The statistical behaviors of two-layered random-phase interfaces in two-dimensional Widom-Rowlinson's model are investigated. The phase interfaces separate two coexisting phases of the lattice Widom-Rowlinson model; when the chemical potential μ of the model is large enough, the convergence of the probability distributions which describe the fluctuations of the phase interfaces is studied. In this paper, the backbones of interfaces are introduced in the model, and the corresponding polymer chains and cluster expansions are developed and analyzed for the polymer weights. And the existence of the free energy for two-layered random-phase interfaces of the two-dimensional Widom-Rowlinson model is given.

1. Introduction

We investigate the statistical behaviors of random interfaces between the two coexisting phases of the Widom-Rowlinson model (W-R model) when the chemical potential μ is large enough; especially we consider the two-layered interfaces behaviors of the model in this paper. The lattice system interfaces in two dimensions are known to fluctuate widely, for example, see [1–4] for the W-R model and [5–13] for the Ising spin system. There are two types of particles (either A or B) in the lattice W-R system, and there is a strong repulsive interaction between particles of the different types. Namely, a B particle cannot occupy a site within distance 2 from a site where an A particle has occupied and vice versa. This means that different types of particles are separated by the empty sites. In [2], under some special conditions for the interfaces (with specified values of the area enclosed below interfaces and the height difference of two endpoints) and the chemical potential μ large enough, it shows the weak convergence of the probability distributions (which describe the fluctuations of such interfaces) to certain conditional Gaussian distribution. According to the dynamic system of the W-R model and the results of [2], the thickness of the random interface (or the intermediate “belt”) between the two coexisting phases of the model is expected to become thinner as μ becomes larger, so we believe that the interfaces of the W-R model behave like those of the Ising model to some extent. We are also interested in the fluctuation behaviors of two or more random interfaces that one interface lies above the other one, such model presents the coexistence of three or more phases, which is the multilayer interacting interface model. In the present paper, the two-layered lattice W-R model is considered, and the convergence of the probability distributions which describe the fluctuations of two-layered random interfaces is exhibited.

The interface behavior of the lattice W-R model has close relation with the wetting phenomenon of the well-known Potts model, for example, see [14–17]. Wetting may occur when three or more phases coexist; it consists of the appearance of a thick (macroscopic) layer of the C phase at an interface between the A and the B phases. Much research work has been devoted to the study of the wetting behavior for the q-state Potts model. Derrida and Schick [16] show that the interface is wetted by the disordered phase as the transition is approached by the mean field approximation. According to the method of low-temperature expansions, Bricmont and Lebowitz [14] exhibit the wetting of the interface between two ordered phases by the disordered one with q being large. At the transition point, De Coninck et al. [15] display the similar wetting behavior by the correlation inequalities, and Messager et al. [17] present an analysis of the order-disorder transition for large q based on the theory of cluster expansion and surface tension of the Potts model. For the interface of the lattice W-R system, we think that the wetting phenomenon may appear when the positive chemical potential μ is small, which means that the layer of empty sites that separate A and B particles may have some thickness. Whereas the results of the present paper may imply that the interface of the W-R model will not be wetted for large parameter μ, since we think that the layer between the two coexisting phases is expected to become thinner as μ becomes larger.

We consider the two-layered phase interfaces in the two-dimensional Widom-Rowlinson model on the rectangle ΛL,M, where ΛL,M=[1,L-1]×[-M,M]⊂ℤ2. Suppose that the particles in ΛL,M are of two types and there is a strong repulsive interaction between particles of the different types. Let σ denote a configuration on {-1,0,+1}ΛL,M, where σ(x)=+1 denotes that the site x is occupied by an A particle, σ(x)=-1 denotes that x is occupied by a B particle, and σ(x)=0 denotes that there is no particle at x. We say that a configuration σ is feasible if σ(x)σ(y)≥0 for all pairs x,y∈ℤ2 with |x-y|≤2, where |·| is the Euclidean distance. Let ΩL,M denote the set of all feasible configurations in ΛL,M, so there is a finite diameter hard-core exclusion between A particles and B particles on ΩL,M. Next, the two-layer interface model is defined for satisfying the following three conditions.

(i) The Hamiltonian of the two-layered model is given by HL,M(σ)=-∑x∈ΛL,Mμ(σ(x)2-1),
for all σ∈ΩL,M, where μ(>0) denotes the chemical potential.

(ii) Let b1=b1(L)>0, let b2=b2(L)>0, and assume that M>b1, M>b2. We define a boundary condition ωb as ωb(x)=ωb(x1,x2)={+1,ifx1=0,|x2|>b1andifx1=L,|x2|>b2andif1≤x1≤L-1,|x2|=M+1,0,ifx1=0,|x2|=b1andifx1=L,|x2|=b2,-1,ifx1=0,|x2|<b1andifx1=L,|x2|<b2,
for each x=(x1,x2)∈∂ΛL,M=[0,L]×[-M-1,M+1]∖ΛL,M.

(iii) Suppose that there is a connected “−1” particles path from the left side of [0,L]×[-M-1,M+1] to the right side.

The Hamiltonian of (i) is the same as that of the W-R model, and conditions (ii) and (iii) (here we suppose that b1(L)=c1L, b2(L)=c2L, for some 0<c1,c2<1) ensure that we have the two-layered interface model. Let ΩL,Mb(⊂ΩL,M) be the corresponding configuration space with the conditions (i)–(iii), such that the configuration σ×ωb is feasible, where σ×ωb(x)=σ(x) for x∈ΛL,M and σ×ωb(x)=ωb(x) for x∈∂ΛL,M. For a fixed configuration σ∈ΩL,Mb, let S0(σ) denote the set of points in ΛL,M such that the configuration σ takes 0 value. The connected components of S0(σ) are called contours; among these contours, there are two contours Γu(σ) and Γl(σ) which are defined as the interfaces of the model. Γu(σ) is the upper interface with the starting point (0,b1) and the ending point (L,b2); Γl(σ) is the lower interface with the starting point (0,-b1) and the ending point (L,-b2). Let SL,Mb,u={Γu(σ);σ∈ΩL,Mb},SL,Mb,l={Γl(σ);σ∈ΩL,Mb}
be the set of upper interfaces and the set of lower interfaces, respectively. The conditional Gibbs distribution on ΩL,Mb with the boundary condition ωb is given by PL,Mb(σ)=(ZL,Mb)-1exp{-μ|S0(σ)|},
where |S| denotes the cardinality of a set S, and ZL,Mb is the corresponding partition function.

