion +

+

␳共Xc共Fr,␦R␧兲X共⌸r,␦R␧兲兲 ⱕ CR−5/8 ,

共39兲

where C = C共⌳ , ␤兲 is a positive constant. Using this bound and Lemma 2 we find that for all z from interval 共38兲,

冉

⬁

¯ e␤B+1 1 zu 兩Sˆ2共r + tn + x0兲兩 ⱕ 兺 ¯ m=2 1 − zu ¯ e␤B+1 eu

冊

m+1 +

+

关␳共Xc共Fr,␦R␧兲X共⌸r,␦R␧兲兲兴m ⱕ CR−5/4 ,

共40兲

with C = C共⌳ , ␤ , z兲. Hence ˆ A1共R兲兩 ⱕ 2 兩K

冕

␴共dr兲

⳵⌳R

冕

␦R␧

˜ A1共r + tn,R兲兩 ⱕ 兩⳵ ⌳兩CR−1/8 . dt兩K

共41兲

0

Now passing to KA1 1共R兲 from 共37兲 we have KA1 1共R兲 =

冕

⳵⌳R

␴共dr兲

冕 冕 冕 ␦R␧

dt

X0

0

⫻S1共r + tn + x 兲 −

P0共dx0兲1X共⌸+

r,␦R␧

0

⫻1X共⌸+

r,␦R␧

兲共r

⳵⌳R

␴共dr兲

冕

␦R␧

兲共r

+ tn + x0兲

dttkR共r兲

0

冕

X0

P0共dx0兲

+ tn + x0兲S1共r + tn + x0兲.

共42兲

From bound 共39兲 and Lemma 2 it follows that the second summand on the right-hand side of Eq. 共42兲 is o共1兲. Therefore, combining Eqs. 共37兲, 共41兲, and 共42兲, we have KA1共R兲 =

冕

⳵⌳R

␴共dr兲

冕

␦R␧

dtKA2 1共r + tn,R兲 + o共1兲 = KA2 1共R兲 + o共1兲,

共43兲

0

where KA2 1共r + tn,R兲 =

冕

X0

⫻

P0共dx0兲1X共⌸+

r,␦R␧

冕

+

+

Xc共Fr,␦R␧兲X共⌸r,␦R␧兲

兲共r

+ tn + x0兲

␳共dy兲WM共⌸+

r,␦R␧

˜ z兲共r 兲共g

+ tn + x0,y兲.

共44兲

We cannot use directly Main Lemma to get rid of the restrictions to the cylinder ⌸r,+ ␦R␧ in Eq. 共44兲. First observe that with the help of identity 共13兲 from I, we have

冕

X

␳共dy兲

=

冕 冕

冕

M

ⱕ

Mc共B

␦R␧共r+tn兲兲

W␳共d␻兲

Mc共B

冉兺

y苸␻

␦R␧共r+tn兲兲

˜ z共r + tn + x0,y, ␻兲兩 W␳共d␻兲兩g

冊

˜ z共r + tn + x0, ␻兲兩 1Mc共B␦R␧共r+tn兲兲共␻ \ 兵y其兲 兩g

W␳共d␻兲兩gz共r + tn + x0, ␻兲兩.

共45兲

Then applying Main Lemma we get KA2 1共r + tn,R兲 = KA3 1共r + tn,R兲 + o共R−2兲 with

共46兲

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J. Math. Phys. 51, 073302 共2010兲

Suren Poghosyan

KA3 1共r + tn,R兲 =

冕

X0

P0共dx0兲1X共⌸+兲共r + tn + x0兲 r

冕

+ + Xc共Fr,␦R␧兲X共⌸r,␦R␧兲

˜ z兲共r + tn + x0,y兲. ␳共dy兲WM共⌸+兲共g r

共47兲 Since ␾ 苸 C1, writing y = 共␰ , ␩兲 + y 0 in the local coordinates, we have ˜ z兲共r + tn + x0,y兲 = WM共⌸+兲共g ˜ z兲共r + tn + x0,共␰, ␩0兲 + y 0兲 WM共⌸+兲共g r

r

+

⳵ ˜ z兲共r + tn + x0,共␰, ¯␩0兲 + y 0兲 WM共⌸+兲共g r ⳵␩

⫻共␩ − ␩0兲,

共48兲

where ␩0 = −inf具y 0 , n典 = −inft具y 0共t兲 , n典 and ¯␩0 = ␩0 + ␪共␩ − ␩0兲 , 0 ⬍ ␪ ⬍ 1. According to this we decompose KA3 1共r + tn , R兲 as KA3 1共r + tn,R兲 = KA4 1共r + tn,R兲 + KA5 1共r + tn,R兲,

共49兲

where

KA4 1共r + tn,R兲 =

冕

X0

P0共dx0兲1X共⌸+兲共r + tn + x0兲 r

冕 冕 ⬁

⫻

d␩

0

X0

P0共dy 0兲1Xc共F+

r,␦R␧

冕

兩␰兩ⱕ␦R␧

兲X共⌸

+ 兲 r,␦R␧

d␰

共共␰, ␩兲 + y 0兲

˜ z兲共r + tn + x0,共␰, ␩0兲 + y 0兲 ⫻WM共⌸+兲共g

共50兲

r

and

KA5 1共r + tn,R兲 =

冕

X0

P0共dx0兲1X共⌸+兲共r + tn + x0兲 r

冕

兩␰兩ⱕ␦R␧

d␰

冕

⫻

冕

⫻

⳵ ˜ z兲共r + tn + x0,共␰, ¯␩0兲 + y 0兲. WM共⌸+兲共g r ⳵␩

X0

P0共dy 0兲1Xc共F+

r,␦R␧

兲X共⌸

+ 兲 r,␦R␧

⬁

d ␩ 共 ␩ − ␩ 0兲

0

共共␰, ␩兲 + y 0兲 共51兲

To estimate KA5 1 we use the following lemma 共see the proof in Appendix兲. Lemma 3: If ␾ 苸 C1 satisfies conditions (2)–(4), then for all z from the interval 0 ⬍ z ⬍ 2␲关e␤B+1 max共储␾储1 , 储ⵜ␾储1兲兴−1 , the derivative of the two-point truncated correlation function satisfies the following bound:

冏

冏

⳵ W共gz兲共x0 + 共␰, ␩兲,y兲 ⱕ D, ⳵␩

共␰, ␩兲 苸 R2 ,

where D = D共␾ , ␤ , z兲 does not depend on x0 + 共␰ , ␩兲 , y . Using Fubini’s theorem and arguments similar to those which were used to prove bound 共41兲 from I, we have

073302-11

J. Math. Phys. 51, 073302 共2010兲

Asymptotic expansion

KA5 1共r + tn,R兲 =

冕

X0

P0共dx0兲1X共⌸+兲共r + tn + x0兲 r

⫻1sup兩y0+␰兩ⱕ␦R␧共y 0兲 ⫻

冕

冕

兩␰兩ⱕ␦R␧

d␰

冕

X0

P0共dy 0兲

⳵ ˜ z兲共r + tn + x0,共␰, ¯␩0兲 + y 0兲 WM共⌸+兲共g r ⳵␩

 0兲−具y 0,n典兴 sup关f r,R共␰+y

␩0

d␩共␩ − ␩0兲.

