Analysis of Two-Layered Random Interfaces for Two Dimensional Widom-Rowlinson's Model

Abstract

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 Λ_𝐿,𝑀, where Λ_𝐿,𝑀=[1,𝐿−1]×[−𝑀,𝑀]⊂ℤ². Suppose that the particles in Λ_𝐿,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 Ω_𝐿,𝑀 denote the set of all feasible configurations in Λ_𝐿,𝑀, so there is a finite diameter hard-core exclusion between 𝐴 particles and 𝐵 particles on Ω_𝐿,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 𝐻_𝐿,𝑀(𝜎)=−_{𝑥∈Λ𝐿,𝑀}𝜇𝜎(𝑥)²−1,(1.1) for all 𝜎∈Ω_𝐿,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 𝜔^𝑏 as 𝜔^𝑏(𝑥)=𝜔^𝑏𝑥¹,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,(1.2) for each 𝑥=(𝑥¹,𝑥²)∈𝜕Λ_𝐿,𝑀=[0,𝐿]×[−𝑀−1,𝑀+1]⧵Λ_𝐿,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 Ω^𝑏_𝐿,𝑀(⊂Ω_𝐿,𝑀) be the corresponding configuration space with the conditions (i)–(iii), such that the configuration 𝜎×𝜔^𝑏 is feasible, where 𝜎×𝜔^𝑏(𝑥)=𝜎(𝑥) for 𝑥∈Λ_𝐿,𝑀 and 𝜎×𝜔^𝑏(𝑥)=𝜔^𝑏(𝑥) for 𝑥∈𝜕Λ_𝐿,𝑀. For a fixed configuration 𝜎∈Ω^𝑏_𝐿,𝑀, let 𝑆⁰(𝜎) denote the set of points in Λ_𝐿,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 Γ^𝑢(𝜎) and Γ^𝑙(𝜎) which are defined as the interfaces of the model. Γ^𝑢(𝜎) is the upper interface with the starting point (0,𝑏₁) and the ending point (𝐿,𝑏₂); Γ^𝑙(𝜎) is the lower interface with the starting point (0,−𝑏₁) and the ending point (𝐿,−𝑏₂). Let 𝒮^𝑏,𝑢_𝐿,𝑀=Γ^𝑢(𝜎);𝜎∈Ω^𝑏_𝐿,𝑀,𝒮^𝑏,𝑙_𝐿,𝑀=Γ^𝑙(𝜎);𝜎∈Ω^𝑏_𝐿,𝑀(1.3) be the set of upper interfaces and the set of lower interfaces, respectively. The conditional Gibbs distribution on Ω^𝑏_𝐿,𝑀 with the boundary condition 𝜔^𝑏 is given by 𝑃^𝑏_𝐿,𝑀(𝜎)=𝑍^𝑏_𝐿,𝑀⁻¹exp−𝜇𝑆⁰(𝜎),(1.4) where |𝑆| denotes the cardinality of a set 𝑆, and 𝑍^𝑏_𝐿,𝑀 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(Γ^𝑢) and 𝜋(Γ^𝑙) of interfaces Γ^𝑢 and Γ^𝑙 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(Γ^𝑢) of Γ^𝑢 to represent Γ^𝑢; that is, among self-avoiding paths connecting the starting point (0,𝑏₁) with the ending point (𝐿,𝑏₂) in Γ^𝑢, we select a self-avoiding path (called backbone 𝜋(Γ^𝑢)) by an “order” given in the following definition (6). Since the backbone 𝜋(Γ^𝑢) 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 Γ^𝑢 as Π_Γ^𝑢=Thesetofself-avoidingpathinΓ^𝑢connecting0,𝑏₁with𝐿,𝑏₂.(2.1) 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.(2.2) More precisely, let 𝜋_𝑥={𝑥₁,…,𝑥_𝑛} and 𝜋_𝑦={𝑦₁,…,𝑦_𝑚} be the two self-avoiding paths in Γ. Let 𝑘=min_𝑖≥1{𝑖∶𝑥_𝑖≠𝑦_𝑖} be the first number 𝑖 such that 𝑥_𝑖≠𝑦_𝑖. We define that 𝜋_𝑥>𝜋_𝑦 if the direction of the ordered edge {𝑥_𝑘−1,𝑥_𝑘} is preferred to the direction of the ordered edge {𝑦_𝑘−1,𝑦_𝑘}. Let 𝜋(Γ^𝑢) be the unique maximal element of Π_Γ^𝑢 with respect to this order, and call 𝜋(Γ^𝑢) the backbone of Γ^𝑢. Similarly, let 𝜋(Γ^𝑙) represent the backbone of Γ^𝑙. In this paper, our study is mainly focused on the backbone of the phase separation belt.