2. Backbones and Partition Functions

The general theory of interfaces between the coexisting phases (which is based on its microscopic description) has been intensively studied, for example, see [1, 2, 6, 7, 18]. De Coninck et al. [18] introduce the SOS approximants for the Potts surface tensions and present a connection between the orientation-dependent surface tension of the Potts model and the corresponding surface tension of the SOS model. And they show that an SOS model is applied for the construction of the Potts crystal shapes. In this section, a similar approximation method is developed; that is, the interface of the W-R model can be approximated by its corresponding backbone. Next we introduce the definitions of the backbones π(Γu) and π(Γl) of interfaces Γu and Γl in the two-layered W-R model and state the main results of the present paper. According to the above definitions of interfaces of the model, the interface of the two-layered model is an intermediate belt between the two coexisting phases, whereas the interface of the two-dimensional Ising model is an open polygon passing through the starting point to the ending point. This means that the methods and techniques of the partition function representation and the partition function cluster expansion, which are applied in analyzing the Ising model, cannot be directly used in analyzing the two-layered W-R model. In the present paper, we define a backbone π(Γu) of Γu to represent Γu; that is, among self-avoiding paths connecting the starting point (0,b1) with the ending point (L,b2) in Γu, we select a self-avoiding path (called backbone π(Γu)) by an “order” given in the following definition (6). Since the backbone π(Γu) is an open polygon, this will help us to study the interfaces of the model. Now we define the set of self-avoiding paths in Γu as ΠΓu=Thesetofself-avoidingpathinΓuconnecting(0,b1)with(L,b2).
Among these paths, we select a self-avoiding path according to the following order; the order is defined with preference among four directions: up>down>right>left.
More precisely, let πx={x1,…,xn} and πy={y1,…,ym} be the two self-avoiding paths in Γ. Let k=mini≥1{i:xi≠yi} be the first number i such that xi≠yi. We define that πx>πy if the direction of the ordered edge {xk-1,xk} is preferred to the direction of the ordered edge {yk-1,yk}. Let π(Γu) be the unique maximal element of ΠΓu with respect to this order, and call π(Γu) the backbone of Γu. Similarly, let π(Γl) represent the backbone of Γl. In this paper, our study is mainly focused on the backbone of the phase separation belt.

Let U1,…,Un be the different connected subsets of ΛL,M, we say that the subsets {Uj} are compatible if they are connected components of the set ∪1≤j≤nUj. We also say that {Uj} are compatible with a connected set G if {G,Uj} are compatible for every 1≤j≤n. Next we define the hole of a connected set of ℤ2, we say that a set D⊂ℤ2 is *-connected if, for every x,y∈D, there exist a sequence x=z0,z1,…,zn=y in D such that |zi-zi-1|≤2 for every 1≤i≤n. A hole of a connected set F⊂ℤ2 is a finite *-connected component of Fc=ℤ2∖F. Since there may be some holes inside an interface of the W-R model (note that the interface of the Ising model has no such holes), there is a large difference of the partition function expansion between the W-R model and the Ising model. Therefore, the partition function ZL,Mb defined in (1.4) can be rewritten by the similar formulas in [1–3] as following: ZL,Mb=∑Γu∈SL,Mb,u,Γl∈SL,Mb,l∑{Uj}2N(Γu)+N(Γl)e-μ(|Γu|+|Γl|)∏j2N(Uj)e-μ|Uj|,
where the second summation is taken over compatible families {Uj}, which are compatible with Γu∪Γl, |Γu| is the number of points in Γu, and N(Γu) is the number of holes in Γu, similar to Γl, N(Γl), and |U|, N(U). Then for some large μ0>0 and μ>μ0, according to the theory of the cluster expansions (see [19]), we have ZL,MbZL,M+=∑Γu∈SL,Mb,u,Γl∈SL,Mb,l×exp{-μ(|Γu|Γl||+|Γl|)+ln2(N(Γu(Γl))+N(Γl))-∑Λ⊂ΛL,M:Λi(Γu∪Γl)Φ(Λ)},
where ZL,M+ is the partition function with the plus boundary condition, Λi(Γu∪Γl) denotes that the set Λ is incompatible with the interfaces Γu∪Γl, and Φ(Λ) is a translation invariant function, which satisfies the following estimate: ∑Λ;Λ∋0|Φ(Λ)|e(μ-μ0)|Λ|<1.
Moreover, if μ is sufficiently large, we have limM→∞ZL,MbZL,M+=∑Γu∈SLb,u,Γl∈SLb,l×exp{-μ(|Γu|Γl||+|Γl|)+ln2(N(Γu(Γl))+N(Γl))-∑Λ⊂ΛL,∞:Λi(Γu∪Γl)Φ(Λ)},
where ΛL,∞=[1,L-1]×(-∞,∞)∩ℤ2 and SLb,u=∪M>0SL,Mb,u,SLb,l=∪M>0SL,Mb,l.
Let W(Γu,Γl)=exp{-μ(|Γu|Γl||+|Γl|)+ln2(N(Γu(Γl))+N(Γl))-∑Λ⊂ΛL,∞:Λi(Γu∪Γl)Φ(Λ)},
which is called the weight of the partition function expansion (2.6). From [19], the last part in (2.6) can be expanded as follows: exp{-∑Λ⊂ΛL,∞:Λi(Γu∪Γl)Φ(Λ)}=∑n=0∞∑Λ1,…,Λn⊂ΛL,∞Λνi(Γu∪Γl)∏ν=1n(e-Φ(Λν)-1).

In the definition of boundary condition ωb (see (1.2)), for a fixed b1≥0, let 𝒮L=(∪b𝒮Lb,u,∪b𝒮Lb,l), then, for the interfaces vector (Γu,Γl)∈𝒮L, the height of the last ending point of (Γu,Γl) is defined as the vector (h(Γu(Γl)),h(Γl))=(h(π(Γu)(Γl)),h(π(Γl))).
The aim of this paper is to study the statistical behaviors of the free energy of the height vector (h(Γu),h(Γl)). The following theorem shows the limit existence of the free energy of the height for the interface vector (Γu,Γl).