共52兲

0 0 共t兲 denotes the projection of y 0共t兲 onto the tangent to ⳵⌳R : y 共t兲 = y 0共t兲 We recall that  y 0 = y 0 − 具y 共t兲 , n典. Due to bound 共40兲 from I,

y 0兲 − 具y 0,n典兴 − ␩0其2 ⱕ C共⌳, ␤兲R−3/2 . 兵sup关f r,R共␰ +

共53兲

Therefore, applying Lemma 3 we get 兩KA5 1共r + tn,R兲兩 ⱕ DC共⌳, ␤兲R−11/8 .

共54兲

Combining Eqs. 共43兲, 共46兲, 共49兲, and 共54兲 we find KA1共R兲 =

冕

␴共dr兲

⳵⌳R

冕

␦R␧

dtKA4 1共r + tn,R兲 + o共1兲 = KA4 1共R兲 + o共1兲,

共55兲

0

where KA4 1共r + tn , R兲 is given by Eq. 共50兲. Observe that KA4 1共R兲 =

冕

⳵⌳R

␴共dr兲

冕

⬁

dtKA4 1共r + tn,R兲 + o共1兲.

共56兲

0

Indeed, by Main Lemma,

冏冕

⬁

␦R␧

冏 冕 冕

dtKA4 1共r + tn,R兲 ⱕ

⬁

␦R␧

dt

X0

P0共dx0兲

冕

Xc共Bt共r+tn兲兲

˜ z兩共r + tn + x0,y兲 ␳共dy兲WM共⌸+兲兩g r

ⱕ CR−15/8 , which implies Eq. 共56兲. The integral in KA4 1共R兲 over the loops 共␰ , ␩兲 + y 0 苸 Xc共Fr,+ ␦R␧兲X共⌸r,+ ␦R␧兲 we approximate by the integral over the loops which cross the parabola, tangent to ⳵⌳R, without crossing the tangent line. Arguing as we did above to estimate KA5 1共r + tn , R兲, we can write that for all t ⬎ 0, KA4 1共r + tn,R兲 =

冕

X0

P0共dx0兲1X共⌸+兲共r + tn + x0兲 r

冕

兩␰兩ⱕ␦R␧

d␰

冕

X0

P0共dy 0兲

˜ z兲共r + tn + x0,共␰, ␩0兲 + y 0兲 ⫻1sup兩␰+y0兩ⱕ␦R␧共␰,y 0兲WM共⌸+兲共g r

⫻兵sup关f r,R共␰ + y 0兲 − 具y 0,n典兴 − ␩0其.

共57兲

y 0兲 − 具y 0 , n典兴 − ␩0其 by 21 kR共r兲共␰ + y 0共 ␶共n兲兲兲2 we get Approximating 兵sup关f r,R共␰ + KA4 1共r + tn,R兲 = KA6 1共r + tn,R兲 + KA7 1共r + tn,R兲, where

共58兲

073302-12

J. Math. Phys. 51, 073302 共2010兲

Suren Poghosyan

KA6 1共r + tn,R兲 =

冕

X0

P0共dx0兲1X共⌸+兲共r + tn + x0兲 r

冕

兩␰兩ⱕ␦R␧

d␰

冕

X0

P0共dy 0兲

˜ z兲共r + tn + x0,共␰, ␩0兲 + y 0兲 ⫻1sup兩␰+y0兩ⱕ␦R␧共␰,y 0兲WM共⌸+兲共g r

1 ␶共n兲兲兲2 ⫻ kR共r兲共␰ + y 0共 2

共59兲

and the correction KA7 1共r + tn,R兲 =

冕

X0

P0共dx0兲1X共⌸+兲共r + tn + x0兲 r

冕

兩␰兩ⱕ␦R␧

d␰

冕

X0

P0共dy 0兲

˜ z兲共r + tn + x0,共␰, ␩0兲 + y 0兲 ⫻1sup兩␰+y0兩ⱕ␦R␧共␰,y 0兲WM共⌸+兲共g

再

r

冎

1 y 0兲 − 具y 0,n典兴 − ␩0 − kR共r兲共␰ + y 0共 ␶共n兲兲兲2 . ⫻ sup关f r,R共␰ + 2

共60兲

Here ␶共n兲 satisfies the condition inf具y 0 , n典 = 具y 0共␶共n兲兲 , n典. As in I we also choose ␶R共␰兲 so that y 0兲 − 具y 0,n典兴 = f r,R共␰ + y 0共 ␶R共␰兲兲兲 − 具y 0共␶R共␰兲兲,n典. sup关f r,R共␰ +

共61兲

We want to show that

冕

⳵⌳R

␴共dr兲

冕

⬁

dtKA7 1共r + tn,R兲 = o共1兲.

