J Stat Phys (2016) 162:761–791 DOI 10.1007/s10955-015-1421-8
A Cluster Expansion Approach to the Heilmann–Lieb Liquid Crystal Model Diego Alberici1
Received: 15 June 2015 / Accepted: 24 November 2015 / Published online: 22 December 2015 © Springer Science+Business Media New York 2015
Abstract A monomer-dimer model with a short-range attractive interaction favoring collinear dimers is considered on the lattice Z2 . Although our choice of the chemical potentials results in more horizontal than vertical dimers, the horizontal dimers have no long-range translational order—in agreement with the Heilmann–Lieb conjecture (Heilmann and Lieb in J Stat Phys 20(6):679–693, 1979). Keywords expansion
Monomer-dimer · Liquid crystal · Heilmann–Lieb conjecture · Cluster
1 Introduction A liquid crystal, at low temperatures, displays a long-range order in the orientation of its molecules, while there is no complete ordering in their positions. In this paper we present a model characterized by these two features. In particular we consider a monomer-dimer model on the two-dimensional lattice Z2 characterized by different chemical potentials for horizontal and vertical dimers (μh > μv to fix ideas) and by a short-range potential J > 0 that favors collinear dimers. We prove that when the parameters satisfy μh > −J and μv < −
5 J, 2
(1.1)
the system has the properties of a liquid crystal. Onsager [14] was the first to propose hard-rods models in order to explain the existence of liquid crystals. In 1970 Heilmann and Lieb [8,9] studied systems of monomer and dimers (hard-rods of length 2) interacting only via the hard-core potential, and proved the absence of phase transitions in great generality. Then in 1972 they [10] proposed two monomer-
B 1
Diego Alberici
[email protected] Department of Mathematics, University of Bologna, Piazza di Porta San Donato 5, Bologna, Italy
123
762
D. Alberici
dimer models (named I and II) on the lattice Z2 , where short-range attractive interactions among parallel dimers are considered beyond the hard-core interaction. Heilmann and Lieb claimed that these systems are liquid crystals. In particular they proved the presence of a phase transition, by means of a reflection positivity argument: at low temperature there is orientational order. Moreover they conjectured the absence of complete translational ordering for their models. A proof of this conjecture for the model I was announced in [10] by Heilmann and Kjær, but never appeared. Letawe, in her thesis [12], claimed to prove the conjecture by cluster expansion methods, even if the result has never been published in a journal. Letawe’s polymers are built starting from contours and the major difficulty seems to arise when she has to deal with a polymer lying in the interior of another one: these two polymers would not be independent. To overcome the problem, ratios of partition functions with different (horizontal or vertical) boundary conditions Z v /Z h are introduced, but it is not proved that these ratios are sufficiently small to guarantee the convergence of the cluster expansion. Numerical simulations related to the Heilmann–Lieb conjecture are performed in [15]. We also mention that, in absence of attractive interaction, systems of sufficiently long hard-rods were proved to display a phase transition and behave like liquid crystals by Disertori and Giuliani [3], using a two scales cluster expansion and the Pirogov-Sinai theory. In the present paper we study a model obtained from the model I of Heilmann and Lieb [10], but while they suppose μh = μv =: μ and μ > −J,
(1.2)
we assume very different horizontal and vertical potentials as in (1.1). This choice of the parameters allows us to work with cluster expansion methods, by defining our polymers starting from regions of vertical dimers, instead of contours. The cluster expansion method permits to rewrite the logarithm of the partition function of a polymer system as a power series of the polymer activities. This expansion entails analyticity results and simplifies considerably the study of the correlation functions, which can be expressed in terms ratios of partition functions. Clearly the cluster expansion cannot hold in general on the whole space of parameters: it converges only when the polymer activities are small enough to compete with the entropy. A rigorous study of the conditions of convergence dates back to [6,7,16], by means of Kirkwood-Salsburg type of equations. In this paper we use a criterion proposed by Kotecky and Preiss [11] in 1986. Afterwards this criterion was compared to the previous ones, was improved and simplified in [1,2,4,5,13,17] (for a clear and modern treatment we suggest for example the last work). The paper is organized as follows. In the Sect. 2 we introduce the model and we state the main results about its liquid crystal properties. In the Sect. 3 we show how to rewrite the partition function as a suitable polymer partition function, following in part the ideas of [12]: our polymers turn out to be connected families of regions of vertical dimers and lines of horizontal dimers and monomers. In the Sect. 4 we prove that the Kotecky–Preiss condition for the convergence of the cluster expansion is verified when the parameters satisfy (1.1) and the temperature is sufficiently low. Finally in the Sect. 5 we use the previous sections to prove the results stated in the Sect. 2. The Appendix A contains the study of a 1-dimensional monomer-dimer model, that is needed in the Sect. 3. For the sake of completeness, in the Appendix B we state the general results of cluster expansion needed in the paper.
123
A Cluster Expansion Approach…
763
2 Definitions and Main Results: The Model and Its Liquid Crystal Properties A monomer-dimer configuration on Z2 can be represented by a bonds1 occupation vector 2 α ∈ {0, 1} B(Z ) with the hard-core constraint α(x,y) ≤ 1 ∀x ∈ Z2 . (2.1) y∼x
If α(x,y) = 1, we say that there is a dimer on the bond (x, y), or also that there is a dimer at the site x; if instead α(x,y) = 0 for all y ∼ x, we say that there is a monomer on the site x. Dimers on Z2 may have two different orientations: vertical (v-dimers) or horizontal (h-dimers), according to the orientation of the occupied bond2 . The model studied in the present paper favors one orientation of the dimers (the horizontal one), both via a chemical potential and via a short-range imitation. Let be a finite sub-lattice of Z2 . Consider a horizontal boundary condition3 , namely we assume that every site of Z2 \ has a h-dimers (with either free or fixed positions). h the set of monomer-dimer configurations on (we allow also dimers toward Denote by D 4 the exterior ) which are compatible with the selected horizontal boundary condition. The Hamiltonian, or energy, of a monomer-dimer configuration is defined as ofwith sites ofwith v + μh −μ H := μh2+J # sites v−dimer 2 # monomer ¯ ¯ (2.2) sites of with h−dimer sites of withv−dimer J + 2 # but h−neighbor also to a + # but v−neighbor also to a . v−dimer or a monomer
h−dimer or a monomer
We assume that the parameters appearing in the Hamiltonian satisfy μh > −J , μh ≥ μv ,
J > 0.
(2.3) chosen5
, then the In this way, if the horizontal boundary condition with free positions is h (i.e. the configurations minimizing the energy under the given condition) ground states in D are exactly the configurations where every site has a h-dimer. The partition function of the system is h Z := e−β H (α) (2.4) h α∈D
where the parameter β > 0 is the inverse temperature. Remark 2.1 We want to show that the Hamiltonian (2.2) essentially corresponds to the model I introduced by Heilmann and Lieb in [10], except for the important fact that we allow the horizontal and vertical dimer potentials μh , μv to be different, while they take μh = μv = μ. 1 Two sites x = (x , x ), y = (y , y ) ∈ Z2 are neighbors (x ∼ y) if |x − y | + |x − y | = 1. A pair of v v h v h v h h sites (x, y) is a bond if x, y are neighbors. B(Z2 ) denotes the set of bonds. 2 Two sites x = (x , x ), y = (y , y ) ∈ Z2 are h-neighbors if x = y and |x − y | = 1, they are v v h v h v h h v-neighbors if xh = yh and |xv − yv | = 1. A bond (x, y) ∈ B(Z2 ) is horizontal if x, y are h-neighbors, it is
vertical if x, y are v-neighbors. 3 Theexternal boundary of is ∂ ext := {x ∈ Z2 \ | x neighbor of y ∈ }. The internal boundary of ¯ := ∪ ∂ ext . is instead ∂ ≡ ∂ int := {x ∈ | x neighbor of y ∈ Z2 \ }. We set 4 Namely we allow dimers having one endpoint in and one in Z2 \ . 5 Also fixed positions work, provided that the positions of the two h-dimers at the endpoints of each horizontal
line of allow a pure dimer configuration on that line.
123
764
D. Alberici
Fig. 1 The same monomer-dimer configuration on the lattice and the corresponding energies in accordance to the Hamiltonian (2.2) (on the left) and to the Hamiltonian (2.5) (on the right). A horizontal boundary condition is drawn in grey
We can introduce another Hamiltonian (that maybe is written in a more natural way; see Fig. 1): := − μh #{h-dimers in } − μv #{v-dimers in } − J # pairs of neighboring H (2.5) collinear dimers in The monomer-dimer model I in [10] is given by the Hamiltonian (2.5) with μh = μv = μ, when is a rectangular lattice of even sides lengths with periodic boundary conditions (torus). It is easy to show that when is a torus the two Hamiltonians (2.2), (2.5) describe the same model; indeed they only differ by an additive constant which does not affect the Gibbs measure: + μh + J || = H H (2.6) 2 since in with in with in with || − 2 #{h-dimers in } = || − # sites = # sites + # sites ; h-dimer monomer v-dimer 2 #{v-dimers in } = # || − 2 # = #
pairs of neighboring collinear dimers in
sites in with monomer
+#
sites in with v-dimer
= || − #
;
sites in with h-dimer (v-dimer) and h-neighbor (v-neighbor) to another h-dimer (v-dimer)
sites in with h-dimer (v-dimer) and h-neighbor (v-neighbor) also to something different
.
On the other hand when has horizontal boundary conditions the two Hamiltonians (2.2), (2.5) are not exactly equivalent. Indeed it holds6
+ μh + J || + J # sites in ∂vint = H H (2.7) without h-dimer 2 2 6 ∂ , ∂ denote respectively the vertical, horizontal component of the boundary; e.g. ∂ := {x ∈ v h v | xh-neighbor of y ∈ Z2 \ } and ∂h := {x ∈ | xv-neighbor of y ∈ Z2 \ }.