Theorem 2.1.

For some δ>0 and a complex vector (ξ,ζ)∈ℂ2, when the chemical potential μ is large enough, one has the existence of the free energy of the heights for the two-layered W-R model as follows:
φ(ξ,ζ)=limL→∞1Lln∑Γu,Γl∈SLeμ(ξh(Γu)+ζh(Γl))W(Γu,Γl),
where the function φ(ξ,ζ) is analytic in (ξ,ζ)∈ℂ2 if the real parts Reξ<1-δ/μ and Reζ<1-δ/μ.

3. Polymer Chains and Cluster Expansions

Since the formula (2.11) heavily depends on the polymer representation of the two-layered W-R model partition function and the theory of the cluster expansions, we study and analyze the polymer representation of the partition function in this section. And we show the estimate of the free energy for the height of the last ending point of (Γu,Γl). Further, we show that the asymptotical behavior of the backbone π(Γu) can represent that of the corresponding interface Γu when μ is large enough; this means that the statistical properties of the interface Γu are similar to those of its backbone π(Γu). From the definition of π(Γu), we define its polymer and polymer chains and develop a new polymer representation of the two-layered model partition function; we also obtain some estimates for the polymer weights.

For the Ising model, the polymers are defined by “cutting" the interface into elementary pieces at the line {x1=n+1/2}(n∈ℤ) of dual lattice, see [5–7]. However, this cutting procedure is invalid for the interface of the W-R model, since the interface of the Ising model is a line, but the interface of the W-R model is a belt. So it needs to develop a new technique to cut the interface of the W-R model; that is, we hope to give a new definition of polymers. Higuchi et al. [2] introduce a new definition of polymers for one-layered interface of the W-R model; here we modify the definitions in [2] and define the polymer chains for the two-layered interface model.

For the interface vector (Γu,Γl)∈𝒮L, let (0,b1) and (L,ku) be the starting and the ending points of Γu; let π(Γu) be the backbone of Γu connecting (0,b1) and (L,ku). Similarly, let (0,-b1) and (L,kl) be the starting and the ending points of Γl; let π(Γl) be the backbone of Γl connecting (0,-b1) and (L,kl). We decompose Γu∖π(Γu) into connected components {Cj}j=1s and decompose Γl∖π(Γl) into connected components {Dj}j=1r. Then, from (2.6) and (2.9), we have∑Γu,Γl∈SLeμ(ξh(Γu)+ζh(Γl))W(Γu,Γl)=∑ku-kl>1-∞≤kl,ku≤+∞∑π;(0,b1)→(L,ku)π′;(0,-b1)→(L,kl)πliesaboveπ′∑C1,…,Cs;compatibleCνiπ,Cν∩π=∅π;backboneofπ∪C1∪⋯∪Cs∑D1,…,Dr;compatibleDνiπ′,Dν∩π=∅π′;backboneofπ′∪D1∪⋯∪Ds×exp{μξku-μ|π|+N(π,C1,…,Cs)ln2-μ∑ν=1s|Cν|}×exp{μζkl-μ|π′|+N(π′,D1,…,Dr)ln2-μ∑ν=1r|Dν|}∑Λ1,…,Λt;connectedΛαiπ∪C1∪⋯∪CsorΛαiπ′∪D1∪⋯∪Dr∏α=1t(e-Φ(Λα)-1),
where N(π,C1,…,Cs) denotes the number of holes of π∪∪ν=1sCν, N(π′,D1,…,Dr) denotes the number of holes of π′∪∪ν=1rDν, and the second summation taken over “π lies above π′”, which is consistent with the definitions (i)–(iii) in Section 1.

Next we give the definitions of polymers of the two-layered W-R model. Let a≤c be the positive integers, η=(γu,γl,C1,…,Cp,D1,…,Dq,Λ1,…,Λv),
is called a polymer with base [a,c] if η satisfies the following conditions (1)–(4).

γu,γl are self-avoiding paths in {a≤x1≤c}, γu starts from (a,au) and ending at a point (c,cu) in {x1=c}, γl starts from (a,al) and ending at a point (c,cl) in {x1=c}, where au, al, cu, cl are fixed integers satisfying au-al>1 and cu-cl>1, and γu lies above γl.

{Cν}ν=1p is a compatible family of connected subsets of {x∈ΛL,∞;a≤x1≤c} such that (i) Cν∩V(γu)=∅, where V(γu) is the set of vertices in γ; (ii) Cν∪V(γu) is connected; (iii) γu is the backbone of γu∪C1∪⋯∪Cp with starting point (a,au) and ending point (c,cu). Similarly, {Dν}ν=1q and γl have the same properties.

{Λα}α=1v is a collection of connected subsets of {x∈ΛL,∞;a≤x1≤c} such that

Λαiγu∪∪ν=1pCνorΛαiγl∪∪ν=1qDν.

For a≤j<c, j∈ℕ, the line ℓj={x1=j+1/2} intersects at least two edges of γu∪ℰ(∪ν=1pCν∪∪α=1vΛα)∪ℰ(γu,∪ν=1pCν∪∪α=1vΛα) or at least two edges of γl∪ℰ(∪ν=1qDν∪∪α=1vΛα)∪ℰ(γl,∪ν=1qDν∪∪α=1vΛα). Here, for B⊂ℤ2, ℰ(B) denotes the set of the nearest neighbor edges of B; ℰ(γ,B) is the set of edges that connect γ with the set B. Also, we identify an edge {x,y} of ℤ2 with the line segment connecting x and y.