共62兲

0

Since ⳵⌳ 苸 C3, we have that 兩f r,R共␰兲兩 ⱕ C兩␰兩2R−1

and

兩 f r,R共␰兲 − 21 kR共r兲␰2兩 ⱕ C兩␰兩3R−2

共63兲

for ␰ , 兩␰兩 ⱕ ␦R␧. By the choice of ␶R共␰兲 and ␶共n兲, obviously

␶R共␰兲兲兲 − f r,R共␰ + y 0共 ␶共n兲兲兲 ⱕ CR−3/4 , 0 ⱕ 具y 0共␶R共␰兲兲,n典 − 具y 0共␶共n兲兲,n典 ⱕ f r,R共␰ + y 0共 y 0兩 ⱕ ␦R␧. Now ␶共n兲 is P0-almost surely unique 共see 2.8 D in Ref. 7; a for all y 0 , ␰ with sup兩␰ + direct proof is given in Ref. 5兲, hence ␶R共␰兲 → ␶共n兲 as R → ⬁. Using Eq. 共63兲 and the definitions of ␶共n兲 and ␶R共␰兲, we have y 0兲 − 具y 0,n典兴 + inf具y 0,n典 − 21 kR共r兲共␰ + y 0共 ␶共n兲兲兲2 sup关f r,R共␰ + ⱕ f r,R共␰ + y 0共 ␶R共␰兲兲兲 − 21 kR共r兲共␰ + y 0共 ␶共n兲兲兲2 ⱕ CR−2兩␰ + y 0共 ␶R共␰兲兲兩3 + 21 kR共r兲关共␰ + y 0共 ␶R共␰兲兲兲2 − 共␰ + y 0共 ␶共n兲兲兲2兴 ⱕ CR−13/8 + 21 R−1¯k⌳兩共␰ + y 0共 ␶R共␰兲兲兲2 − 共␰ + y 0共 ␶共n兲兲兲2兩. On the other hand sup关f r,R共␰ + y 0兲 − 具y 0,n典兴 + inf具y 0,n典 − 21 kR共r兲共␰ + y 0共 ␶共n兲兲兲2 ⱖ f r,R共␰ + y 0共 ␶共n兲兲兲 − 21 kR共r兲共␰ + y 0共 ␶共n兲兲兲2 ⱖ − CR−2兩␰ + y 0共 ␶共n兲兲兩3 ⱖ − CR−13/8 . Thus,

073302-13

J. Math. Phys. 51, 073302 共2010兲

Asymptotic expansion

兩sup关f r,R共␰ + y 0兲 − 具y 0,n典兴 + inf具y 0,n典 − 21 kR共r兲共␰ + y 0共 ␶共n兲兲兲2兩 ⱕ CR−13/8 + 21 R−1¯k⌳兩共␰ + y 0共 ␶R共␰兲兲兲2 − 共␰ + y 0共 ␶共n兲兲兲2兩

共64兲

Substituting this into Eq. 共60兲 we get 兩KA7 1共r + tn,R兲兩 ⱕ CR−13/8

冕

X0

冕 冕 ␦R␧

P0共dx0兲

−␦R␧

d␰

X0

P0共dy 0兲1sup兩␰+y0兩ⱕ␦R␧共␰,y 0兲

˜ z兩共r + tn + x0,共␰, ␩0兲 + y 0兲 + R−1¯k⌳ ⫻WM共⌸+兲兩g ⫻

r

冕

X0

P0共dx0兲

冕 冕 ␦R␧

−␦R␧

d␰

X0

P0共dy 0兲1sup兩␰+y0兩ⱕ␦R␧共␰,y 0兲

˜ z兩共r + tn + x0,共␰, ␩0兲 + y 0兲兩共␰ + y 0共 ␶R共␰兲兲兲2 ⫻WM共⌸+兲兩g r

0

˜ A1共r + tn,R兲. − 共␰ + y 共␶共n兲兲兲2兩 = KˆA7 1共r + tn,R兲 + K 7

共65兲

Let us fix any t ⬎ 0. According to Main Lemma, KˆA7 1共r + tn,R兲 ⱕ CR−13/8

冕

X0

P0共dx0兲

冕

˜ z兩共r + tn + x0,y兲 ⱕ CR−13/8共1 + t兲−l . ␳共dy兲WM兩g

Xc共Bt共r+tn兲兲

共66兲 Since ␶R共␰兲 → ␶共n兲 as R → ⬁, by Lebesgue dominated convergence theorem, R−1¯k⌳

冕

⳵⌳R

␴共dr兲

冕

⬁

˜ A1共r + tn,R兲 → 0. dtK 7

共67兲

0

To apply the Lebesgue theorem, we observe that due to Main Lemma,

冕 冕 冕 ⬁

dt

X0

0

ⱕ

P0共dx0兲

⬁

0

冕

⬁

d␰兩␰兩s

−⬁

⬁

dt 兺 共k + 1兲s k=0

冕

X0

+ x0,共␰, ␩0兲 + y 0兲 ⱕ

冕

X0

˜ z兩共r + tn + x0,共␰, ␩0兲 + y 0兲 P0j 共dy 0兲WM兩g

P0共dx0兲

冕

⬁

d␰1kⱕ兩␰兩⬍k+1

−⬁

冕

⬁

0

⬁

dt 兺 共k + 1兲s k=0

冕

˜ z兩共r + tn + x0,y兲 ⱕ C共␤,z,l兲, ⫻WM兩g

X0

冕

X0

P0共dx0兲

˜ z兩共r + tn P0j 共dy 0兲WM兩g

冕

Xc共Bmax共k,t兲共r+tn兲兲

␳ j共dy兲

j,s = 0,1,2.

共68兲

From Eqs. 共66兲 and 共67兲 we get Eq. 共62兲 which together with Eqs. 共55兲 and 共58兲 implies KA1共R兲 = KA6 1共R兲 + o共1兲 =

冕

⳵⌳R

␴共dr兲

冕

⬁

dtKA6 1共r + tn,R兲 + o共1兲,

共69兲

0

where KA6 1共r + tn , R兲 is given by Eq. 共59兲. Thus applying once more Main Lemma to KA6 1共R兲 we get KA1共R兲 = Jc2 + o共1兲, with the constant term

共70兲

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J. Math. Phys. 51, 073302 共2010兲

Suren Poghosyan

Jc2 =

1 2

冕 冕

⳵⌳

⫻

X0

␴共dr兲k共r兲

冕 冕 ⬁

dt

X0

0

P0共dx0兲1X共⌸+兲共r + tn + x0兲 r

冕

⬁

d␰

−⬁

˜ z兲共r + tn + x0,共␰, ␩0兲 + y 0兲共␰ + y 0共 P0共dy 0兲WM共⌸+兲共g ␶共n兲兲兲2 . r

共71兲

Note that due to Eq. 共68兲 the integral on the right-hand side of Eq. 共71兲 is absolutely convergent in the interval 0 ⬍ z ⬍ 共2l␤e␤B+1␭储␾l储1兲−1 .