123
A Cluster Expansion Approach…
765
when the following conventions are adopted in the definition (2.5): if only half a dimer is in while the other half is in Z2 \ , it counts 21 ; if only one dimer of a pair of neighboring collinear dimers is in , while the other one is in Z2 \ , this pair counts 21 . The monomer-dimer model that we have introduced, in a certain region of the parameters corresponding to large horizontal potential, small vertical potential and low temperature, behaves like a liquid crystal. This means that the model exhibits an order in the orientation of the molecules (dimers), while there is no complete order in their positions. The following results will give a precise mathematical meaning to these statements. First we introduce some observables attached to the sites, asking questions as “Is there a horizontal dimer at site x?”, “If so, is it positioned to the left or to the right of x?”. To measure the absence or presence of some kind of order, at a microscopic level we study the expectations and the covariances of these quantities according to the Gibbs measure, while at a macroscopic level we introduce a suitable order parameter and study its expectation and possibly its variance7 . Define the following local observables8
f h,x := 1 x has a h-dimer , f v,x := 1 x has a v-dimer ; (2.8)
f l,x := 1 x has a left-dimer , f r,x := 1 x has a right-dimer . (2.9) Clearly f h,x = f l,x + f r,x and f h,x + f v,x ≤ 1. In the following we denote the Gibbs expectation of any observable f by 1 f h := h f (α) e−β H (α) . Z h α∈D
We denote by N the minimal distance between any two vertical components of the boundary of and our only assumption on the shape of is that N → ∞ as Z2 . To fix ideas one could think that is a rectangle (in this case N would be simply its horizontal side length), but actually we will need to consider also non-simply connected regions. There exists β0 > 0 depending on μh , μv , J only and N0 (β) depending on β, μh , J only such that the following results hold true. Theorem 2.2 (Microscopic expectations) Assume that J > 0, μh + J > 0 and 2μv +5J < 0. Let β > β0 . Let ⊂ Z2 finite having N > N0 (β). Let x ∈ such that dist h (x, ∂) > N0 (β). Then f l,x h ≥
μh +J μh +J 1 1 − e−β 2 , f r,x h ≥ − e−β 2 . 2 2
(2.10)
As a consequence: f h,x h ≥ 1 − 2 e−β | f r,x h − f l,x h | ≤ 2 e
μh +J 2
μ +J −β h2
;
(2.11)
.
(2.12)
Theorem 2.3 (Microscopic covariances) Assume that J > 0, μh + J > 0 and 2μv +5J < 0. Let β > β0 . Let ⊂ Z2 finite such that N > N0 (β). Let x, y ∈ such that dist h (x, ∂) > N0 (β), dist h (y, ∂) > N0 (β) and dist h (x, y) > N0 (β). Then: 7 When the expectation of the order parameter is zero but the variance is not, a small perturbation can lead to
a spontaneous order of the system.
8 We say that the site x has a left-dimer if there is a dimer on the bond x, x − (1, 0) , a right-dimer if there
is a dimer on the bond x, x + (1, 0) .
123
766
D. Alberici
9m − m (dist 2 (x,y)−1) Z , (2.13) e 4 16 9m − m (dist 2 (x,y)−1) Z , (2.14) | f r,x f r,y h − f r,x h f r,y h | ≤ e 4 16 9m − m (dist 2 (x,y)−1) Z . (2.15) | f l,x f r,y h − f l,x h f r,y h | ≤ e 4 16 The definition of m is clarified in the Appendix (lemma 5.5); anyway it can be sufficient to | f l,x f l,y h − f l,x h f l,y h | ≤
know that m = e−β
μh +3J 2
(1 + o(1)) as β → ∞.
The density of lattice sites occupied by h-dimers/v-dimers is respectively: 1 1 νh := f h,x , νv := f v,x . || || x∈
(2.16)
x∈
A parameter measuring the orientational order of the dimers is orient. := νh − νv .
(2.17)
Corollary 2.4 (Orientational order parameter) Assume that J > 0, μh + J > 0 and 2μv + 5J < 0. Let β > β0 . Let ⊂ Z2 finite, having N > 2 N0 (β). Then μh +J N0 (β) orient. h ≥ 1 − 2 1 − 4 e−β 2 . (2.18) N Hence lim lim inf orient. h = 1.
(2.19)
β ∞ Z2
The corollary 2.4 shows that fixing β sufficiently large and then choosing sufficiently big (more precisely the distance N between vertical components of ∂ must be large enough), the average density of sites occupied by h-dimers is arbitrarily close to 1: in other terms the system is oriented along the horizontal direction. The majority of sites is occupied by h-dimers. But there can still be some freedom, indeed we may distinguish the h-dimers in two classes according to their positions: a h-dimer is called even (resp. odd) if its left endpoint has even (resp. odd) horizontal coordinate. The density of lattice sites occupied by even/odd h-dimers is respectively:
1 2 νeven := || x∈ f r,x , x∈ 1 x has an even h-dimer = || xh even
(2.20) 1 2 νodd := || x∈ f l,x . x∈ 1 x has an odd h-dimer = || xh even
A parameter measuring the translational order of the h-dimers is transl. := νeven − νodd .
(2.21)
Corollary 2.5 (Translational order parameter. Part I) Assume that J > 0, μh + J > 0 and 2μv + 5J < 0. Let β > β0 . Let ⊂ Z2 finite such that N > 2 N0 (β). Then N0 (β) −β μh +J N0 (β) 2 2e +2 | transl. h | ≤ 1 − 2 (2.22) N N Hence lim lim sup | transl. h | = 0.
β ∞ Z2
(2.23)
Corollary 2.6 (Translational order parameter. Part II) Assume that J > 0, μh + J > 0 and 2μv + 5J < 0. Let β > β0 . Let ⊂ Z2 finite such that N > 2 N0 (β). Then
123
A Cluster Expansion Approach…
767
h
2 (transl. )2 − transl. h ≤ Hence for fixed β > β0
9m N0 (β) 1 N0 (β) 6 − 8 . (2.24) + m || (1 − e− 4 )2 N N
h
2 lim (transl. )2 − transl. h = 0.
Z2
(2.25)
The Corollaries 2.5, 2.6 show that fixing β sufficiently large and then choosing sufficiently big (in particular the distance between different components of ∂v must be big enough), the mean value and the variance of the difference between the density of even hdimers and the density of odd h-dimers are arbitrarily close to zero. In other terms, at large but finite β, there is not a spontaneous translational order for the h-dimers. Remark 2.7 The bounds (2.22) hold for any kind of horizontal boundary conditions, but in some particular cases it is possible to obtain a better result by a symmetry argument. Assume that is a rectangle with N + 1 sites in each horizontal side. If N + 1 is odd, by choosing horizontal dimers with free positions at the boundary one obtains transl. h = νeven h − νodd h = 0
(2.26)
for all parameters β, J, μh , μv . To prove it consider the reflection on with respect to the h → Dh . vertical axis at distance N2 from ∂v : this transformation induces a bijection T : D It is easy to check that H (T (α)) = H (α), νeven (T (α)) = νodd (α), νodd (T (α)) = νeven (α) h. for all α ∈ D On the other hand if N + 1 is even, by choosing periodic boundary conditions one still obtains per.
transl.
= 0
(2.27)
for all parameters β, J, μh , μv . To prove it one can consider the reflection on with respect per. to two vertical axis at distance N 2+1 from each other: it induces a bijection from D to itself having all the previous properties.
3 Polymer Representation h as a polymer partition In this section we show how to rewrite the partition function Z function of type (5.16). This representation will be suitable for applying the cluster expansion machinery (see Appendix B) in a regime of large horizontal potential, small vertical potential and low temperature. We start by isolating the “few” vertical dimers. Associate to each monomer-dimer conh the set figuration α ∈ D
V = V (α) := {x ∈ | x has a v-dimer according to α}. Partition V into its connected components (as a sub-graph of the lattice9 Z2 ): V =
n
Si , Si ∈ S ∀ i , dist Z2 (Si , S j ) > 1 ∀ i = j
i=1 9 On any graph the distance between two objects is defined as the length of the shortest path connecting them. In particular dist Z2 (S, S ) := inf x∈S, y∈S dist Z2 (x, y) for all S, S ⊂ Z2 and dist Z2 (x, y) := |xh − yh | + |xv − yv | for all x = (xh , xv ), y = (yh , yv ) ∈ Z2 .
123
768
D. Alberici
Fig. 2 A monomer-dimer configuration on and the corresponding regions S1 , S2 , S3 and lines L 1 , . . . , L 15 ∈ L (∪i Si ). Given the positions of the regions, the configurations on the lines are mutually independent: the arrows represent the energy contributions of type J/2. A horizontal boundary condition is drawn
where the family S is defined by def
S ∈ S ⇔ S ⊆ , S = ∅, S connected (as a sub-graph of Z2 ), every maximal vertical segment of S has an even number of sites, Sdoes not contains those sites of∂vint that necessarily have a h-dimer because of the boundary conditions.
(3.1)
The knowledge of the set V (or equivalently of S1 , . . . , Sn ) does not determine completely the configuration α of the system, since on \ V there can be both h-dimers and monomers. Anyway a fundamental feature of the model is that the system on \ V can be partitioned into independent 1-dimensional systems. Introduce the family L (V ) defined by def
L ∈ L (V ) ⇔ L is a maximal horizontal line of \ V.
(3.2)
The Hamiltonian (2.2) rewrites as
n μh −μv |Si | + 2J |∂h Si | + 2J |∂v Si ∩ ∂| H = i=1 2 sites of L with h-dimer μh +J sites of L with J but h-neighbor also to a + + # # . L∈L (∪i Si ) monomer 2 2 monomer or to ∪i Si
Hence the partition function (2.4) rewrites as (see Fig. 2) h = Z
1 n! n≥0
123
n
i=1 S1 ,...,Sn ∈S dist(Si ,S j )>1 ∀i = j
e
−β
μh −μv 2
|Si | +
J J 2 |∂h Si | + 2 |∂v Si ∩∂|
ZL L∈L (∪i Si )
(3.3)
A Cluster Expansion Approach…
769
where Z L is the monomer-dimer partition function of the line L, considered as a sub-lattice of the 1-dimensional lattice Z, with suitable boundary conditions: e−β HL (α L ) e Il,xl (αxl ) e Ir,xr (αxr ) . (3.4) Z L := α L ∈D L
An explanation of the notations introduced in (3.4) is required. D L denotes the set of monomer-dimer configurations on L (dimers can only be horizontal, external dimers at the endpoints of L are allowed); J sites of L with dimer μh + J sites of L with + # but h-neighbor also to ; HL := # monomer 2 2 a monomer xl , xr denote respectively the left, right endpoint of the line L (which eventually may coincide): observe 10 that because of (3.2)
∪i ∂rext Si ∩ ∂l \ ∪i ∂l Si , xl (L) = (3.5) L∈L (∪i Si )
xr (L) =
∪i ∂lext Si ∩ ∂r \ ∪i ∂r Si ;
(3.6)
L∈L (∪i Si )
finally 11 if xl if xl if xl if xl
∈ ∪i ∂rext Si ⇒ Il,xl := −∞ −β 2J 0
∈ ∂l , on xl −(1, 0) it is fixed a l-dimer ⇒ Il,xl := −∞ 0 −β 2J
∈ ∂l , on xl −(1, 0) it is fixed a r-dimer ⇒ Il,xl := 0 −∞ −∞
∈ ∂l , on xl −(1, 0) there is a free h-dimer ⇒ Il,xl := 0 0 −β 2J
(3.7)
and, similarly, if xr if xr if xr if xr
∈ ∪i ∂lext Si ⇒ Ir,xr := −β 2J −∞ 0
∈ ∂r , on xr +(1, 0) it is fixed a r-dimer ⇒ Ir,xr := 0 −∞ −β 2J
∈ ∂r , on xr +(1, 0) it is fixed a l-dimer ⇒ Ir,xr := −∞ 0 −∞
∈ ∂r , on xr +(1, 0) there is a free h-dimer ⇒ Ir,xr := 0 0 −β 2J .