We call (γu,γl) the backbone of η. For two disjoint self-avoiding paths γ1u,γ2u (similarly for γ1l,γ2l) such that the starting point of γ2u is the nearest neighbor of the endpoint of γ1u, we can define the concatenation γ1u∘γ2u of these paths by simply connecting them. Let η1=(γ1u,γ1l,C1,…,Cp,D1,…,Dq,Λ1,…,Λv),η2=(γ2u,γ2l,C1′,…,Cu′,D1′,…,Dw′,Λ1′,…,Λz′)
be two polymers with bases [a,c] and [a′,c′](a≤a′), respectively. We say that η1 and η2 are compatible if either of the following conditions holds:

c+1<a′,

a′=c+1, the backbone of

Γ̃=γ1u∪C1∪⋯∪Cp∪(γ2u+(0,h(γ1u⋃γ1l)⋃γ1l)⋃γ1l)∪(C1′+(0,h(γ1u⋃γ1l)⋃γ1l)⋃γ1l)∪⋯∪(Cu′+(0,h(γ1u⋃γ1l)⋃γ1l)⋃γ1l)⋃γ1l∪D1∪⋯∪Dq∪(γ2l+(0,h(γ1l)))∪(D1′+(0,h(γ1l)))∪⋯∪(Dw′+(0,h(γ1l))),
is the concatenation (γ1u∘(γ2u+(0,h(γ1u))),γ1l∘(γ2l+(0,h(γ1l)))), and connected components of the set Γ̃∖(γ1u∘(γ2u+(0,h(γ1u)))∪γ1l∘(γ2l+(0,h(γ1l)))) are {C1,…,Cp,C1′,…,Cu′} and {D1,…,Dq,D1′,…,Dw′}. Here, h(γ) is the height of the endpoint of γ.

Note that the previous work (see [2]) has presented the similar formula as that of the above (3.5) for the one-layered interface W-R model, so we can derive the formula (3.5) for the two-layered interface model from the corresponding work in Section 2 of [2].

The family {ηp}p=0n+1 is compatible if ηp and ηp′(p≠p′) are compatible. Let η=(π(Γu),π(Γl),C1,…,Cs,D1,…,Dr,Λ1,…,Λt).
An edge e={x,y} of η is not admissible if it is a horizontal edge in ℰ(π(Γu),∪ν=1sCν∪∪α=1tΛα)∪ℰ(π(Γl),∪ν=1rDν∪∪α=1tΛα), such that

the left vertex x is in a connected E of ∪ν=1sCν∪∪ν=1rDν∪∪α=1tΛα, and the right vertex y is in V(π(Γu))∪V(π(Γu));

further, there exists a horizontal edge e′={x′,y′} of η such that x′∈V(π(Γu))∪V(π(Γu)) and y′∈E, where x′ is the left vertex of e′.

Other edges of η are admissible.

We say that the line ℓj={x1=j+1/2}(0≤j≤L-1) is the cutting line of η if ℓj intersects only two admissible edges of η=(π(Γu),π(Γl),{Cν}ν=1s,{Dν}ν=1r,{Λα}α=1t). Here, one of two admissible edges is connected to π(Γu); the other is connected to π(Γl).

Let ℓ0<ℓj1<⋯<ℓjn<ℓjn+1=ℓL-1 be all the cutting lines of (π(Γu),π(Γl), {Cν}ν=1s,{Dν}ν=1r, {Λα}α=1t). For each m∈{0,1,…,n+1}, there are only two edges emu={Bmu,Am+1u} and eml={Bml,Am+1l} of π(Γu), and π(Γl), respectively, which intersect ℓjm. Let γmu be the portion of π(Γu) starting from Amu and ending at Bmu; let γml be the portion of π(Γl) starting from Aml and ending at Bml. Also let {Cν(m)}ν=1s(m), {Dν(m)}ν=1r(m), and {Λα(m)}α=1t(m) be the set of elements of {Cν}ν=1s, {Dν}ν=1r, and {Λα}α=1t, respectively, such that they are subsets of [jm-1+1,jm]×(-∞,∞)∩ℤ2. Then Amu=(jm-1+1,pu), Aml=(jm-1+1,pl) for some pu,pl∈ℤ, where pu>pl. Thus, we obtain the mth polymer ηm by setting ηm=(γmu-(0,pu),γml-(0,pu),{Cν(m)-(0,pu)}ν=1s(m),{Dν(m)-(0,pu)}ν=1r(m),{Λα(m)-(0,pu)}α=1t(m)).
By the above definitions, {η0,η1,…,ηn+1} are compatible.

For a polymer ηm=(γmu,γml,{Cν(m)},{Dν(m)},{Λα(m)}),
let hmu=h(ηmu)=h(γmu) be the height of the endpoint of the self-avoiding path γmu; similarly let hml=h(ηml)=h(γml). Then the heights h(π(Γu)) and h(π(Γl)) are given by h(π(Γu))=∑m=0n+1h(γmu),h(π(Γl))=∑m=0n+1h(γml).
Now we introduce a statistical weight of a polymer η′=(γu,γl,{Cν}ν=1p,{Dν}ν=1q,{Λα}α=1v),
by setting Ψ(η′)=exp{μ∑ν=1q-μ|γu|-μ|γl|+N*(γu,C1,…,Cp)ln2+N*(γl,D1,…,Dq)ln2-μ∑ν=1p|Cν|-μ∑ν=1q|Dν|}×∏α=1v(e-Φ(Λα)-1),
where N*(γu,C1,…,Cp)=N(γu,C1,…,Cp)+Nl̂(γu,C1,…,Cp)+Nr̂(γu,C1,…,Cp),
and Nl̂(γu,C1,…,Cp) is the number of new holes created by V(γu)∪∪ν=1pCν and the line {x1=l̂-1}, where base(η′)=[l̂,r̂]; Nr̂(γu,C1,…,Cp) is the number of new holes created by V(γu)∪∪ν=1pCν and the line {x1=r̂+1}. Similarly, we can give the definition of N*(γl,D1,…,Dq).

A polymer η is called simple if base(η) is one point and η=(γu,γl,∅,∅,∅). Thus, the weight Ψ(η) is given by Ψ(η)=e-μ|γu|-μ|γl|.
A polymer η is called decorated if it is not simple. A decorated polymer η=(γu,γl,{Cν},{Dν},{Λα}) with base(η)=[l̂,r̂] is said to be r-active if there exists a simple polymer η1=(γ1u,γ1l,∅,∅,∅) with base(η1)={r̂+1} such that η1 is incompatible with η, or the concatenation of γu and γ1u together with ∪νCν produces a new hole, or the concatenation of γl and γ1l together with ∪νDν produces a new hole. η is said to be l-active if there exists a simple polymer η2=(γ2u,γ2l,∅,∅,∅) with base(η2)={r̂-1} such that η2 is incompatible with η, the concatenation of γu and γ2u together with ∪νCν produces a new hole, or the concatenation of γl and γ2l together with ∪νDν produces a new hole. If η is both r-active and l-active, we call it bi-active.