共72兲

Lemma 2 and formulas 共36兲 and 共39兲 imply that for R large enough, 兩WMc共F+

r,␦R␧

兲M共⌸

+ 兲 r,␦R␧

˜ z兲共r + tn + x0兲兩 ⱕ C共⌳, ␤,z兲R−5/8 共g

On the other hand, using bound 共53兲 with ␰ = 0, we have

冏冕

␦R␧

dt共1 − tkR共r兲兲

0

ⱕ2

冕

X0

冕

X0

P0共dx0兲1Xc共F+

r,␦R␧

兲X共⌸

+ 兲 r,␦R␧

共r + tn + x0兲

共73兲

冏

˜ 0兲 − 具x0,n典兴 + inf具x0,n典其 P0共dx0兲1sup兩x˜0兩⬍␦R␧共x0兲兵sup关f r,R共x

ⱕ C共⌳, ␤兲R−3/4 .

共74兲

Hence 兩KA2共R兲兩 ⱕ C共⌳ , ␤ , z兲R−3/8. Thus, we conclude from Eqs. 共32兲 and 共70兲 that KA共R兲 = Jc2共R兲 + o共1兲.

共75兲

C. Analysis of KB„R…: The third contribution to the constant term

The treatment of KB共R兲 in many aspects is similar to that of KA共R兲. Approximating the integral in KB共r + tn , R兲 关see Eqs. 共29兲 and 共76兲兴 over the configurations 共r + tn + x0 , ␻兲, where the loop r + tn + x0 leaves ⌳R by the integral over 共r + tn + x0 , ␻兲, where the loop r + tn + x0 crosses the parabola tangent to ⳵⌳R but not the tangent line, we get the contribution Jc3. Thus, the constant term is the sum Jc1 + Jc2 + Jc3. If the potential is rotation invariant each term of this sum is factorized into a potential dependent factor times the integral of the curvature along the boundary ⳵⌳R, which by the Gauss–Bonnet theorem is the Euler–Poincaré characteristic of ⌳ multiplied by 2␲. From Eqs. 共29兲 and 共30兲 and Main Lemma we get KB共R兲 =

冕

⳵⌳R

␴共dr兲

冕

␦R␧

dt共1 − tkR共r兲兲KB1 共r + tn,R兲 + o共1兲,

共76兲

0

where KB1 共r + tn,R兲 =

冕

X0

P0共dx0兲1Xc共F+

r,␦R␧

兲X共⌸

+ 兲 r,␦R␧

˜ z兲共r + tn + x0兲. 共r + tn + x0兲WM共⌸+兲共g r

共77兲

Similarly to Eq. 共48兲 we have ˜ z兲共r + tn + x0兲 = WM共⌸+兲共g ˜ z兲共r + t0n + x0兲 + WM共⌸+兲共g r

r

⳵ ˜ z兲共r + ¯t0n + x0兲共t − t0兲, WM共⌸+兲共g r ⳵t 共78兲

where t0 = −inf具x , n典 ,¯t0 = t0 + ␪共t − t0兲. Hence 0

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J. Math. Phys. 51, 073302 共2010兲

Asymptotic expansion

KB1 共r + tn,R兲 = KB2 共r + tn,R兲 + KB3 共r + tn,R兲,

共79兲

where KB2 共r + tn,R兲 =

冕

X0

P0共dx0兲1Xc共F+

r,␦R␧

兲X共⌸

+ 兲 r,␦R␧

˜ z兲共r + t0n + x0兲 共r + tn + x0兲WM共⌸+兲共g r

共80兲

and KB3 共r + tn,R兲 =

冕

P0共dx0兲1Xc共F+

r,␦R␧

X0

兲X共⌸

+ 兲 r,␦R␧

共r + tn + x0兲

⳵ ˜ z兲共r + ¯t0n + x0兲共t − t0兲. WM共⌸+兲共g r ⳵t 共81兲

Using Lemma 3, bound 共53兲 with ␰ = 0 and invoking arguments which were used to derive Eq. 共54兲, we can write

冏冕

␦R␧

冏 冕 冕

dt共1 − tkR共r兲兲KB3 共r + tn,R兲 ⱕ 2

0

⫻共r + tn + x0兲

冏

⬁

dt

0

X0

P0共dx0兲1Xc共F+

r,␦R␧

冏

⳵ ˜ z兲共r + ¯t0n + x0兲 兩共t − t0兲兩 ⱕ 2D WM共⌸+兲共g r ⳵t

兲X共⌸

冕

X0

+ 兲 r,␦R␧

P0共dx0兲

˜ 0兲 − 具x0,n典兴 + inf具x0,n典其2 ⱕ C共⌳, ␾, ␤,z兲R−3/2 . ⫻兵sup关f r,R共x

共82兲

Hence KB3 共R兲 =

冕

⳵⌳R

␴共dr兲

冕

␦R␧

dt共1 − tkR共r兲兲KB3 共r + tn,R兲 = o共1兲.

共83兲

0

It remains to separate the last contribution to the constant term coming from KB2 共R兲. We split in two parts

KB2 共R兲

KB2 共R兲 =

冕

⳵⌳R

␴共dr兲

冕

␦R␧

dtKB2 共r + tn,R兲 −

0

冕

⳵⌳R

␴共dr兲kR共r兲

冕

␦R␧

dttKB2 共r + tn,R兲.

共84兲

0

With the help of Lemma 2 and bound 共53兲, one can easily show that the second summand on the ␧ right-hand side of Eq. 共84兲 is o共1兲. By Main Lemma we can replace the integral 兰0␦R dt by 兰⬁0 dt in the first summand of Eq. 共84兲 with correction o共1兲. On the other hand, by Fubini’s theorem,

冕

⬁

dtKB2 共r + tn,R兲 =

0

冕

X0

˜ z兲共r + t0n + x0兲兵sup关f r,R共x ˜ 0兲 − 具x0,n典兴 + inf具x0,n典其. P0共dx0兲WM共⌸+兲共g r

共85兲 Therefore, KB2 共R兲 =

冕

⳵⌳R

␴共dr兲

冕

⬁

dtKB2 共r + tn,R兲 + o共1兲.