(3.8)
The one-dimensional systems described by Z L , L ∈ L (∪i Si ), are studied in the Appendix A. h , the weight of the regions (S , . . . , S ) is not a product of the In the form (3.3) of Z 1 n weights of each region Si , because of the lines L connecting different regions. Therefore the regions Si ∈ S are not a good choice for a polymer representation of the model. In order to decouple some regions from some other ones, it is possible to do a simple trick. It is convenient to deal in different ways with the endpoints lying on ∂ ext Si and those on ∂; hence given a line L ∈ L (∪i Si ) we set
(2.5) εl,xl := 1 xl ∈ (∪i ∂rext Si ) ∩ , ηl,xl := 1− εl,xl = 1 (xl ∈ (∂l ) \ ∪i ∂l Si );
(2.6) εr,xr := 1 xr ∈ (∪i ∂lext Si ) ∩ , ηr,xr := 1− εr,xr = 1 (xr ∈ (∂r ) \ ∪i ∂r Si ). 10 ∂ , ∂ denote respectively the left, right component of the vertical boundary; e.g. ∂ := {x ∈ | x − l r l (1, 0) ∈ Z2 \ } and ∂r := {x ∈ | x + (1, 0) ∈ Z2 \ }.
11 The possible states of a site x ∈ L are three: “l”=left-dimer namely a dimer on the bond x, x − (1, 0) ,
“r” = right-dimer namely a dimer on the bond x, x + (1, 0) , “m”=monomer. Here we think Il,xl , Ir,xr as
vectors: Il,xl = Il,xl (l) Il,xl (r ) Il,xl (m) and Ir,xr = Ir,xr (l) Ir,xr (r ) Ir,xr (m) .
123
770
D. Alberici
Using the notations of the Appendix A, given a line L ∈ L (∪i Si ) we introduce the two vectors representing the boundary conditions outside its endpoints x l , xr : ⎛ ⎞ e Ir,xr (l) μ +J ⎜ ⎟ e Ir,xr (r ) Bl,xl := e Il,xl (l) e Il,xl (r ) e−β h4 + Il,xl (m) , Br,xr := ⎝ ⎠; μh +J e−β 4 + Ir,xr (m) then to shorten the notation we set 1 1 (1) bl,xl := √ Bl,xl E r(1) , br,xr := √ E l Br,xr . λ1 λ1 Now define R L :=
ZL |L| ηl,x
εl,x
η
λ1 bl,xl l br,xr,xrr
ε
r,xr − bl,xll br,x r
(3.9)
and, using L as an abbreviation for L (∪i Si ), rewrite the quantity elementary algebraic tricks: η ZL εl,x εr,xr η l,x R L + bl,xll br,x bl,xl l br,xr,xrr |L| = r L∈L λ1 L∈L ηl,xl ηr,xr bl,xl br,xr RL = K ⊆L
L∈L
L∈K
L∈L \K
L∈L
εl,x bl,xll
Z L by means of
εr,xr br,x r
.
By identities (3.5), (3.6) it holds L∈L
⎛ ηl,x bl,xl l
L∈L \K
η br,xr,xrr
= ⎝ ⎛
⎜ εl,x εr,xr ⎜ bl,xll br,x r = ⎜ ⎝
⎞⎛ bl,x ⎠ ⎝
x∈∂l \∪i ∂l Si
⎞ br,x ⎠
x∈∂r \∪i ∂r Si
⎞ ⎛ ⎟ ⎟ bl,x ⎟ ⎠
x∈(∪i ∂rext Si )∩ x∈ / supp K
⎜ ⎜ ⎜ ⎝
⎞ ⎟ ⎟ br,x ⎟ ; ⎠
x∈(∪i ∂lext Si )∩ x∈ / supp K
By substituting into the previous formula and thinking K = {L 1 , . . . , L p }, we find out12 ZL L∈L
|L|
λ1
=
bl/r, x
x∈∂v \∪i ∂v Si
1 × p! p≥0
L 1 ,...,L p ∈L L h = L k ∀h =k
p k=1
RLk
br/l, x .
(3.10)
x∈(∪i ∂vext Si )∩ x ∈∪ / k Lk
12 In the first product on the r.h.s. of (3.10) the shorten notation b l/r,x means: take bl,x if x ∈ ∂l , take br,x
if x ∈ ∂r ; notice that ∂l and ∂r are disjoint for N > 1. In the last product instead the shorten notation br/l,x means: take br,x if x ∈ ∂lext Si only, take bl,x if x ∈ ∂rext Si only, and take the product br,x bl,x in the case that x belongs to both ∂lext Si and ∂rext S j .
123
A Cluster Expansion Approach…
771
Fig. 3 The first three pictures represent three different examples of polymers P ∈ P . The set represented in the last picture is not a unique polymer since it is not connected in Z2 (even if it is connected in Z2 )
Now substitute (3.10) into (3.3), using also the fact that || =
n
i=1 |Si |
+
and obtain: ||
h = λ Z 1
×
p≥0
1 n!
n
S1 ,...,Sn ∈ S i=1 dist(Si ,S j )>1 ∀i= j
1 p!
L∈L (∪i Si )
|L|,
bl/r, x
x∈∂v
n≥0
×
e
−β
μ −μ v |S | + J |∂ S | h i 2 2 h i |S | λ1 i
p
L 1 ,...,L p ∈ L (∪i Si ) L k = L h ∀k=h
k=1
RLk
x∈∂v ∩∂v Si
J
e−β 2 bl/r, x
(3.11)
br/l, x .
x∈(∪i ∂vext Si )∩ x ∈∪ / k Lk
n p The next step is to partition i=1 Si ∪ k=1 L k into connected components as a sub-graph of Z2 , where Z2 is the lattice obtained from Z2 by removing all the vertical bonds incident to the lines L k : n i=1
Si ∪
p k=1
Lk =
q
supp Pt ,
t=1
Pt ∈ P ∀t, dist Z2 (supp Pt , supp Ps ) > 1 ∀t = s where the family P (yes, it is finally our family of polymers! see Fig. 3) is defined by:
P := P ≡ (Si )i∈I , (L k )k∈K | (Si )i ∈ PS , (L k )k ∈ PL (∪i Si ) , (3.12)
(Si )i∈I ∈ PS
⎧ ⎪ ⎨0 ≤ |I | < ∞ def ⇔ Si ∈ S ∀i ⎪ ⎩ dist Z2 (Si , S j ) > 1 ∀i = j,
(3.13)
123
772
D. Alberici
(L k )k∈K
⎧ 0 ≤ |K | < ∞, |I | + |K | ≥ 1 ⎪ ⎪ ⎪ ⎨ L ∈ L (∪ S ) ∀k def k i i ∈ PL (∪i∈I Si ) ⇔ ⎪ L = L k h ∀k = h ⎪ ⎪ ⎩ (∪i Si ) ∪ (∪k L k ) connected in Z2 .
(3.14)
The identity (3.11) now rewrites as h Z = C
1 q! q≥0
q
(Pt )
P1 ,...,Pq ∈P t=1
δ(Pt , Ps )
(3.15)
t
by setting, for all P, P ∈ P with P = (Si )i∈I , (L k )k∈K ,
||
C := λ1 (P) :=
1 |I |!
×
i∈I
1 |K |!
bl/r, x ,
(3.16)
x∈∂v
e
−β
μ −μ v |S | + J |∂ S | h i 2 2 h i |S | λ1 i
RLk
k∈K
#
δ(P, P ) :=
J
x∈∂v ∩∂v Si
e−β 2 bl/r, x
,
br/l, x ext x∈( i∈I ∂v Si )∩ x∈ / k∈K L k
1, if dist Z2 (P, P ) > 1 . 0, otherwise
(3.17)
(3.18)
h , up to a factor C , admits a The identity (3.15) finally shows that the partition function Z polymer representation of the form (5.16). It is convenient to bound the polymer activity by a simpler quantity. Using the proposition 5.8 plus the lemmas 5.6, 5.7 and the fact that |∂h Si | ≥ 2, one finds: v 1 −β μh −μ 1 −m|L k | |Si | + J 2 (P) ≤ (P) := e e γL k (3.19) |I |! |K |! i∈I
k∈K
with the γ L ’s defined by the equation (5.15).
4 Convergence of the Cluster Expansion h as a polymer partition function In the previous section we rewrote our partition function Z up to a factor C [see formula (3.15)]. In this section we will find a region of the parameters space μh , μv , J where the condition (5.17) is verified by our model at low temperature, so that the general theorem 5.9 about the convergence of the cluster expansion will apply to our case.
Theorem 4.1 Assume that J > 0, μh + J > 0 and 2μv + 5J < 0. By choosing a(P) :=
123
m | supp P| ∀ P ∈ P 2
(4.1)
A Cluster Expansion Approach…
773
the conditions
m ∀ x ∈ , 8
(4.2)
(P) ea(P) ≤ a(P ∗ ) ∀ P ∗ ∈ P
(4.3)
(P) ea(P) ≤
P∈P supp Px
P∈P δ(P,P ∗ )=0
hold true, provided that β > β0 and N > N0 (β) (N is the minimum distance between two vertical components of ∂). Here β0 > 0 depends on μh , μv , J only, while N0 (β) depends on β, μh , J only; they do not depend on , P ∗ , x. Corollary 4.2 Assume that J > 0, μh + J > 0 and 2μv + 5J < 0. Suppose also that β > β0 and N > N0 (β). Denote by CP the set of clusters13 composed by polymers of P . Then the partition function (2.4) rewrites as ⎞ ⎛ ∗ h (4.4) = C exp ⎝ U ((Pt )t )⎠ Z where we denote
∗
(Pt )t ∈CP
:=
(Pt )t ∈CP
1 q≥0 q!
q
(Pt )t=1 ∈CP
U (P1 , . . . , Pq ) := u(P1 , . . . , Pq )
and
q
(Pt ).