A polymer chain is a family of decorated polymers 𝒞={η1,…,ηm} such that

{η1,…,ηm} are compatible;

if base(ηu)=[l̂u,r̂u], 1≤u≤m, then l̂u+1=r̂u+1 or r̂u+2 for each u;

if l̂u+1=r̂u+2 for some u, then ηu is r-active and ηu+1 is l-active.

Let 𝒞1 and 𝒞2 be two polymer chains. We say that 𝒞1 and 𝒞2 are compatible if 𝒞1∪𝒞2 is a compatible family of polymers, but now it is not a polymer chain. For example, if base(𝒞1)=[l̂u,r̂u] and base(𝒞2)=[l̂u′,r̂u′] have l̂u′-r̂u>2, then 𝒞1∪𝒞2 is compatible polymers, but not a polymer chain.

Let 𝒦=𝒦L be the set of all decorated polymers with base in [0,L], and let 𝒞𝒫L denote the set of polymer chains with base in [0,L], then we have the following Lemma 3.1.

Lemma 3.1.

Let Q(ξ,ζ) be the generating function of the heights of the endpoints of a simple polymer
Q(ξ,ζ)=e-μe-μ∑ku-kl>1-∞≤kl,ku≤+∞eμξkue-|ku|μeμζkle-|kl|μ,
then one has
1Q(ξ,ζ)L∑Γu,Γl∈SLeμ(ξh(Γu)+ζh(Γl))W(Γu,Γl)=∑C1,…,Cz∈CPL;compatible∏i=1zΨ̂(Ci;ξ,ζ),
where Ψ̂(𝒞i,ξ,ζ) is the weight function of polymer chains which is given (3.17).

Proof of Lemma <xref ref-type="statement" rid="lem1">3.1</xref>.

Considering a polymer chain 𝒞={η1,…,ηm}, let base(𝒞)=base(η1)∪⋯∪base(ηm). Further, for a polymer η, from (3.9) and (3.11), we define
Ψ̂(η;ξ,ζ)=eμξh(ηu)eμζh(ηl)Ψ(η)Q(ξ,ζ)-|base(η)|,
where |base(η)|=r̂-l̂+1 for base(η)=[l̂,r̂]. By the formula (3.16), for a polymer chain 𝒞={η1,…,ηm}, we put
Ψ̂(Ci;ξ,ζ)=∏u=1mΨ̂(ηu;ξ,ζ)×Jl̂(η1)Jr̂(ηm)∏u=1m-1J(ηu,ηu+1),
where 𝒥l̂, 𝒥r̂, 𝒥 are defined in the following way. For base(η)=[l̂,r̂] and base(η̅)=[c,d] with c>r̂, we define
Jl̂(η)={∑η′cηl̂-1Ψ̂(η′;ξ,ζ)2N(η′,η)-Nl̂(γu,C1,…,Cp)-Nl̂(γl,D1,…,Dq),ifηisl-active,1,otherwise,
where ∑η′cηl̂-1 means over polymers η′=(γ′u,γ′l,∅,∅,∅) with base{l̂-1} which are compatible with η, and N(η′,η) is the number of new holes created by the concatenation of γ′u and γu together with ∪νCν or by the concatenation of γ′l and γl together with ∪νDν. Similarly,
Jr̂(η)={∑η′cηr̂+1Ψ̂(η′;ξ,ζ)2N(η,η′)-Nr̂(γu,C1,…,Cp)-Nr̂(γl,D1,…,Dq),ifηisr-active,1,otherwise,
and 𝒥(η,η̅) is defined in the following two cases.

From the formulas (3.16) and (3.18)–(3.21), we show the weight expression of Ψ̂(𝒞i;ξ,ζ). According to the polymer representation of the partition function which is introduced in this section, and, by (3.1)–(3.14), we can show the existence of (3.15). This completes the proof of Lemma 3.1.

4. Proof of the Main Results

In the first part of this section, we do some preparations for the main results by some lemmas. Then we present the proof of Theorem 2.1.

Lemma 4.1.

Let I⊂ℤ be a fixed interval; if μ1 is large enough, then for some δ>0, one has
∑base(γu)=Ie-(δ/3)Nv(γu)-(μ1-2δ/3)(Nh(γu)+1)≤R(μ1,δ)|I|1-R(μ1,δ),
where γu is a upper backbone, Nv(γu) is the number of vertical edges in γu, Nh(γu) is the number of horizontal edges in γu, and
R(μ1,δ)=2e-(μ1-2δ/3)(1+e-δ/3)(1-e-δ/3)-1.
The upper bound of (4.1) also exists for the lower backbone γl.

Proof of Lemma <xref ref-type="statement" rid="lem2">4.1</xref>.

We separate γu into fragments by the following method. Let γu={x0,x1,…,xn} be a self-avoiding path with base(γu)=I. For j0=0 and i≥1, we let
ji=min{j>ji-1;{xj-1,xj}isahorizontaledge}.
Each vertical part {xji-1,xji-1+1,…,xji-1} of γu with the direction of the exit vector {xji-1,xji} is called a fragment. For a fragment f={x̅0,x̅1,…,x̅p} with exit direction e(f), we define
W(f)=e-(δ/3)Nv(f)-(μ1-2δ/3)=e-pδ/3-(μ1-2δ/3).
Then the decomposition of γu into fragments {f1,…,fr} leads to the identity
e-(δ/3)Nv(γu)-(μ1-2δ/3)(Nh(γu)+1)=∏j=1rW(fj).
Therefore, if μ1 is sufficiently large, we have
∑γu;base(γu)=Ie-(δ/3)Nv(γu)-(μ1-2δ/3)(Nh(γu)+1)=∑r=|I|∞∑f1,…,fr∏j=1rW(fj)≤∑r=|I|∞(2∑k=-∞∞e-(δ/3)|k|)r×e-(μ1-2δ/3)r=R(μ1,δ)|I|1-R(μ1,δ),
where the last equality comes from [7]. Following the same proving procedure, we can obtain the existence of the formula (4.1) for γl. This completes the proof of Lemma 4.1.