共86兲

0

˜ 0兲 − 具x0 , n典兴 + inf具x0 , n典其 by 21 kR共r兲 ⫻ 共x0共 Now we approximate 兵sup关f r,R共x ␶共n兲兲兲2 and estimate the correction. We proceed in a similar way as we did in the proof of Eqs. 共62兲 and 共69兲. It follows from Eq. 共86兲 that

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Suren Poghosyan

冕

⬁

0

1 dtKB2 共r + tn,R兲 = kR共r兲 2

冕

X0

˜ z兲共r + t0n + x0兲 P0共dx0兲WM共⌸+兲共g r

⫻共x0共 ␶共n兲兲兲2 +

再

冕

X0

˜ z兲共r + t0n + x0兲 P0共dx0兲WM共⌸+兲共g r

˜ 0兲 − 具x0,n典兴 + inf具x0,n典 ⫻ sup关f r,R共x

冎

1 ␶共n兲兲兲2 , − kR共r兲共x0共 2

共87兲

where ␶共n兲 is defined by inf具x0 , n典 = 具x0共␶共n兲兲 , n典. Then applying inequality 共64兲 with ␰ = 0 we get

冕

⳵⌳R

␴共dr兲

冕

X0

再

˜ z兲共r + t0n + x0兲 sup关f r,R共x ˜ 0兲 − 具x0,n典兴 P0共dx0兲WM共⌸+兲共g r

冎

1 + inf具x0,n典 − kR共r兲共x0共 ␶共n兲兲兲2 = o共1兲. 2

共88兲

Thus,

冕

⳵⌳R

␴共dr兲

冕

⬁

dtKB2 共r + tn,R兲 = Jc3 + o共1兲,

共89兲

0

where Jc3 =

1 2

冕

⳵⌳

␴共dr兲k共r兲

冕

X0

˜ z兲共r + t0n + x0兲共x0共 P0共dx0兲WM共⌸+兲共g ␶共n兲兲兲2 . r

共90兲

Observe that the integral in Eq. 共90兲 absolutely converges for z from interval 共72兲. Now combining Eqs. 共75兲, 共76兲, 共79兲, 共83兲, and 共86兲, we find that K共R兲 = Jc2 + Jc3 + o共1兲, Jc2

共91兲

Jc3

with and given by Eqs. 共71兲 and 共90兲, respectively. The desired expansion 共6兲 of the log-partition function follows from formulas 共15兲, 共23兲, 共25兲, and 共91兲. We observe that the constant term, c共⌳, ␾,z兲 = Jc1 − Jc2 − Jc3 =

冕

⳵⌳

− ⫻ −

␴共dr兲k共r兲

冕 冕 冕

1 2

1 2

⬁

dtt

X0

0

⳵⌳

X0

冕 冕 冕 冕

P0共dx0兲W共1Mc共⌸+兲˜gz兲共r + tn + x0兲

⬁

␴共dr兲k共r兲

dt

X0

0

r

P0共dx0兲1X共⌸+兲共r + tn + x0兲 r

冕

⬁

d␰

−⬁

˜ z兲共r + tn + x0,共␰, ␩0兲 + y 0兲共␰ + y 0共 P0共dy 0兲WM共⌸+兲共g ␶共n兲兲兲2

⳵⌳

r

␴共dr兲k共r兲

冕

X0

˜ z兲共r + t0n + x0兲共x0共 P0共dx0兲WM共⌸+兲共g ␶共n兲兲兲2 , r

共92兲

is an analytic function in z for 兩z兩 from interval 共72兲. If ␾ is rotation invariant, the Ursell function and the measure Wz␳ also are rotation invariant, therefore, the integral 兰⬁0 dt兰X0 P0共dx0兲W共1Mc共⌸+兲˜gz兲共r + tn + x0兲 does not depend on the orientation r

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Asymptotic expansion

of the normal n共r兲 and can be evaluated with respect to any fixed unit vector d1 and the half-plane ⌸d+ = 兵u 苸 R␯ 兩 具u , d1典 ⱖ 0其. In a similar way the corresponding potential dependent factors in the 1 integrals Jc1 , Jc2 , Jc3 do not depend on r 苸 ⌳ and can be evaluated with the help of an arbitrary fixed pair of orthogonal unit vectors d1 , d2. Thus, the terms b共⌳ , ␾ , z兲 and c共⌳ , ␾ , z兲 of expansion 共6兲 have simpler form, b共⌳, ␾,z兲 = 兩⳵ ⌳兩b共␾,z兲

c共⌳, ␾,z兲 = 2␲␹共⌳兲c共␾,z兲,

and

共93兲

where b共␾,z兲 = −

冕 冕 ⬁

dt

X0

0

P0共dx0兲W共1Mc共⌸+ 兲˜gz兲共td1 + x0兲

共94兲

d1

and c共␾,z兲 =

冕 冕 ⬁

dtt

X0

0

1 2

P0共dx0兲W共1Mc共⌸+ 兲˜gz兲共td1 + x0兲 − d1

⫻1X共⌸+ 兲共td1 + x0兲 d1

冕 冕 ⬁

d␰

−⬁

X0

冕 冕 ⬁

dt

X0

0

P0共dx0兲

˜ z兲共td1 + x0,y 0 P0共dy 0兲WM共⌸+ 兲共g d1

+ 共␰,− inf具y 0,d1典兲兲关␰+ 具y 0共␶共d1兲兲,d2典兴2 −

1 2

冕

X0

P0共dx0兲

˜ z兲共− inf具x0,d1典d1 + x0兲具x0共␶共d1兲兲,d2典2 . ⫻WM共⌸+ 兲共g

共95兲

d1

Here ␶共d1兲 is defined by inf具y 0 , d1典 = 具y 0共␶共d1兲兲 , d1典. In conclusion we note that the second equality in Eq. 共93兲 is a consequence of the Gauss– Bonnet theorem: 兰⳵⌳␴共dr兲k共r兲 = 2␲␹共⌳兲. Theorem 1 is proven. It is worth to note that the expansion of the log-partition function ln ⌶id共⌳R , z兲 of the ideal gas is obtained from expansion 共6兲 by setting ␾ ⬅ 0. In this case ln Zid共⌳R,z兲 = R2兩⌳兩␤zpid + R兩⳵ ⌳兩zbid + ␲␹共⌳兲zcid ,

共96兲

with pid = 兰X0 P0共dx0兲, bid = 兰X0 P0共dx0兲inf具x0 , d1典 and cid = 兰X0 P0共dx0兲关共inf具x0 , d1典兲2 0 2 − 具x 共␶共d1兲兲 , d2典 兴. Let us show how to get the constant term of expansion 共96兲. Eqs. 共92兲 and 共95兲 imply c c − Jid,3 =z cid共⌳,z兲 = Jid,1

−z ⫻

1 2

冕

冕

X0

⳵⌳

冕

⳵⌳

␴共dr兲k共r兲

␴共dr兲k共r兲 ·

冕

X0

冕

X0

P0共dx0兲

冕

−inf具x0,d1典

0

P0共dx0兲具x0共␶共d1兲兲,d2典2 = z

1 2

冕

dtt

⳵⌳

␴共dr兲k共r兲

P0共dx0兲关共inf具x0,d1典兲2 − 具x0共␶共d1兲兲,d2典2兴 = ␲␹共⌳兲cid共z兲.