(4.5)
t=1
Remind that C is defined by (3.16), is defined by (3.17) and u is defined by (5.19), (3.18). Furthermore for all E ⊆ P it holds ∗
|U (Pt )t | ≤ | (P)| ea(P) (4.6) (Pt )t ∈CP ∃t: Pt ∈E
P∈P P∈E
where a is defined by (4.1). Proof The corollary follows from the general theory of cluster expansion (theorem 5.9), h admits a polymer representation (3.15) and satisfies the Kotecky–Preiss condition since Z ((4.3), | | ≤ ). For ease of reading, in the following of this section we will denote 1 ∗ 1 ∗ := and := n! p! n p n p (Si )i
(Si )i=1 ∈PS
(L k )k
(L k )k=1 ∈PL (∪i Si )
where PS , PL (∪i Si ) are the projections of the polymer set P defined in (3.13), (3.14). The next lemmas provide the entropy estimates that will be needed in the proof of theorem 4.1. Lemma 4.3 If ∪i Si = ∅, namely n ≥ 1, then ∗ 1 ≤ 4 i |Si | .
(4.7)
(L k )k 13 As explained in the Appendix B, using the definition (3.18) for δ, a family of polymers (P , . . . , P ) is a q 1 q cluster iff ∪t=1 supp Pt is connected in Z2 .
123
774
D. Alberici ( p)
p
Proof Fix p ≥ 0 and denote by PL (∪i Si ) the set of (L k )k=1 ∈ PL (∪i Si ). p ( p) Given (L k )k=1 ∈ PL (∪i Si ), each line L k has at least one endpoint on ∪i ∂vext Si , since (∪i Si ) ∪ (∪k L in Z2 . Therefore the number of ways to choose each k ) have to be connected ext L k is at most i |∂v Si | ≤ 2 i |Si |. Since the L k , k = 1, . . . , p, must be all distinct, it follows that $ $ $ $ ( p) |Si | 2 |Si | − 1 · · · 2 |Si | − p + 1 . $PL (∪i Si )$ ≤ 2 i
Therefore ∗ (L k )k
1 =
i
i
$ 2 1 $$ $ ( p) i |Si | = 22 i |Si | . $PL (∪i Si )$ ≤ p! p p p
Lemma 4.4 Let x ∈ Z2 . For all s ≥ 2 16 4s # S ⊂ Z2 connected | |S| = s, S x ≤ 4 . 3
(4.8)
Proof Given a connected graph G and one of its vertices x, there exists a walk in G that starts from x and crosses each edge exactly twice14 . Therefore # S ⊂ Z2 connected | |S| = s, S x 2 ≤ 2s e=s−1 # S connected sub-graph of Z | |edges of S| = e, S x 2 ≤ 2s e=s−1 # walks in Z that start from x and have lenght 2e 2e ≤ 44s+2 . ≤ 2s e=s−1 4 3 Lemma 4.5 Let A ⊂ Z2 finite. For all s ≥ 2, 1 ≤ d < ∞ 32 # S ⊂ Z2 connected | |S| = s, dist h (S, A) = d ≤ |A| 44s . 3
(4.9)
Here dist h (S, A) := inf x∈S, y∈A dist h (x, y) and the horizontal distance between x = (xh , xv ), y = (yh , yv ) ∈ Z2 is defined as # |xh − yh | if xv = yv dist h (x, y) := . (4.10) +∞ if xv = yv Proof Observe that dist h (S, A) = d if and only if there exists a horizontal line L, |L| = d +1, having one endpoint on ∂v A and the other one on ∂v S. Therefore: # S ⊂ Z2 connected | |S| = s, dist h (S, A) = d # S ⊂ Z2 connected | |S| = s, ∂v S other endpt. of L ≤ ≤
L horiz. line, |L|=d+1, ∂v A one endpt. of L 2|∂v A| # S ⊂ Z2 connected
| |S| = s, S 0 ≤ 2|A|
For the last inequality we have used the lemma 4.4. 14 This can be easily proven by induction on the number of edges.
123
16 3
44s .
A Cluster Expansion Approach…
775
Lemma 4.6 Let n ≥ 1. Let T be a tree over the vertices {1, . . . , n}. Let si ≥ 2 for all i = 1, . . . , n and di j ≥ 2 for all (i, j) ∈ T . Then given A ⊂ Z2 and 1 ≤ d < ∞ n # (Si )i=1 ∈ PS | dist h (S1 , A) = d, |Si | = si ∀i, dist h (Si , S j ) = di j ∀(i, j) ∈ T (4.11) n 32 4s degT (i) i s ; 4 ≤ |A| i=1 i 3 while given x ∈ Z2
n # (Si )i=1 ∈ PS | S1 x, |Si | = si ∀i, ≤
n i=1
32 3
44si
dist h (Si , S j ) = di j ∀(i, j) ∈ T deg (i) . si T
(4.12)
Here degT (i) denotes the degree of the vertex i in the tree T . Proof Let start by proving the inequality (4.11) by induction on n. If n = 1, then the tree T is trivial and (4.11) is already provided by the lemma 4.5. Now let n ≥ 2, assume that (4.11) holds for at most n − 1 vertices and prove it for n. It is convenient to think that the tree T is rooted at the vertex 1 and denote by j ← i the relation “vertex j is son of vertex i in T ” and by T (i) the sub-tree of T induced by the vertex i together with its descendants. Then, denoting by NT ,1 A, d; (si )i∈T , (di j )(i, j)∈T the cardinality on the l.h.s. of (4.11), it holds
NT ,1 A, d; (si )i∈T , (di j )(i, j)∈T
= S1 ∈S , |S1 |=s1 v←1 NT (v),v S1 , d1v ; (si )i∈T (v) , (di j )(i, j)∈T (v) . disth (S1 ,A)=d
Since T (v) has at most n − 1 vertices, the induction hypothesis gives 32
deg (i) NT (v),v S1 , d1v ; (si )i∈T (v) , (di j )(i, j)∈T (v) ≤ s1 . 44si si T (v) 3 i∈T (v)
Then by substituting in the previous identity, bounding degT (v) (i) by degT (i) and using the lemma 4.5, one obtains: 32
deg (i) . NT ,1 A, d; (si )i∈T , (di j )(i, j)∈T ≤ |A| 44si si T 3 i∈T
This concludes the proof of (4.11).
In order to prove the inequality (4.12), denote by NT ,1 x; (si )i∈T , (di j )(i, j)∈T the cardinality on the l.h.s. of (4.12) and observe that
NT ,1 x; (si )i∈T , (di j )(i, j)∈T =
NT (v),v S1 , d1v ; (si )i∈T (v) , (di j )(i, j)∈T (v) . S1 ∈S , |S1 |=s1 v←1 S1 x
By (4.11) we already know that 32
deg (i) NT (v),v S1 , d1v ; (si )i∈T (v) , (di j )(i, j)∈T (v) ≤ s1 44si si T (v) . 3 i∈T (v)
123
776
D. Alberici
Then by substituting in the previous identity, bounding degT (v) (i) by degT (i) and using the lemma 4.4, one obtains: 32
deg (i) 44si si T , NT ,1 x; (si )i∈T , (di j )(i, j)∈T ≤ 3 i∈T
which proves (4.12).
Proof (of the theorem 3.1) According to the definition (3.18), the condition δ(P, P ∗ ) = 0 implies that supp P∩[supp P ∗ ]1 = ∅, where [A]1 := {x ∈ Z2 | dist Z2 (x, A) ≤ 1}. Therefore (P) ea(P) ≤ (P) ea(P) P∈P δ(P,P ∗ )=0
≤
x∈[supp P ∗ ]1 P∈P supp Px 4 | supp P ∗ | maxx∈
(P) ea(P) .
P∈P supp Px
Thus, by choosing a(P) := m2 | supp P| for all P ∈ P , the inequality (4.3) will be a consequence of (4.2). We have to prove the inequality (4.2). It is worth to write down explicitly the quantity we will work with (see the definitions (3.19) and (4.1)): p n μ −μ 1 1 − m |L k | − β h 2 v − m2 |Si |−β J a(P) 2 (P) e = e e γL k n! p! i=1
k=1
n , (L ) p for all P ∈ P , P = (Si )i=1 k k=1 . Notice that if supp P x, the site x may belong either to a region Si or to a line L k ; hence we can split the sum on the l.h.s. of (4.2) into two parts: (P) ea(P) = 1 + 2 with: (4.13) P∈P supp Px
1 :=
∗ (Si )i ∪i Si x
2 :=
i
∗ (L k )k
i
∗
(Si )i
e
μ −μ − β h 2 v − m2 |Si |−β J
e
μ −μ − β h 2 v − m2 |Si |−β J
m
e− 2 |L k | γ L k
(4.14)
k
∗
m
e− 2 |L k | γ L k .