Lemma 4.2.

There is a large chemical potential μ2>0; if μ>μ2, then for any ξ,ζ∈ℂ with the conditions Reξ<1-δ/μ and Reζ<1-δ/μ, and for each polymer η0, one has
∑η∈KL;ηiη0ec(η)+d(η)|Ψ̂(η;ξ,ζ)|≤c(η0),
where 𝒦L is the set of all decorated polymers with the base in [0,L], c(η)=3|base(η)|, and
d(η)={(μ-μ2)|base(η)|+δ6|γu|+δ6|γl|-(μ-μ1-1),if|base(η)|≥2,(μ-μ2)|base(η)|+δ6|γu|+δ6|γl|,if|base(η)|=1,
where μ1 is given in Lemma 4.1.

Proof of Lemma <xref ref-type="statement" rid="lem3">4.2</xref>.

In order to verify the convergence and analyticity, we have to show that there exist two functions
c,d:KL={ξ;decoratedpolymer}⟶[0,∞),
such that the above estimate (4.7) exists. First we consider the statistical properties of
η=(γu,γl,C1,…,Cp,D1,…,Dq,Λ1,…,Λv),
and we have
|γu|=Nv(γu)+Nh(γu)+1,|γl|=Nv(γl)+Nh(γl)+1.
By the definition of the decorated polymers, if base(η) is one point, then
Nh(γu)+Nh(γl)+∑ν=1p|Cν|+∑ν=1q|Dν|+∑α=1v|Λα|≥1,
since {Cν} or {Dν} or {Λα} is nonempty if base(η) is one point. If |base(η)|≥2, then we have
Nh(γu)+Nh(γl)+∑ν=1p|Cν|+∑ν=1q|Dν|+∑α=1v|Λα|≥3(|base(ξ)|-1).
Let (γu,γl) be the backbone of some decorated polymer with the base I=[l̂,r̂]. Next we estimate the following function
G(γu,γl)=∑η;(γu,γl)isthebackboneofη|eμξh(γu)eμζh(γl)Ψ(η)|.
From (2.4) and |Φ(Λ)|≤e-(μ-μ0)|Λ|<1 for large parameter μ0, we have
|e-Φ(Λ)-1|≤e(-μ-μ0-1)|Λ|.

(i) If l̂=r̂, that is |I|=1, then
N*(γu,C1,…,Cp)=N*(γl,D1,…,Dq)=0.
From the (3.11), (4.14), and (4.15), we have
G(γu,γl)≤e-μ|γu|eμh(γu)Reξ∑{Cν};Cνiγue-μ∑ν|Cν|×e-μ|γl|eμh(γl)Reζ∑{Dν};Dνiγle-μ∑ν|Dν|×∑{Λα};Λαiγu∪C1∪⋯∪CporΛαiγl∪D1∪⋯∪Dqe-(μ-μ0-1)∑α|Λα|≤e-μ|γu|+μh(γu)Reξ-μ|γl|+μh(γl)Reζ-(μ-μ1-1)∑{Cν};Cνiγue-μ1∑ν|Cν|×∑{Dν};Dνiγle-μ1∑ν|Dν|∑{Λα};Λαiγu∪C1∪⋯∪CporΛαiγl∪D1∪⋯∪Dqe-(μ1-μ0)∑α|Λα|.
The summation over {Λα} is estimated as follows:
∑{Λα};Λαiγu∪C1∪⋯∪CporΛαiγl∪D1∪⋯∪Dqe-(μ1-μ0)∑α|Λα|≤∑t=0∞1t!∑Λ1iγu∪C1∪⋯∪CporΛ1iγl∪D1∪⋯∪Dq⋯∑Λtiγu∪C1∪⋯∪CporΛtiγl∪D1∪⋯∪Dqe-(μ1-μ0)∑α|Λα|≤exp{4|γu∪C1∪⋯∪Cp∪γl∪D1∪⋯∪Dq|∑Λ∋0;connectede-(μ1-μ0)|Λ|}=exp{(|γu|+|γl|+∑ν|Cν|+∑ν|Dν|)g1(μ1,μ0)}.
There exist constants K1,κ>0 such that the number Nm of connected sets of m points in ℤ2 which contain the origin is bounded as
Nm≤K1κm(m≥1),
then we know that the function
g1(μ1,μ0)=4∑Λ∋0;connectede-(μ1-μ0)|Λ|,
goes to zero exponentially fast as μ1→∞. Thus, from (4.17) and (4.18) we obtain
G(γu,γl)≤e-(μ-g1(μ1,μ0))|γu|+μh(γu)Reξ-(μ-g1(μ1,μ0))|γl|+μh(γl)Reζ×e-(μ-μ1-1)×∑{Cν};Cνiγue-(μ1-g1(μ1,μ0))∑ν|Cν|∑{Dν};Dνiγle-(μ1-g1(μ1,μ0))∑ν|Dν|≤e-(μ-g1(μ1,μ0)-g2(μ1,μ0))|γu|+μh(γu)Reξ×e-(μ-g1(μ1,μ0)-g2(μ1,μ0))|γl|+μh(γl)Reζ×e-(μ-μ1-1),
where
g2(μ1,μ0)=4∑D∋0;connectede-(μ1-g1(μ1,μ0))|D|.