Thus, one recovers the familiar case of large volume asymptotic expansion of Brownian integrals 关see Eqs. 共2.55兲 and 共2.56兲 in Ref. 9 with F = 0兴. IV. CONCLUDING REMARKS

The present paper together with I gives the three first terms of the asymptotic expansion of the log-partition function of a quantum gas in two-dimensional domain. We can apply the same arguments in the d-dimensional case, d ⬎ 2, and get a similar result,

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Suren Poghosyan

ln Z共⌳R,z兲 = Rd␤ p共␾,z兲兩⌳兩 + Rd−1b共␾,z兲兩⳵ ⌳兩 + Rd−2c1共⌳兲c2共␾,z兲 + o共Rd−2兲. Here p共␾ , z兲 and b共␾ , z兲 are given by formulas similar to Eqs. 共18兲 and 共94兲 and c1共⌳兲 = 共d − 1兲兰⳵⌳␴共dr兲km共r兲, where km共r兲 is the mean curvature of ⳵⌳ at the point r 苸 ⳵⌳. At the same time, we are not able to get more terms of the expansion. This is a familiar case also for the ideal gas, and it is not clear whether the reason is technical or not 共cf. Ref. 9, Sec. VII兲. Note that for classical gas all the nondecreasing terms of the asymptotic expansion of the log-partition function were obtained in Ref. 12. Reference 2 extends this result pushing the expansion beyond the constant term. We remark that in contrast to the classical case the log-partition function of the ideal quantum gas has nontrivial asymptotic expansion 共96兲. The reason is that a Brownian path may have nonempty intersection both with the domain ⌳ and its exterior. To prove boundedness property of the derivative of the two-point truncated correlation functions 共Lemma 3兲, we used the tree identity1 which involves the function u共x , y兲, and here for the sake of simplicity, we assumed the boundedness of the interaction ␾. One can release this restrictive condition by developing further the techniques of Refs. 16 and 10 which use the algebraic approach of Ruelle17 with tree-graph bounds involving the function e−u共x,y兲 − 1 which is bounded due to the stability of ␾. Relaying on the results of Refs. 5 and 13, we conjecture that the method developed in this paper allows to obtain a similar result in the case of a quantum gas with Bose–Einstein statistics. Finally, we note that the geometrical factors of all the three terms of expansion 共7兲 are the same as in the Hadwiger’s theorem from the integral geometry. It states that any real valued, additive function ␺ on the space of compact convex subsets ⌳ in R2 which is convex continuous 共with respect to the Housdorff metric兲 and invariant with respect to the Euclidean motions is a linear combination of the area, the length of the boundary, and the Euler–Poincaré characteristic of ⌳ : ␺共⌳兲 = c2兩⌳兩 + c1兩⳵⌳兩 + c0␹共⌳兲 with constant coefficients c2 , c1 , c0. But in contrast to this, expansion 共7兲 is an asymptotic expansion and the connections are not clear. We note that since ln Z共⌳ , z兲 as a function of convex bodies is not additive: ln Z共⌳1 艛 ⌳2 , z兲 ⫽ ln Z共⌳1 , z兲 + ln Z共⌳2 , z兲 − ln Z共⌳1 艚 ⌳2 , z兲, the Hadwiger’s theorem cannot be applied to ln Z共⌳ , z兲.

ACKNOWLEDGMENTS

The author is grateful to the referee for his useful comments that prompted a clearer layout of the paper. I also thank the German Academic Exchange Service for the generous support of this research project. This work is dedicated to 65th anniversary of Hans Zessin.

APPENDIX: PROOFS OF THE LEMMAS

Proof of Main Lemma: We note first that the measure P0k has the large deviation property, 2

P0k 共sup兩y 0兩 ⱖ R兲 ⱕ C1e−C2R ,

Ci = Ci共k, ␤兲,

i = 1,2.

共A1兲

4

Indeed, since P has the large deviation property, 共A1兲 follows from an application of the Schwarz inequality, 0

0 共X0兲兲1/2共P0共sup兩y 0兩 ⱖ R兲兲1/2 . P0k 共sup兩y 0兩 ⱖ R兲 ⱕ 共P2k

One can prove Main Lemma by repeating the arguments for the proof of Lemma 5.3 from Ref. 16. Below we give a simpler proof based on results of Ref. 14. Under our conditions on the interaction potential, we can apply Theorem 2.3 of Ref. 14 with X = X , ␮共dx兲 = z␳共dx兲 , a共x兲 = 1 , b共x兲 = ␤B, where B is the stability constant. Then the two-point truncated correlation function W共gz兲共x , y兲 satisfies the following inequality:

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J. Math. Phys. 51, 073302 共2010兲

Asymptotic expansion ⬁

兩W共gz兲共x,y兲兩 ⱕ W兩gz兩共x,y兲 ⱕ 共ze

␤B+1 2

兲

兺 共ze␤B+1兲m兩u兩ⴰm共x,y兲,

共A2兲

m=0

where 兩u兩ⴰ0共x , y兲 = 兩u共x , y兲兩 with u共x , y兲 defined by Eq. 共9兲 and 兩u兩ⴰm共x,y兲 =

冕

m

Xm

␳共dx1兲 ¯ ␳共dxm兲 兿 兩u共xi,xi+1兲兩,

m ⱖ 1,

x0 ⬅ x,

xm+1 ⬅ y.

i=0

By Fubini’s theorem 兰X␳k共dy兲兩u共x , y兲兩 ⱕ ␤ P0k 共X0兲储␾储1 for all x 苸 X. Therefore,

冕

X

␳k共dy兲兩u兩ⴰm共x,y兲 =

冕

X

␳k共dy兲

冕

Xm

␳共dx1兲 . . . ␳共dxm兲

m

⫻ 兿 兩u共xi,xi+1兲兩 ⱕ 关␤ P0k 共X0兲储␾储1兴m,

共A3兲

m = 0,1, . . . .

i=0

Let us show that for all R ⬎ 0 , a ⬎ 0, the following inequality

冕

Xc共BR+a共0兲兲

␳k共dy兲W兩gz兩共x,y兲 ⱕ

C 共1 + a兲l

共A4兲

holds uniformly in x 苸 X共BR共0兲兲. We observe that Eq. 共A4兲 implies Main Lemma due to Lemma 5.3 of Ref. 16. Setting Sn共a兲 = 兰Xc共BR+a共0兲兲␳k共dy兲兩u兩ⴰn共x , y兲, we first prove by induction in n that Sn共a兲 ⱕ C共n + 1兲关2l␤储␾l储1 P0k 共X0兲兴n+1共1 + a兲−l .