(4.15)
(L k )k k ∪k L k x
During all the proof o(1) will denote any function ω = ω(β, μh , J ) such that ω → 0 as β → ∞ and ω depends only on β, μh , J (in particular it does not depend on the choices of ⊂ Z2 , x ∈ Z2 , P ∈ P ). I. Study of the term 1 . n We fix a family of regions (Si )i=1 that contains the point x; we also assume that PL (∪i Si ) is non-empty, otherwise the contribution to 1 is zero. By the Lemma 4.3 it holds m ∗ m e− 2 |L k | γ L k ≤ 4 i |Si | max e− 2 |L k | γ L k (4.16) (L k )k
k
(L k )k
k
where the maximum is taken over all (L k )k ∈ PL (∪i Si ). The factor γ L k can take two values (see formula (5.15)), both smaller than 1 for β sufficiently large (uniformly with
123
A Cluster Expansion Approach…
777
respect to L k ), since each line L k must have at least one endpoint on ∪i ∂vext Si to ensure that (∪i Si ) ∪ (∪k L k ) is connected in Z2 . n S to contain the point x. It is convenient to consider Obviously n ≥ 1 in order for ∪i=1 i separately the case n = 1 and the case n ≥ 2: 1 = 1 + 1 . The case n = 1 is easy to deal with, simply by bounding the r.h.s. of (4.16) by 4|S| and using the lemma 4.4. Precisely: 1 := ≤ ≤
e
S∈S Sx
e
μ −μ − β h 2 v − m2 |S|−β J
μ −μ − β h 2 v − m2 |S|−β J
∗ (L k )k
m
e− 2 |L k | γ L k
k
4|S|
S∈S Sx
μh −μv m 16 4s − β 2 − 2 s−β J 3 4 e
(4.17) 4s
s≥2 even 10 −β (μh −μv +J ) (1 + o(1)). = 16 3 4 e p Now assume n ≥ 2. Fix a family of lines (L k )k=1 ∈ PL (∪i Si ). We can consider the graph G ≡ G (Si )i , (L k )k with vertices i ∈ {1, . . . , n} and edges k ∈ {1, . . . , p}: the edge k joins the two vertices i, j iff the line L k has one endpoint on ∂vext Si and the other one on ∂vext S j . In the graph G there can be multiple edges, loops and pseudo-edges with a single
endpoint. The graph G is connected (it follows from definition 3.14), hence G admits at least m one spanning sub-tree T . And clearly, since each factor e− 2 |L k | γ L k is smaller than 1, p
m
e− 2 |L k | γ L k ≤
m
e− 2 |L k | γ L k ≤
where γ S,S :=
1
2e
max
−β J
(L k )k
+e
μ +J −β h2
e− 2 (disth (Si ,S j )−1) γ Si ,S j m
(i, j)∈T
k∈T
k=1
(disth
m
e− 2 |L k | γ L k ≤
k
(S,S )−1)
(1 + o(1)). Therefore: m e− 2 (disth (Si ,S j )−1) γ S ,S
max
Now using (4.16) and (4.18) we can bound 1 : μ −μ ∗ 1 n − β h 2 v − m2 |Si |−β J e 1 := i=1 n! n≥2
≤
n (Si )i=1 ∪i Si x
n≥2 T tree over {1,...,n}
×
(i, j)∈T
e
1 n!
i
T tree over {1,...,n} (i, j)∈T
n
n (Si )i=1 ∪i Si x − m2 (disth (Si ,S j )−1)
i=1 e
(L k )k
(4.18)
j
m
k
μ −μ − β h 2 v − m2 −log 4 |Si |−β J
e− 2 |L k | γ L k (4.19)
γ Si ,S j
n where in the sums we keep implicit that (Si )i=1 ∈ PS . 15 Substitute into (4.19) the entropy bound (4.12). Since ∪i Si x, but not necessarily S1 x, an extra factor n appears. Moreover observe that |Si | is even and ≥ 2 (see the 15 The families of regions (S )n i i=1 such that dist h (Si , S j ) = ∞ for at least one edge (i, j) ∈ T give zero
contribution to the sum, therefore we do not need to worry about them.
123
778
D. Alberici
definition (3.1)) and dist h (Si , S j ) ≥ 2. Then: n 1 ≤ n!
n
(si )i=1,...,n (di j )i j∈T s even ≥2 di j ≥2 i μ −μ − β h 2 v − m2 −log 4 si −β J
n≥2 T tree over {1,...,n}
×
n
i=1 e
32 i=1 3
degT (i)
44si si
(4.20)
(i, j)∈T e
− m2 (di j −1)
γdi j
μh +J
where γd := 21 e−β J + e−β 2 (d−1) (1 + o(1)). Given n ≥ 2 and δ1 , . . . , δn ≥ 1, the number of trees T over the vertices {1, . . . , n} with given degrees degT (i) = δi ∀i = 1, . . . , n is exactly16 (n − 2)! (δ1 − 1)! · · · (δn − 1)!
n if i=1 (δi − 1) = n − 2 and zero otherwise. Furthermore the number of edges of T is n − 1. Therefore the bound (4.20) leads to n μ −μ − β h 2 v − m2 −5 log 4 s sδ 32 −β J 1 ≤ n≥2 3 e s≥2 e δ≥1 (δ−1)! even (4.21) n−1 − m2 (d−1) × e γ . d d≥2 The sum over s gives, as β → ∞,
e
s≥2 even
=
μ −μ − β h 2 v − m2 −5 log 4 s
se
sδ (δ−1)!
δ≥1 μh −μv m − β 2 − 2 −5 log 4−1 s
= 2 e2 410 e−β(μh −μv ) (1 + o(1)).
(4.22)
s≥2 even
The sum over d gives, as β → ∞, − m (d−1) e 2 γd d≥2
= =
m
e− 2 (d−1)
d≥2 1 e−β J m 2 1−e− 2
e−β J 2
+
m
e− 2 (d−1) e−β
μh +J 2
(d−1)
(1 + o(1))
(4.23)
d≥2
μh +J + o(1) (1 + o(1)) = eβ 2 (1 + o(1)) m
μh +3J
where we used the fact that 1 − e− 2 = 21 e−β 2 (1 + o(1)) (see Lemma 5.5). Substituting (4.22), (4.23) into (4.21), one obtains n 226 e2 μ +J μh +J −β(μh −μv )+β h2 1 ≤ (1 + o(1)) e−β 2 (1 + o(1)). (4.24) e 3 n≥2
Assume µh − µv >
µh + J 2 .
Then for β sufficiently large (4.24) becomes: 26 2 μh +J 2 μh +J 1 ≤ 2 3e e−β(μh −μv )+β 2 e−β 2 (1 + o(1)) =
252 e4 9
e−β 2(μh −μv )+β
μh +J 2
(1 + o(1)).
16 This is an improvement of the well-known Cayley’s formula.
123
(4.25)
A Cluster Expansion Approach…
779
II. Study of the term 2 . n The ideas are not far from those seen for 1 . We fix a family of regions (Si )i=1 and we assume that there exists (L k )k ∈ PL (∪i Si ) such that ∪k L k x, otherwise the contribution to 2 is zero. Clearly the line L x ∈ L (∪i Si ) that contains x is unique. It is convenient to consider separately four cases: 2 = 2 + 2 + 2 + 2 . In 2 we assume n = 0, namely ∪i Si = ∅; then L x have to be a maximal horizontal line of . In 2 we assume n = 1, namely there is a unique region S and L x may have one endpoint on ∂vext S and one on ∂v or both on ∂vext S. In 2 we assume n ≥ 2 and L x has one endpoint on ∪i ∂vext Si and one on ∂v or both on the same ∂vext Si . In 2 we assume n ≥ 2 and L x has one endpoint on ∂vext Si and one on ∂vext S j with i = j. By methods similar to those already seen for 1 , one can prove that m
2 ≤ e− 2 N (1 + o(1));
(4.26)
√ 225 2 −β (μh −μv )+β μh +2J 2 (1 + o(1)); e 3 √ 252 e4 2 −β 2(μh −μv )+β 2μh +3J 2 ≤ (1 + o(1)); e 9
2 ≤ 2
2 ≤
(4.27)
(4.28)
252 e4 −β 2(μh −μv )+β(μh +2J ) (1 + o(1)). e 9
(4.29)
We refer to the Arxiv version of the paper for the details. In conclusion, by using the estimates (4.17), (4.25), (4.26), (4.27), (4.28), (4.29), and the fact that m = e−β that: 1 m
μh +3J 2
P∈P supp Px
(1 + o(1)) (see lemma 5.5), if we assume µh − µv > (P) ˜ ea(P)
μh + 3J = eβ 2 1 + 1 + 2 + 2 + 2 + 2 (1 + o(1)) μh + J μh + 2J 24 52 4 ≤ 23 e−β(μh −μv ) + β 2 + 2 9 e e−β 2(μh −μv ) + β 2 + + 2 3 e−β(μh −μv ) + β 25.5
µh + J 2 , we find
2μh + 5J 2
+
252.5 e4 9
e−β 2(μh −μv ) + β
1 m
m
e− 2 N (4.30)
3μh + 6J 2
3μh +7J
+ 2 9 e e−β 2(μh −μv ) + β 2 (1 + o(1)) m 5J 25.5 = m1 e− 2 N + 2 3 eβ(μv + 2 ) (1 + o(1)) 52 4
where N is the minimum distance between two different vertical components of ∂ and o(1) → 0 as β → ∞ (uniformly with respect to N ). Now we assume that µv + 52J < 0. Thus there exists β0 > 0 such that for all β > β0 the 5J
function 1 + o(1) on the r.h.s. of (4.30) is < 2 and the term 2 3 eβ(μv + 2 ) ≤ 1/32. There m exists17 also N0 (β) such that for all N > N0 (β) the term m1 e− 2 N ≤ 1/32. Therefore if 25.5
17 N = 2 log 32 . 0 m m
123
780
D. Alberici
μv + 5J 2 < 0 (which entails also the previous condition μh −μv > (4.30) implies that m (P) ˜ ea(P) ≤ 8
μh +J 2 ), then the inequality
P∈P supp Px
for β > β0 and N > N0 (β). This concludes the proof.
5 Proofs of the Liquid Crystal Properties In this section we will finally prove that the model behaves like a liquid crystal, as stated in the Sect. 2, by means of the cluster expansion results obtained in the previous sections.
5.1 Proof of the Theorem 2.2 We will prove the inequality (2.10) for f l,x . That one for f r,x can be proved analogously; then (2.11) and (2.12) follow since f x = f l,x + f r,x . Observe that f l,x h =
h Z \x h Z
,
h is the partition function over the lattice \x with horizontal boundary conditions where Z \x including a left-dimer at the site x. Since N > N0 (β) and disth (x, ∂) > N0 (β), both partition functions satisfy the hypothesis of the corollary 4.2. Hence by the cluster expansion (4.4) the partition functions rewrite as
∗ h Z = C exp U (Pt )t , CP (Pt )t ∈
∗ h Z \x = C\x exp U\x (Pt )t . (Pt )t ∈CP\x
By applying the definition (3.16), C\x br,x−(1,0) bl,x+(1,0) = . C λ1 Now consider a polymer P ∈ P ∪ P\x . Keeping in mind the definitions of polymer (3.12) and polymer activity (3.17), observe that18 if dist h (supp P, x) > 1 ⇒ P ∈ P ∩ P\x , (P) = \x (P). Therefore:
∗
U\x (Pt )t −
(Pt )t ∈CP\x
≥ −
∗
∗
U (Pt )t
(Pt )t ∈CP
|U\x (Pt )t | −
(Pt )t ∈CP\x ∃t: disth (supp Pt , x)≤1
∗
|U (Pt )t | .