(ii) If l̂<r̂; that is, |I|≥2, then we have
N*(γu,C1,…,Cp)≤Nh(γu)+∑ν=1p|Cν|,N*(γl,D1,…,Dq)≤Nh(γl)+∑ν=1q|Dν|.
From the formula (4.13), we have
G(γu,γl)≤e-μ|γu|eμh(γu)Reξe-μ|γl|eμh(γl)Reζ∑{Cν};Cνiγue-μ∑ν|Cν|2N*(γu,C1,…,Cp)×∑{Dν};Dνiγle-μ∑ν|Dν|2N*(γl,D1,…,Dq)∑{Λα};Λαiγu∪C1∪⋯∪CporΛαiγl∪D1∪⋯∪Dqe-(μ-μ0-1)∑α|Λα|≤e-μ|γu|+μh(γu)Reξ-μ|γl|+μh(γl)Reζ-(μ-μ1-1)(3|I|-Nh(γu)-Nh(γl)-3)2Nh(γu)2Nh(γl)×∑{Cν};Cνiγue-(μ1-ln2)∑ν|Cν|∑{Dν};Dνiγle-(μ1-ln2)∑ν|Dν|×∑{Λα};Λαiγu∪C1∪⋯∪CporΛαiγl∪D1∪⋯∪Dqe-(μ1-μ0)∑α|Λα|.
Further, according to the formula (4.18), we have
G(γu,γl)≤e-(μ-g1(μ1,μ0))|γu|+μh(γu)Reξ-(μ-g1(μ1,μ0))|γl|+μh(γl)Reζe-(μ-μ1-1)(3|I|-Nh(γu)-Nh(γl)-3)×∑{Cν};Cνiγue-(μ1-g1(μ1,μ0)-ln2)∑ν|Cν|2Nh(γu)∑{Dν};Dνiγle-(μ1-g1(μ1,μ0)-ln2)∑ν|Dν|2Nh(γl)≤e-(μ-g1(μ1,μ0)-g3(μ1,μ0))|γu|+μh(γu)Reξ×e-(μ-g1(μ1,μ0)-g3(μ1,μ0))|γl|+μh(γl)Reζ×e-(μ-μ1-1)(3|I|-Nh(γu)-Nh(γl)-3)+Nh(γu)ln2+Nh(γl)ln2,
where
g3(μ1,μ0)=4∑D∋0;connectede-(μ1-g1(μ1,μ0)-ln2)|D|.

We choose μ1(>μ0) sufficiently large, such that
g1(μ1,μ0)<δ4,g2(μ1,μ0)<δ4,g3(μ1,μ0)<δ4.
Assume that Reξ<1-δ/μ and Reζ<1-δ/μ. Since
Nv(γu)≥|h(γu)|,Nv(γl)≥|h(γl)|,
then from the formula (4.11), for |I|≥2, we have
G(γu,γl)≤e-(δ/2)Nv(γu)-(μ1-δ/2)(Nh(γu)+1)e-(δ/2)Nv(γl)-(μ1-δ/2)(Nh(γl)+1)e-(μ-μ1-1)(3|I|-1).
For |I|=1, we have
G(γu,γl)≤e-(δ/2)Nv(γu)-(δ/2)Nv(γu)-2(μ1-δ/2)e-3(μ-μ1-1).

Let c(η), d(η) be defined in (4.7). Since c(η) and d(η) only depend on the backbone (γu,γl), let
c(η)=c(γu,γl),d(η)=d(γu,γl).
Then, for a fixed interval I, we have
∑η;(γu,γl)isthebackboneofηec(η)+d(η)|eμξh(γu)eμζh(γl)Ψ(η)|=G(γu,γl)ec(γu,γl)+d(γu,γl)≤e-(δ/3)Nv(γu)-(μ1-2δ/3)(Nh(γu)+1)e-(δ/3)Nv(γl)-(μ1-2δ/3)(Nh(γl)+1)e-(2μ+μ2-3μ1-6)|base(γu,γl)|,
where base(γu,γl)=base(η) for any η such that (γu,γl) is the backbone of η. Then
∑base(γ)=Iec(γu,γl)+d(γu,γl)G(γu,γl)≤e-(2μ+μ2-3μ1-6)|I|∑base(γ)=Ie-(δ/3)Nv(γu)-(μ1-2δ/3)(Nh(γu)+1)e-(δ/3)Nv(γl)-(μ1-2δ/3)(Nh(γl)+1).
By Lemma 4.1, if Reξ<1-δ/μ,Reζ<1-δ/μ, and μ>μ1, where μ1 is sufficiently large, we have
∑base(ξ)=Iec(γu,γl)+d(γu,γl)G(γu,γl)≤e-(2μ+μ2-3μ1-6)|I|(R(μ1,δ)|I|1-R(μ1,δ))2.
From (3.14), there is a μ3>0 such that
|Q(ξ,ζ)|≥e-2μ-μ3,
then we have
∑base(ξ)=I|Ψ̂(η;ξ,ζ)|ec(γu,γl)+d(γu,γl)≤e-(2μ+μ2-3μ1-6)|I|(R(μ1,δ)|I|1-R(μ1,δ))2e(2μ+μ3)|I|=e-(μ2-3μ1-μ3-6)|I|(R(μ1,δ)|I|1-R(μ1,δ))2.
Let μ1 be large enough, we take μ2>3μ1+μ3+6 and μ>μ2. For a fixed η0 with the base(η0)=[l̂,r̂], then we have
∑ηiη0|Ψ̂(η;ξ,ζ)|ec(η)+d(η)≤∑x∈[l̂-1,r̂+1]∑I∋x(R(μ1,δ)|I|1-R(μ1,δ))2=(r̂-l̂+3)(1-R(μ1,δ))2∑k=1∞kR(μ1,δ)2k≤3|base(η0)|(1-R(μ1,δ))2×R(μ1,δ)2(1-R(μ1,δ)2)2≤c(η0).
This finishes the proof of Lemma 4.2.

Next we give the proof of Theorem 2.1, where the technique of polymer chains and Lemmas 3.1–4.2 are applied to show the limit existence of the free energy for the two-layered lattice W-R model.

Proof of Theorem <xref ref-type="statement" rid="thm1">2.1</xref>.

From Lemma 3.1, we have
1Q(ξ,ζ)L∑Γu,Γl∈SLeμ(ξh(Γu)+ζh(Γl))W(Γu,Γl)=∑C1,…,Cz∈CPL;compatible∏i=1zΨ̂(Ci;ξ,ζ).
In the following part, we will show the existence of the following limit:
φ̂(ξ,ζ)=limL→∞1Lln∑C1,…,Cz∈CPL;compatible∏i=1zΨ̂(Ci;ξ,ζ),
where φ̂(ξ,ζ) is analytic for Reξ<1-δ/μ and Reζ<1-δ/μ. In this case, from the formulas (2.11), (4.38), and (4.39) we have
φ(ξ,ζ)=φ̂(ξ,ζ)+lnQ(ξ,ζ),
which is analytic in this region. According to (4.40), in order to demonstrate Theorem 2.1 (or to prove the existence of (2.11)), it is sufficient to show the limit existence of the above (4.39).