共A5兲

For n = 0 and x 苸 X共BR共0兲兲 we have S0共a兲 =

冕

Xc共BR+a共0兲兲

冋

ⱕ ␤ P0k 共X0兲

␳k共dy兲1X共Bc

R+a/2

冕

兩u兩⬎a/2

共0兲兲共y兲兩u共x,y兲兩

+

冕

Xc共BR+a共0兲兲

冉

du兩␾共u兲 + 储␾储1 P0k sup兩y 0兩 ⬎

a 2

冊册

␳k共dy兲1Xc共Bc

R+a/2

共0兲兲共y兲兩u共x,y兲兩

冉 冊

ⱕ C␤储␾l储1 P0k 共X0兲 1 +

a 2

−l

.

Hence Eq. 共A5兲 is valid. Assume that Eq. 共A5兲 holds true for all n = 0 , 1 , . . . , m and consider Sm+1共a兲. From Eq. 共A3兲 and induction hypothesis 共A5兲, it follows that for all x 苸 X共BR共0兲兲,

冕

Sm+1共a兲 =

X共BR+a/2共0兲兲

␳共dxm+1兲兩u兩ⴰm共x,xm+1兲

⫻兩u共xm+1,y兲兩 + ⫻

冕

Xc共BR+a共0兲兲

冕

Xc共BR+a/2共0兲兲

冕

冕

X

sup

冕

xm+1苸X X

冕 冕

xm+1苸X共BR+a/2共0兲兲 Xc共B R+a共0兲兲

␳k共dxm+1兲兩u兩ⴰm共x,xm+1兲 +

⫻␳k共dxm+1兲兩u兩ⴰm共x,xm+1兲 sup which implies Eq. 共A5兲.

␳k共dy兲

␳共dxm+1兲兩u兩ⴰm共x,xm+1兲

␳k共dy兲兩u共xm+1,y兲兩 ⱕ

⫻␳k共dy兲兩u共xm+1,y兲兩

Xc共BR+a共0兲兲

Xc共BR+a/2共0兲兲

␳k共dy兲兩u共xm+1,y兲兩,

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Suren Poghosyan

Now with the help of Eqs. 共A2兲 and 共A5兲 and the monotone convergence theorem, we conclude that for z , 0 ⬍ z ⬍ 2le␤B+1␤储␾l储1 P0k 共X0兲,

冕

⬁

Xc共B

R+a共0兲兲

␳k共dy兲W兩gz兩共x,y兲 ⱕ Cze

␤B+1

共m + 1兲关2lze␤B+1␤储␾l储1 P0k 共X0兲兴m+1共1 + a兲−l 兺 m=0

ⱕ C1ze␤B+1共1 + a兲−l , with C1 = C1共␾ , ␤ , z , k , l , d兲 ⬎ 0. This completes the proof of Eq. 共A4兲. Proof of Lemma 2: Let ␻ = 兵x1 , . . . , xm其. Then, with the help of Proposition 6.1, part 共b兲, from Ref. 14 and Stirling’s formula, we can write ⬁

˜ z兩兲共␻兲 ⱕ 兺 W共兩g

n=0

共ze2␤B兲n+m 兺 n! T苸T共1,. . .,m,m+1,. . .,m+n兲

冕

兿

Xn 兵i,j其苸T

兩u共xi,x j兲兩

⬁

⫻ dxm+1 ¯ dxm+n ⱕ ⱕ

冋

¯ e2␤B兲n+m 1 共zu 共m + n兲m+n−2 兺 n! ¯u n=0

¯ e2␤B+1 共m − 1兲! zu ¯ ¯ e2␤B+1 eu 1 − zu

册

m

.

Here, we used the well-known fact that the number of trees with n + m vertices is 共n + m兲n+m−2, and the last line is a consequence of the formula ⬁

兺 n=0

共m + n − 1兲! n 共m − 1兲! t = , n! 共1 − t兲m

兩t兩 ⬍ 1.

from Ref. 17, Sec. 4.4. This completes the proof of Lemma 2. Proof of Lemma 3: To prove Lemma 3 we use the tree identity from Ref. 1,

兺 兿

␥苸Cn 兵i,j其苸␥

共e−ui,j − 1兲 =

兺 兿

T苸Tn 兵i,j其苸T

共− ui,j兲

冕

冉

冊

d␭T共兵sij其兲exp − 兺 sijui,j . i⬍j

Here Cn is the set of connected graphs with n vertices, the real numbers, ui,j , 1 ⱕ i ⬍ j ⱕ n, satisfy the stability condition 兺1ⱕi⬍jⱕnui,j ⱖ −bn , sij = s ji , 0 ⱕ sij ⱕ 1 , ␭T depends on the tree T and is a probability measure supported on the set si,j , 1 ⱕ i ⬍ j ⱕ n, such that sijui,j ⱖ − bn. 兺 i⬍j

共A6兲

The details can be found in Ref. 1. We will apply this identity with ui,j = u共xi , x j兲 and b = ␤B, where B is the stability constant for ␾. Setting x1 = x01 + 共␰1 , ␩1兲, due to the tree identity we have

⳵ ⳵ ␩1

g共x1, . . . ,xn兲 =

兺

T苸Tn

⫻ ⫻

冕

⳵

冉

d␭T共兵sij其兲exp − 兺 siju共xi,x j兲

兿

⳵ ␩1 兵i,j其苸T

冉

i⬍j

共− u共xi,x j兲兲 +

⳵ − 兺 siju共xi,x j兲 ⳵ ␩1 i⬍j

兺

T苸Tn

冊兿

兵i,j其苸T

冕

冊 冉

d␭T共兵sij其兲exp − 兺 siju共xi,x j兲 i⬍j

共− u共xi,x j兲兲.