(Pt )t ∈CP ∃t: disth (supp Pt , x)≤1
18 The condition dist (supp P, x) > 1 guarantees that supp P ⊆ \ x and that the polymer P does not h include any line L k having one endpoint on x ± (1, 0), nor any region Si containing these points.
123
A Cluster Expansion Approach…
781
h , Zh , And by the inequalities (4.6) and (4.2) applied to both Z \x
∗
(Pt )t ∈CP ∃t: disth (supp Pt , x)≤1
∗
|U (Pt )t | ≤
(P) ea(P) ≤ 3
P∈P disth (supp P,x)≤1
|U\x (Pt )t | ≤
(Pt )t ∈CP\x ∃t: disth (supp Pt , x)≤1
(P) ea(P) ≤ 2
P∈P\x disth (supp P,x)≤1
m ; 8
m . 8
In conclusion one obtains: f l,x h =
h Z \x h Z
exp −5 m8
μh +J = 21 1 − e−β 2 (1 + o(1)) , ≥
br, x−(1,0) bl, x+(1,0) λ1
(1)
where the last identity follows from the fact that λ1 br,x−(1,0) bl,x+(1,0) = E l Br,x−(1,0) (1) Bl,x+(1,0) E r = √1 (1 − a2 (1 + o(1))) √1 (1 − a2 (1 + o(1))) (by lemma 5.7, since there 2
2
h ), λ = 1 + is a left-dimer fixed at x according to Z \x 1
e−5m/8
ab 2
(1 + o(1)) (proposition 5.3), and
5 8 ab (1
= 1− + o(1)) (lemma 5.5). Finally, since o(1) → 0 as β → ∞ and o(1) does not depend on the choice of x and , one may obtain the desired inequality eventually increasing β0 .
5.2 Proof of the Corollary 2.4 Set ϕ,N0 := #{x ∈ | dist h (x, ∂) > N0 } / ||. By the theorem 2.2, bound (2.11), using also f v,x ≤ 1 − f h,x , one obtains: orient. h =
μh +J
1 f h,x h − f v,x h ≥ ϕ,N0 (β) 1 − 4 e−β 2 . ||
x∈
On the other hand: ϕ,N0 ≥
min
L maximal horiz.line of
ϕ L ,N0 =
min
L maximal horiz.line of
N0 (β) |L| − 2N0 (β) = 1−2 . |L| N
5.3 Proof of the Corollary 2.5 Set ϕ,N0 := #{x ∈ | dist h (x, ∂) > N0 } / ||. By the theorem 2.2, bound (2.12), | transl. h | ≤
μh +J 2 | f r,x h − f l,x h | ≤ ϕ,N0 (β) 2e−β 2 + 1 − ϕ,N0 (β) . ||
x∈, xh even
On the other hand we have already observed in the proof of the corollary 2.4 that ϕ,N0 ≥ 1 − 2N0 /N .
123
782
D. Alberici
5.4 Proof of the Theorem 2.3 We will prove the inequality (2.13). (2.14) and (2.15) can be proved analogously. First of all observe that, since 0 ≤ f l,x , f l,y ≤ 1, | f l,x f l,y h
− f l,x h f l,y h |
≤ log
f l,x f l,y h
f l,x h f l,y h
∨
f l,x h f l,y h f l,x f l,y h
.
(5.1)
Now observe that: f l,x f l,y h =
h Z \x,y h Z
h Z \x
, f l,x h =
h Z
, f l,y h =
h Z \y h Z
,
h , Zh , Zh where Z \x \y \x,y are the partition function respectively over the lattices \ x, \ y, \ x, y, with horizontal boundary conditions including a left-dimer respectively at the site x, at the site y, at both sites x, y. Therefore
f l,x f l,y h
f l,x h f l,y h
=
h Zh Z \x,y h h Z \x Z \y
.
(5.2)
Since N > N0 (β), dist h (x, ∂) > N0 (β), dist h (y, ∂) > N0 (β), dist h (x, y) > N0 (β), all four partition functions satisfy the hypothesis of the Corollary 4.2. Hence by the cluster expansion (4.4) the partition functions rewrites as h Z
= C exp
h Z \x
= C\x exp
h Z \y
= C\y
∗
U (Pt )t ,
(P t )t ∈CP ∗
U\x (Pt )t ,
CP\x (Pt )t ∈
∗ exp U\y (Pt )t ,
h = C\x,y exp Z \x,y
(Pt )t ∈CP\y ∗
(5.3)
U\x,y (Pt )t .
(Pt )t ∈CP\x,y
By applying the definition (3.16), it holds C C\x,y = 1. C\x C\y
(5.4)
Now consider a polymer P ∈ P ∪ P\x ∪ P\y ∪ P\x,y . Keeping in mind the definitions of polymer (3.12) and polymer activity (3.17), observe that: if dist h (supp P, x) > 1, dist h (supp P, y) > 1 ⇒ P ∈ P ∩ P\x ∩ P\y ∩ P\x,y , (P) = \x (P) = \y (P) = \x,y (P);
123
A Cluster Expansion Approach…
783
and that19 : if dist h (supp P, x) ≤ 1, dist h (supp P, y) > 1 ⇒
P ∈ P ∩ P\y \ P\x ∪ P\x,y , (P) = \y (P) or
P ∈ P\x ∩ P\x,y \ P ∪ P\y , \x (P) = \x,y (P) or P ∈ P ∩ P\x ∩ P\y ∩ P\x,y , (P) = \y (P), \x (P) = \x,y (P); and the case dist h (supp P, x) > 1, dist h (supp P, y) ≤ 1 is clearly symmetric to the previous one. Therefore: ∗ ∗
U (Pt )t − U\x (Pt )t (Pt )t ∈CP
− ≤
∗
(Pt )t ∈CP\x
∗
U\y (Pt )t +
(Pt )t ∈CP\y ∗
U\x,y (Pt )t
(Pt )t ∈CP\x,y
∗
|U (Pt )t | +
|U\x (Pt )t |
(Pt )t ∈CP (Pt )t ∈CP\x ∃t: disth (supp Pt , x)≤1 ∃t: disth (supp Pt , x)≤1 ∃t : disth (supp Pt , y)≤1 ∃t : disth (supp Pt , y)≤1 ∗ ∗
|U\y (Pt )t | + |U\x,y (Pt )t | + (Pt )t ∈CP\y (Pt )t ∈CP\x,y ∃t: disth (supp Pt , x)≤1 ∃t: disth (supp Pt , x)≤1 ∃t : disth (supp Pt , y)≤1 ∃t : disth (supp Pt , y)≤1
(5.5)
.
It is crucial to observe that given a cluster (Pt )t ∈ CP , since ∪t supp Pt have to be connected in Z2 , | supp Pt | − 1 + dist Z2 (∪t supp Pt , y). dist Z2 (x, y) ≤ dist Z2 (∪t supp Pt , x) + t
Hence, assuming that distZ2 (∪t supp Pt , x) ≤ 1, dist Z2 (∪t supp Pt , y) ≤ 1, it follows (Pt ) t
pt n t v = t n t !1pt ! exp −β μh −μ i=1 |Si | − m k=1 |L k | − β J n t 2
= exp − m4 t | supp Pt | n t m 3 m pt v × t n t !1pt ! exp − β μh −μ − |S | − |L | − β J n i k t i=1 k=1 2 4 4
m ∗ (Pt ) ≤ exp − 4 (dist Z2 (x, y) − 1) t
pt nt for all t and ∗ (Pt ) is defined as the factor appearing in , (L k )k=1 where Pt = (Si )i=1 the product over t at the penultimate step. By defining a∗ (P) := m4 | supp P|, we have that (P) ea(P) : we can follow exactly the proof of ∗ (P) ea∗ (P) is essentially equivalent to the theorem 4.1 up to the inequality (4.30) and prove that the Kotecky–Preiss conditions (4.2), (4.3) hold also with ∗ , a∗ and , a and m/8 (eventually increasing
m/16 in place of ∗ (Pt )t := u (Pt )t β0 ). Therefore, defining U ∗ (Pt ), by the general theory of cluster t ∗ , expansion the inequality (4.6) holds also with U ∗ and a∗ in place of U , and a. As a consequence: 19 The first possibility, namely P polymer only of the lattices that contain x, happens when supp P x or
P includes a region Si containing x − (1, 0). The second possibility, namely P polymer only of the lattices that do not contain x, happens when P includes a line L k with one endpoint on x ± (1, 0). The last possibility happens when P includes a region Si containing x + (1, 0) (and does not verify the other conditions).
123
784
D. Alberici
∗
∗
|U (Pt )t | ≤
|u (Pt )t | t (Pt )
(Pt )t ∈CP ∃t: disth (supp Pt , x)≤1 ∃t : disth (supp Pt , y)≤1
≤
=
(Pt )t ∈CP ∃t: disth (supp Pt , x)≤1 ∃t : disth (supp Pt , y)≤1 ∗
− m4 (distZ2 (x,y)−1) |u (Pt )t | t ∗ (Pt ) e (Pt )t ∈CP ∃t: disth (supp Pt , x)≤1 ∃t : disth (supp Pt , y)≤1 ∗
m ∗ (Pt )t | e− 4 (distZ2 (x,y)−1) |U (Pt )t ∈CP ∃t: disth (supp Pt , x)≤1 ∃t : disth (supp Pt , y)≤1
(4.6)
m
(4.2)
m
≤ e− 4 (distZ2 (x,y)−1)
∗ (P) e P∈P disth (supp P, x)≤1
(5.6)
a∗ (P)
m ≤ e− 4 (distZ2 (x,y)−1) 3 16 .