Next we estimate the weight of the polymer chains. We call a family of intervals
I1=[l̂1,r̂1],…,In=[l̂n,r̂n],
the linked intervals if r̂u<l̂u+1<r̂u+2 for each 1≤u≤n. So that the base of a polymer chain forms the linked intervals. For a fixed polymer chain 𝒞0, let [base(𝒞0)]=[l̂0,r̂0] be the smallest intervals (in length) including base(𝒞0). Let
c*(C)=∑η∈Cc(η),d*(C)=∑η∈Cd(η),
where c(η) and d(η) are defined in Lemma 4.2. Then, for the functions
c*,d*:CP={C;polymerchain}⟶[0,∞),
we show that
∑C∈CPL;CiC0|Ψ̂(C;ξ,ζ)|ec*(C)+d*(C)≤c*(C0),
for any polymer chain 𝒞0 and for any ξ,ζ∈ℂ with Reξ<1-δ/μ and Reζ<1-δ/μ. Noting that the distance of base(𝒞0) and base(𝒞) is less than 2 if 𝒞0 and 𝒞 are incompatible, then we have
∑C;CiC0|Ψ̂(C;ξ,ζ)|ec*(C)+d*(C)=∑C;CiC0|Ψ̂(C;ξ,ζ)|exp{∑η∈Cc(η)+∑η∈Cd(η)}≤∑x∈[l̂0-2,r̂0+2]∑n=1∞∑I1,…,In⊂[0,L];∪Iu∋xlinkedintervals∑η1,…,ηn∈KL;base(ηu)=Iu,1≤u≤n∏u=1n[Ψ̂(ηu;ξ,ζ)ec(ηu)+d(ηu)]Jl(η1)Jr(ηn)∏u=1n-1J(ηu,ηu+1).
From definitions (3.18)–(3.21) and the formula (4.35), there exists μ3′>0 such that |𝒥l|, |𝒥r|, and |𝒥| are all bounded from above by eμ3′ if Reξ<1-δ/μ and Reζ<1-δ/μ. Therefore, from (4.36) or Lemma 4.2, we have that
∑η1,…,ηn∈KL;base(ηu)=Iu,1≤u≤n∏u=1n[Ψ̂(ηu;ξ,ζ)ec(ηu)+d(ηu)]Jl(η1)Jr(ηn)∏u=1n-1J(ηu,ηu+1)≤∏u=1ne-(μ2-3μ1-μ3-2μ3′-6)|Iu|(R(μ1,δ)|Iu|1-R(μ1,δ))2.
Assuming that μ2>3μ1+μ3+2μ3′+6, then, from the inequalities (4.45) and (4.46) and if μ1 is large enough (by following the similar estimate procedure of (4.37) in Lemma 4.2), we obtain
∑C;CiC0|Ψ̂(C;ξ,ζ)|ec*(C)+d*(C)≤(r̂0-l̂0+4)∑n=1∞∑u=1n∑I1,…,In⊂[0,L];∪Iu∋xlinkedintervals∏u=1n(R(μ1,δ)|Iu|1-R(μ1,δ))2≤(r̂0-l̂0+4)R(μ1,δ)2(1-R(μ1,δ))2(1-R(μ1,δ)2)2∑n=1∞n(2R(μ1,δ)2(1-R(μ1,δ)2)2)n-1≤(r̂0-l̂0+4)2≤c*(C0).
Since ∑η∈𝒞0|base(η)|≥max{(2/3)[base(𝒞0)],1} and, by the definition c*(𝒞)=∑η∈𝒞c(η) and c(η)=3|base(η)| (which is given in Lemma 4.2), the above last inequality holds.

According to the above inequality (4.44), we apply the general theory of cluster expansion for the partition function to display the following results; for the details, see [19]. Let 𝒫f(𝒞𝒫) be the collection of all finite subsets of 𝒞𝒫, so that there exists a function
Ψ̂T:Pf(CP)×C2⟶C2,
such that Ψ̂T is analytic for Reξ<1-δ/μ and Reζ<1-δ/μ, and it satisfies
∑C1,…,Cz∈CPL;compatible∏i=1zΨ̂(Ci;ξ,ζ)=exp{∑Δ∈Pf(CPL)Ψ̂T(Δ;ξ,ζ)},∑ΔiC0|Ψ̂T(Δ;ξ,ζ)|ed*(Δ)≤c*(C0),
where d*(Δ)=∑𝒞∈Δd*(𝒞). If Δ is decomposed into two disjoint subsets Δ1 and Δ2, such that, for each pair 𝒞1∈Δ1,𝒞2∈Δ2, {𝒞1,𝒞2} are compatible, then Ψ̂T(Δ;ξ,ζ)=0. We call Δ∈𝒫f(𝒞𝒫L) a cluster if there are no such decomposition Δ=Δ1∪Δ2. Note that Ψ̂T(Δ;ξ,ζ) is invariant under horizontal translation of Δ. For Δ∈𝒫f(𝒞𝒫), we set base(Δ)=∪𝒞∈Δbase(𝒞). Then, by the cluster expansion theory of [19] and (4.49), we have that the limit
φ̂(ξ,ζ)=limL→∞1Lln∑C1,…,Cz∈CPL;compatible∏i=1zΨ̂(Ci;ξ,ζ)=∑Δ∈Pf(CP);[base(Δ)]=[0,k]forsomek≥0Ψ̂T(Δ;ξ,ζ),
exists and is analytic for Reξ<1-δ/μ,Reζ<1-δ/μ if μ>μ2.

From (4.40) and (4.50), we complete the proof of Theorem 2.1.

Acknowledgments

The authors were supported in part by National Natural Science Foundation of China Grant nos. 70471001, 70771006, and 10971010, and by the BJTU Foundation Grant no. S11M00010.

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