Interchanging integration and differentiation 共see Lemma 2.2 of Ref. 8兲, we get

冊

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J. Math. Phys. 51, 073302 共2010兲

Asymptotic expansion

⳵ ⳵ ␩1

W共gz兲共x01 + 共␰1, ␩1兲,x2兲 = G1共x1,x2兲 + G2共x1,x2兲,

共A7兲

with ⬁

G1共x1,x2兲 = z

2

zn−2

兺 n=2 共n − 2兲!

冉

冕

␳共dx3兲 ¯ ␳共dxn兲

Xn−2

⫻exp − 兺 siju共xi,x j兲 i⬍j

冊冋 ␩ ⳵

⳵

兿

1 兵i,j其苸T

兺

T苸Tn

冕

d␭T共兵sij其兲

共− u共xi,x j兲兲

册

and ⬁

G2共x1,x2兲 = z2 兺

n=2

zn−2 共n − 2兲!

冕

␳共dx3兲 ¯ ␳共dxn兲

Xn−2

冉

冕

d␭T共兵sij其兲

兿

兵i,j其苸T

冊 ␩冉

− 兺 siju共xi,x j兲 .

冊

兿

⫻共− u共xi,x j兲兲exp − 兺 siju共xi,x j兲 i⬍j

兺

T苸Tn

⳵

⳵

1

冊

i⬍j

Let us estimate G1共x1 , x2兲. Evidently

冉

⳵ ⳵ 兿 共− u共xi,x j兲兲 = j:兵1,j其苸T 兺 − ⳵ ␩1 u共x1,x j兲 ⳵ ␩1 兵i,j其苸T

兵k,l其苸T;兵k,l其⫽兵1,j其

共− u共xk,xl兲兲.

Hence by stability inequality 共A6兲 we have ⬁

兩G1共x1,x2兲兩 ⱕ 兺

n=2

⫻

共ze␤B兲n 兺 兺 共n − 2兲! T苸Tn j:兵1,j其苸T

兿

冕

Xn−2

␳共dx3兲 ¯ ␳共dxn兲

冏

⳵ u共x1,x j兲 ⳵ ␩1

冏

⬁

␤B 2

兵k,l其苸T;兵k,l其⫽兵1,j其

兩u共xk,xl兲兩 ⱕ e␤ max共M,M ⬘兲共ze 兲

兺n

n=2

⫻ 共n − 1兲关z␤e␤B+1 P0共X0兲max共储␾储1,储ⵜ␾储1兲兴n−2 .

共A8兲

Now we consider G2共x1 , x2兲. Since ␾ has bounded derivative,

冏 冉兺 ⳵

⳵ ␩1

−

siju共xi,x j兲

i⬍j

冊冏

n

ⱕ 兺 s1j j=2

冏

⳵ ⳵ ␩1

冏

u共x1,x j兲 ⱕ ␤ M ⬘n.

Therefore, ⬁

␤B 2

兩G2共x1,x2兲兩 ⱕ e␤ MM ⬘共ze 兲 2

兺 n共n − 1兲关z␤e␤B+1P0共X0兲储␾储1兴n−2 .

共A9兲

n=2

Now observe that if z␤e␤B+1 P0共X0兲max共储␾储1 , 储ⵜ␾储1兲 ⬍ 1, both series in the last lines of Eqs. 共A8兲 and 共A9兲 converge, hence Lemma 3 is proven. 1

Brydges, D. C., A Short Course on Cluster Expansions, Phénomènes Critiques, Systèmes Aléatoires, Théories de Jauge 共Elsevier, Amsterdam/North-Holland, Amsterdam, 1986兲, pp. 129–183. 2 Collet, P. and Dunlop, F., “Geometric expansion of the boundary free energy of a dilute gas,” Commun. Math. Phys. 108, 1 共1987兲. 3 Ginibre, J., “Reduced density matrices of quantum gases. I. Limit of infinite volume,” J. Math. Phys. 6, 238 共1965兲; “CII. Cluster property,” J. Math. Phys. 6, 252 共1965兲. 4 Ginibre, J., in Statistical Mechanics and Quantum Field Theory, edited by DeWitt, C. and Stora, R. 共Gordon and Breach, Les Houches, 1971兲, pp. 327–427. 5 Frigio, S. and Poghosyan, S., “Asymptotics of Brownian integrals and pressure. Bose-Einstein statistics,” J. Contemp.

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J. Math. Phys. 51, 073302 共2010兲

Math. Anal. 42, 49 共2007兲. Kac, M., “Can one hear the shape of a drum,” Am. Math. Monthly 73, 1 共1966兲. 7 Karatzas, I. and Shreve, S. E., Brownian Motion and Stochastic Calculus 共Springer, New York, 1988兲. 8 Lang, S., Real Analysis 共Addison-Wesley, London, 1983兲. 9 Macris, N., Martin, P. A., and Pulé, J. V., “Large volume asymptotics of brownian integrals and orbital magnetism,” Ann. Inst. Henri Poincare 66, 147 共1997兲. 10 Minlos, R. A. and Poghosyan, S., “Estimates of Ursell function, group functions and their derivatives,” Theor. Math. Phys. 31, 408 共1977兲. 11 O’Neil, B., Elementary Differential Geometry 共Academic, New York, 1966兲. 12 Pogosian, S., “Asymptotic expansion of the logarithm of the partition function,” Commun. Math. Phys. 95, 227 共1984兲. 13 Poshosyan, S., “Strong cluster properties of Ginibre gas. Quantum statistics,” J. Contemp. Math. Anal. 40, 57 共2005兲. 14 Poghosyan, S. and Ueltschi, D., “Abstract cluster expansion with applications to statistical mechanical systems,” J. Math. Phys. 50, 053509 共2009兲. 15 Poghosyan, S. and Zessin, H., “Asymptotic expansion of the log-partition function for a gas of interacting Brownian loops,” J. Math. Phys. 48, 093301 共2007兲. 16 Poghosyan, S. and Zessin, H., “Decay of correlations of the Ginibre gas obeying Maxwell-Boltzmann statistics,” Markov Processes Relat. Fields 7, 561 共2001兲. 17 Ruelle, D., Statistical Mechanics: Rigorous Results 共World Scientific, Singapore, 1999兲. 6