The same reasoning can be repeated also for the clusters in CP\x , CP\y and CP\x,y . Thus, by (5.2), (5.3), (5.4), 5.5, (5.6), one finally obtains: f l,x f l,y h
f l,x h f l,y h
=
h Zh Z \x,y h h Z \x Z \y
m m ≤ exp e− 4 (distZ2 (x,y)−1) (3 + 2 + 2 + 2) . 16
The same bound can be shown to hold also for the inverse ratio we conclude that:
f l,x h f l,y h , f l,x f l,y h
m
| f l,x f l,y h − f l,x h f l,y h | ≤ e− 4 (distZ2 (x,y)−1)
hence by (5.1)
9m . 16
5.5 Proof of the Corollary 2.6 Since transl. =
2 ||
with
x∈, ( f r,x xh even
(transl. )2
h
− f l,x ), the variance of rewrites as:
2 − transl. h =
4 ||2
C x,y
x,y∈ xh ,yh even
C x,y := f r,x f r,y h − f r,x h f r,y h + f r,x h f l,y h − f r,x f l,y h +
+ f l,x h f r,y h − f l,x f r,y h + f l,x f l,y h − f l,x h f l,y h .
By the theorem 2.3, for x, y ∈ such that dist h (x, ∂) > N0 (β), dist h (y, ∂) > N0 (β) and disth (x, y) > N0 (β), it holds C x,y ≤ 4 Hence: h
2 (transl. )2 − transl. h ≤ 4
9m − m (dist 2 (x,y)−1) Z . e 4 16
9m − m (dist 2 (x,y)−1) Z e 4 + 1 − ϕ,,N0 (β) , 16||2 x,y∈ x= y
123
A Cluster Expansion Approach…
785
where we set ϕ,,N0 :=
#{(x, y) ∈ × |dist h (x, ∂) ∨ dist h (y, ∂ ) ∨ dist h (x, y) > N0 } . || | |
Now observe that ϕ,,N0 ≥
min
L ,L maximal horiz.lines of
ϕ L ,L ,N0 ≥
≥ 1 − 2 NN0 1 − 4 NN0 ,
min
(|L|−2N0 ) (|L |−4N0 ) |L| |L |
L ,L maximal horiz.lines of
hence 1 − ϕ,,N0 ≤ N0 /N (6 − 8N0 /N ). And on the other hand: − m (dist 2 (x,0)−1) − m (dist 2 (x,y)−1) Z Z e 4 ≤ || e 4 x,y∈ x = y
= ||
d≥1 4d
e
− m4 (d−1)
x∈Z2 x=0
= ||
4 . m (1−e− 4 )2
Acknowledgments I thank Prof. Elliott H. Lieb for his invitation at Princeton University, for having proposed me to work on his conjecture and for many useful discussions. I thank Prof. Pierluigi Contucci, Emanuele Mingione and Lukas Schimmer for interesting discussions. Financial support from UniBo Department of Mathematics, from FIRB Grant RBFR10N90W and from PRIN Grant 2010HXAW77 is acknowledged.
Appendix A: 1D Systems Consider a finite line L, that is a finite connected sub-lattice of Z. Consider a monomer-dimer model on L given by the following partition function: ZL = e−β HL (α) e Il (αxl ) e Ir (αxr ) . α∈D L
D L denotes the set of monomer-dimer configurations on L (allowing also external dimers at
the endpoints of L); the Hamiltonian is defined as μh + J sites of L with J sites of L with dimer + # monomer # but neighbor to monomer in L . 2 2 xl , xr denote the left and the right endpoint of L respectively; Il , Ir represent the interaction among the configuration on L and the boundary condition outside its endpoints. This one-dimensional system can be described by a transfer matrix T over the three possible states of a site, l ≡“left-dimer”, r ≡“right-dimer”, m ≡“monomer”: √ ⎞ ⎛ ⎞ ⎛ 0 1 T (l, l) T (l, r ) T (l, m) ab 0 ⎠, (5.7) T ≡ ⎝ T (r, l) T (r, r ) T (r, m) ⎠ := ⎝1 √0 T (m, l) T (m, r ) T (m, m) 0 ab a HL =
μh +J √ where to shorten the notation we set a := e−β 4 the transfer contribution of a √ J monomer20 , b := e−β 2 the transfer contribution of a site with a dimer but neighbor to a monomer. Two vectors are also needed to encode the boundary conditions:
20 The transfer energy of a monomer is half the energy of a monomer because it appears during two “transfers”.
123
786
D. Alberici
√ (m) := e Il (l) e Il (r ) a e Il (m) , Bl ≡ ⎛Bl (l) B⎞ l (r ) Bl⎛ ⎞ e Ir (l) Br (l) Br ≡ ⎝ Br (r ) ⎠ := ⎝ e Ir (r ) ⎠ . √ Ir (m) Br (m) ae
(5.8)
Proposition 5.1 The partition function of the system rewrites as a bilinear form: Z L = Bl T |L|−1 Br .
(5.9)
Proof According to the previous definitions it is clear that for every configuration α ∈ {l, r, m}|L| 1(α ∈ D L ) e−β HL (α) √ 1(α =m) √ 1(α =m) = a 1 T (α1 , α2 ) T (α2 , α3 ) . . . T (α|L|−1 , α|L| ) a |L| . Therefore ZL =
=
α∈{l,r,m}|L|Bl (α1 ) Bl T |L|−1 Br .
T (α1 , α2 ) T (α2 , α3 ) . . . T (α|L|−1 , α|L| ) Br (α|L| )
Assume for the moment that the transfer matrix T is diagonalizable. Denote by λ1 , λ2 , λ3 (1) (2) (3) (1) (2) (3) its eigenvalues and by E r , E r , E r , E l , E l , E l the corresponding right (column) (i) (i) eigenvectors and left (row) eigenvectors, normalized so that E l E r = 1 for i = 1, 2, 3. Corollary 5.2 ZL =
|L|−1
λi
(i)
Bl E r(i) E l Br .
(5.10)
i=1,2,3
Proof Since we are assuming that T is diagonalizable, it holds T = P D P −1 where D is the diagonal matrix of eigenvalues, P is the matrix with the right eigenvectors on the columns, P −1 has the left eigenvectors on the rows. Then T |L|−1 = P D |L|−1 P −1 and Bl T |L|−1 Br = (Bl P) D |L|−1 (P −1 Br ) =
3
|L|−1
(Bl E r(i) ) λi
(i)
(E l Br ).
i=1
Now our purpose is to diagonalise the transfer matrix T when β is large. Proposition 5.3 For all β > 0 the transfer matrix T is diagonalizable over R. Its eigenvalues are λ1 = 1 + ab 2 (1 + o(1)) λ2 = −1 + ab 2 (1 + o(1)) λ3 = a − ab − a 3 b (1 + o(1))
(5.11)
as β → ∞. Proof The eigenvalues λ1 , λ2 , λ3 are the (complex) roots of the characteristic polynomial of T , that is p(λ) := det(λI − T ) = −ab + (λ − a)(λ2 − 1).
123
A Cluster Expansion Approach…
787
For all β > 0 it turns out that p has 3 distinct real roots 21 , hence T is diagonalizable over the reals. As β → ∞, p(λ) → λ(λ2 − 1) hence λ1 → 1, λ2 → −1, λ3 → 0. Thus it is convenient to write λ1 = 1 + ε1 , λ2 = −1 + ε2 , λ3 = a + ε3 with εi → 0 as β → ∞ for i = 1, 2, 3. Now expand the polynomial p in powers of εi and truncate it at the first order: 0 = ⇒ 0 = ⇒ 0 = ⇒
p(λ1 ) = −ab + (1 − a + ε1 ) (2ε1 + ε12 ) = −ab + 2ε1 (1 + o(1)) ε1 = ab 2 (1 + o(1)); p(λ2 ) = −ab + (−1 − a + ε) (−2ε2 + ε22 ) = −ab + 2ε2 (1 + o(1)) ε2 = ab 2 (1 + o(1));
p(λ3 ) = −ab + ε3 (a + ε3 )2 − 1 = −ab − ε3 (1 + o(1)) ε3 = −ab (1 + o(1)).
In order to find the following order of λ3 , now one can write λ3 = a − ab (1 + ε3 ) with ε3 → 0 as β → ∞ and repeat the procedure: 2
2 3) 0 = p(λ −ab = 1 + (1 + ε3 ) a (1 + o(1)) − 1 = a (1 + o(1)) − ε3 (1 + o(1)) ⇒ ε3 = a 2 (1 + o(1)).
Proposition 5.4 The right eigenvectors of the transfer matrix T are ⎛ ⎞ 1 − a2 (1 + o(1)) (1) E r = √1 ⎝1√− a2 (1 + o(1))⎠ 2 ⎛ ab a(1 + o(1)) ⎞ 1 + 2 (1 + o(1)) (2) a ⎠ E r = √1 ⎝−1 √− 2 (1 + o(1)) 2 ab (1 + o(1)) ⎞ ⎛ √ −a√ ab (1 + o(1)) (3) E r = ⎝ − ab (1 + o(1)) ⎠ 1 + a (1 + o(1)) and moreover
(5.12)
√ (2) (2) (2) √ (1 + o(1)) E r (1) + E r (2) + ab E r (3) = ab 2 2 √ √ (3) (3) 2 E r (2) + ab E r (3) = −a ab (1 + o(1)) (i)
as β → ∞. The left eigenvectors are obtained by a simple transformation: E l for i = 1, 2, 3, where ⎛ ⎞ v1
σ ⎝v2 ⎠ := v2 v1 v3 . v3
(i) = σ Er
Proof The right eigenvectors E r associated to the eigenvalue λ are the non-zero solutions of the linear system ⎞ ⎛ λ(λ − a) (λI − T ) E r = 0 ⇔ E r = ⎝ λ√− a ⎠ t , t ∈ R. ab 21 The discriminant of the cubic is = 18a(1−b) + 4a 2 (1−b) + a 2 + 4 − 27a 2 (1−b), which is strictly positive for all 0 ≤ a, b ≤ 1, (a, b) = (1, 0).
123
788
D. Alberici
And the left eigenvectors E l associated to the same eigenvalue λ are the non-zero solutions of the linear system √ E l (λI − T ) = 0 ⇔ E l = λ − a λ(λ − a) ab t , t ∈ R. The desired normalization E l E r = 1 can be obtained by choosing % t = 2λ(λ − a) + ab in both cases. Now to conclude the proof it is sufficient to exploit the estimates of the eigenvalues given by the proposition 5.3. The formula (5.10) together with the estimates of propositions 5.3, 5.4 give us a complete control on the one-dimensional system on L at low temperature, for every choice of the boundary conditions. We concentrate on providing an estimation of the quantity R L defined by (3.9), since it is needed in the Sect. 3. We have to distinguish three cases, according to where the endpoints of L lie. Lemma 5.5 The ratios of the eigenvalues of the transfer matrix T are λ2 = −1 + ab (1 + o(1)) , λ1
λ3 = −a + ab (1 + o(1)) λ2
as β → ∞. In particular setting m := − log |λ2 /λ1 | it holds e−m = 1 − e−β
μh +3J 2
(1 + o(1)) as β → ∞.
(5.13)
Proof It follows immediately from the proposition 5.3. Lemma 5.6 If xl ∈ ∂rext S j , then as β → ∞ (1)
Bl E r
=
(2)
=
(3)
=
Bl E r
Bl E r
√ √b (1 + o(1)) 2√ − √b (1 + o(1)) 2
√ a (1 + o(1)). (1)
(2)
(3)
If xr ∈ ∂lext S j , then the same estimates hold for E l Br , E l Br , E l Br respectively. Proof If xl ∈ ∂rext S j then by (3.7) and (5.8) the vector describing the boundary condition on √ √ (i) the left side of the line L is Bl = 0 b a . Then the estimates for Bl E r , i = 1, 2, 3, are computed using the proposition 5.4. Lemma 5.7 If xl ∈ ∂l , then as β → ∞ #
√1 1 − a (1 + o(1)) if the h-dimer on xl −(1, 0) has fixed position 2 (1) 2 Bl E r = √
a 2 1 − 2 (1 + o(1)) if the h-dimer on xl −(1, 0) has free position
Bl E r(2)
⎧
1 a ⎪ ⎪− √2 1 + 2 (1 + o(1)) if the h-dimeron xl −(1, 0) is fixed to the left ⎨
if the h-dimer onxl −(1, 0) is fixed to the right = √12 1 + a2 (1 + o(1)) ⎪ ⎪ ⎩ ab √ (1 + o(1)) if the h-dimer onxl −(1, 0) has free position 2 2
123
A Cluster Expansion Approach…
Bl E r(3)
789
# √ −a 2 ab (1 + o(1)) if the h-dimer onxl −(1, 0) is fixed to the left √ = −a ab (1 + o(1)) if the h-dimer onxl −(1, 0) is fixed to the right or free (1)
(2)
(3)
If xr ∈ ∂r , then the same estimates hold respectively for E l Br , E l Br , E l Br after substituting: xl − (1, 0) by xr + (1, 0), “left” by “right” and “right” by “left”. Proof If xl ∈ ∂l then by (3.7) and √ (5.8) the vector describing the boundary condition on the left side of the line L is: Bl = 0 1 ab if a left-dimer is fixed on xl −(1, 0); Bl = 1 0 0 √ if a right-dimer is fixed on xl −(1, 0); Bl = 1 1 ab if on xl −(1, 0) there is a h-dimer with (i)
free position. Then the estimates for Bl E r , i = 1, 2, 3, are computed using the proposition 5.4. Proposition 5.8 Denote by o(1) any function ω(β, μh , J ) that goes to zero as β → ∞ and does not depend on the choice of the line L nor on . Then for every line L ∈ L (∪ j S j ), S j ∈ S pairwise disconnected, ⊂ Z2 finite, it holds |R L | ≤ e−m|L| γ L
(5.14)
where the quantity γ L can be chosen as follows: ⎧ −β J μ +J e −β h2 |L| ⎪ (1 + o(1)) if xl ∈ ∪i ∂rext Si , xr ∈ ∪i ∂lext Si + e ⎪ ⎪ 2 ⎪ ⎪ J ⎨ e−β 2 √ (1 + o(1)) if xl ∈ ∪i ∂rext Si , xr ∈ ∪i ∂r (5.15) γ L := 2 ⎪ ext S ⎪ ⎪ ∈ ∪ ∂ , x ∈ ∪ ∂ or vice versa x l i l r i l i ⎪ ⎪ ⎩ 1 + o(1) if xl ∈ ∂l , xr ∈ ∂r Proof • Suppose xl ∈ ∂rext Si and xr ∈ ∂lext S j . The definition (3.9) and the corollary 5.2 give λ1 R L = =
(1) (1) ZL Br |L|−1 − Bl E r E l λ1 |L|−1 (2) (2) λ2 Bl E r E l Br λ1
+
λ3 λ1
|L|−1
(3)
(3)
Bl E r E l Br .
By the lemma 5.5 |λ3 /λ1 | ≤ a |λ2 /λ1 | when β is sufficiently large. Therefore, using also the estimates of lemma 5.6, one finds $ $|L|−1 $ λ2 $ b |L| $ $ |R L | ≤ $ $ +a (1 + o(1)). λ1 2 • Suppose now xl ∈ ∂rext S j and xr ∈ ∂r . The definition (3.9) and the corollary 5.2 give 1/2
λ1
RL = =
(1) ZL − Bl E r (1) E l Br |L|−1 (2) (2) BE E B
|L|−1
λ1
λ2 λ1
l
r
l (1) E l Br
r
+
λ3 λ1
|L|−1
(3)
(3)
Bl E r E l Br (1) E l Br
.
By the lemma 5.5 |λ3 /λ1 | ≤ a |λ2 /λ1 | when β is sufficiently large. Therefore, using also the estimates of lemmas 5.6, 5.7, one finds $ $|L|−1 √ $ λ2 $ b |R L | ≤ $$ $$ √ (1 + o(1)). λ1 2
123
790
D. Alberici
• Suppose now xl ∈ ∂l and xr ∈ ∂r . The definition (3.9) and the corollary 5.2 give RL = =
ZL −1 (1) (1) Bl E r E l Br |L|−1 (2) (2) BE E B
|L|−1
λ1
λ2 λ1
r r l (1) (1) Bl E r E l Br l
+
λ3 λ1
|L|−1
(3)
(3)
(1)
(1)
Bl E r E l Br Bl E r E l Br
.
By the lemma 5.5 |λ3 /λ1 | ≤ a |λ2 /λ1 | when β is sufficiently large. Therefore, using also the estimates of lemma 5.7, one finds $ $|L|−1 $ λ2 $ |R L | ≤ $$ $$ (1 + o(1)). λ1
Appendix B: Cluster Expansion In this Appendix we state the main results about the general theory of cluster expansion used in this paper. The condition that we adopt to guarantee the convergence of the expansion is due to Kotecky–Preiss [11]. For a modern proof we refer to [17]. Let P be a finite set, called the set of polymers. Let : P → C, called the polymer activity, and δ : P × P → {0, 1}, called the polymer hard-core interaction, such that δ(P, P) = 0 and δ(P, P ) = δ(P , P) for all P, P ∈ P . Consider the polymer partition function: Z := P,P ∈P δ(P, P ) P ⊆P P∈P (P) P = P (5.16) q 1 = q≥0 q! P1 ,...,Pq ∈P t
Then: 1 log Z = q! q≥0
P1 ,...,Pq ∈P
q
(Pt ) u(P1 , . . . , Pq )
where the series on the r.h.s. is absolutely convergent and u(P1 , . . . , Pq ) :=
(−1)|E| .
G=(V,E)connected graph V ={1,...,q} E⊆{(t,s) | δ(Pt ,Ps )=0}
123
(5.18)
t=1
(5.19)
A Cluster Expansion Approach…
791
Moreover, for all E ⊆ P 1 q! q≥0
$ $ q $$ $ $ $$ $ $ (P ) | (P)| ea(P) . t $ u(P1 , . . . , Pq ) ≤ $
P1 ,...,Pq ∈P t=1 ∃ t: Pt ∈E
(5.20)
P∈P P∈E
It is worth to observe that if (P1 , . . . , Pq ) is not a cluster then u(P1 , . . . , Pq ) = 0. Therefore only the clusters of polymers (that are infinitely many) give non-zero contributions to the expansion (5.18) of log Z .
References 1. Bovier, A., Zahradnik, M.: A simple inductive approach to the problem of convergence of cluster expansions of polymer models. J. Stat. Phys. 100(3–4), 765–778 (2000) 2. Contucci, P.: An analyticity bound for two-dimensional Ising model at low temperature. J. Stat. Phys. 82(5–6), 1647–1657 (1993) 3. Disertori, M., Giuliani, A.: The nematic phase of a system of long hard rods. Commun. Math. Phys. 323(1), 143–175 (2013) 4. Dobrushin, R.L.: Estimates of semi-invariants for the ising model at low temperatures. In: Dobrushin, R.L., Minlos, R.A., Shubin, M.A., Vershik, A.M. (eds.) Topics in Statistical and Theoretical Physics. F.A. Berezin Memorial Volume, pp. 59–81. American Mathematical Society, Providence, RI (1995) 5. Fernández, R., Procacci, A.: Cluster expansion for abstract polymer models. New bounds from an old approach. Commun. Math. Phys. 274(1), 123–140 (2007) 6. Gallavotti, G., Martin-Löf, A., Miracle-Solé, S.: Some Problems Connected with the Description of Coexisting Phases at Low Temperatures in the Ising Model. Lecture Notes in Physics, vol. 20, pp. 162– 204. Springer, Heidelberg (1973) 7. Gruber, C., Kunz, H.: General properties of polymer systems. Commun. Math. Phys. 22, 133–161 (1971) 8. Heilmann, O.J., Lieb, E.H.: Monomers and dimers. Phys. Rev. Lett. 24(25), 1412–1414 (1970) 9. Heilmann, O.J., Lieb, E.H.: Theory of monomer-dimer systems. Commun. Math. Phys. 25(3), 190–232 (1972) 10. Heilmann, O.J., Lieb, E.H.: Lattice models for liquid crystals. J. Stat. Phys. 20(6), 679–693 (1979) 11. Kotecky, R., Preiss, D.: Cluster expansion for abstract polymer models. Commun. Math. Phys. 103(3), 491–498 (1986) 12. Letawe, I.: Le Modèle de Cristaux Liquides de Heilmann et Lieb, thesis in Physics at Université Catholique de Louvain, supervisor J. Bricmont (1994) 13. Miracle-Solé, S.: On the convergence of cluster expansions. Physica A 279, 244–249 (2000) 14. Onsager, L.: The effects of shape on the interaction of colloidal particles. Ann. N. Y. Acad. Sci. 51, 627–659 (1949) 15. Papanikolaou, S., Charrier, D., Fradkin, E.: Ising nematic fluid phase of hard-core dimers on the square lattice. (2014). arXiv:1310.4173 16. Ruelle, D.: Statistical Mechanics: Rigorous Results, vol. 4. W.A. Benjamin Inc., New York (1969) 17. Ueltschi, D.: Cluster expansion & correlation functions. Mosc. Math. J. 4, 511522 (2004)
123