Let 𝑈₁,…,𝑈_𝑛 be the different connected subsets of Λ_𝐿,𝑀, we say that the subsets {𝑈_𝑗} are compatible if they are connected components of the set ∪_{1≤𝑗≤𝑛}𝑈_𝑗. We also say that {𝑈_𝑗} are compatible with a connected set 𝐺 if {𝐺,𝑈_𝑗} 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,…,𝑧_𝑛=𝑦 in 𝐷 such that |𝑧_𝑖−𝑧_𝑖−1|≤√2 for every 1≤𝑖≤𝑛. A hole of a connected set 𝐹⊂ℤ² is a finite ∗-connected component of 𝐹^𝑐=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 𝑍^𝑏_𝐿,𝑀 defined in (1.4) can be rewritten by the similar formulas in [1–3] as following: 𝑍^𝑏_𝐿,𝑀=_{Γ𝑢∈𝒮𝑏,𝑢𝐿,𝑀,Γ𝑙∈𝒮𝑏,𝑙𝐿,𝑀𝑈𝑗}2^{𝑁(Γ𝑢)+𝑁(Γ𝑙)}𝑒^{−𝜇(|Γ𝑢|+|Γ𝑙|)}_𝑗2^{𝑁(𝑈𝑗)}𝑒^{−𝜇|𝑈𝑗|},(2.3) where the second summation is taken over compatible families {𝑈_𝑗}, which are compatible with Γ^𝑢∪Γ^𝑙, |Γ^𝑢| is the number of points in Γ^𝑢, and 𝑁(Γ^𝑢) is the number of holes in Γ^𝑢, similar to Γ^𝑙, 𝑁(Γ^𝑙), 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 𝑍^𝑏_𝐿,𝑀𝑍⁺_𝐿,𝑀=_{Γ𝑢∈𝒮𝑏,𝑢𝐿,𝑀,Γ𝑙∈𝒮𝑏,𝑙𝐿,𝑀}×exp−𝜇Γ^𝑢+Γ^𝑙+ln2𝑁Γ^𝑢+𝑁Γ^𝑙−_{Λ⊂Λ𝐿,𝑀Γ∶Λ𝑖𝑢∪Γ𝑙}Φ(Λ),(2.4) where 𝑍⁺_𝐿,𝑀 is the partition function with the plus boundary condition, Λ𝑖(Γ^𝑢∪Γ^𝑙) denotes that the set Λ is incompatible with the interfaces Γ^𝑢∪Γ^𝑙, and Φ(Λ) is a translation invariant function, which satisfies the following estimate: _Λ;Λ∋0|Φ(Λ)|𝑒^{(𝜇−𝜇0)|Λ|}<1.(2.5) Moreover, if 𝜇 is sufficiently large, we have lim_𝑀→∞𝑍^𝑏_𝐿,𝑀𝑍⁺_𝐿,𝑀=_{Γ𝑢∈𝒮𝐿𝑏,𝑢,Γ𝑙∈𝒮𝐿𝑏,𝑙}×exp−𝜇Γ^𝑢+Γ^𝑙+ln2𝑁Γ^𝑢+𝑁Γ^𝑙−_{Λ⊂Λ𝐿,∞Γ∶Λ𝑖𝑢∪Γ𝑙}Φ(Λ),(2.6) where Λ_𝐿,∞=[1,𝐿−1]×(−∞,∞)∩ℤ² and 𝒮_𝐿^𝑏,𝑢=∪_𝑀>0𝒮^𝑏,𝑢_𝐿,𝑀,𝒮_𝐿^𝑏,𝑙=∪_𝑀>0𝒮^𝑏,𝑙_𝐿,𝑀.(2.7) Let 𝑊Γ^𝑢,Γ^𝑙=exp−𝜇Γ^𝑢+Γ^𝑙+ln2𝑁Γ^𝑢+𝑁Γ^𝑙−_{Λ⊂Λ𝐿,∞Γ∶Λ𝑖𝑢∪Γ𝑙}Φ(Λ),(2.8) 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,∞Γ∶Λ𝑖𝑢∪Γ𝑙}Φ(Λ)=^∞_{𝑛=0Λ1,…,Λ𝑛⊂Λ𝐿,∞Λ𝜈𝑖Γ𝑢∪Γ𝑙}^𝑛_𝜈=1𝑒^{−Φ(Λ𝜈)}−1.(2.9)

In the definition of boundary condition 𝜔^𝑏 (see (1.2)), for a fixed 𝑏₁≥0, let 𝒮_𝐿=(∪_𝑏𝒮_𝐿^𝑏,𝑢,∪_𝑏𝒮_𝐿^𝑏,𝑙), then, for the interfaces vector (Γ^𝑢,Γ^𝑙)∈𝒮_𝐿, the height of the last ending point of (Γ^𝑢,Γ^𝑙) is defined as the vector ℎΓ^𝑢,ℎΓ^𝑙=ℎ𝜋Γ^𝑢,ℎ𝜋Γ^𝑙.(2.10) The aim of this paper is to study the statistical behaviors of the free energy of the height vector (ℎ(Γ^𝑢),ℎ(Γ^𝑙)). The following theorem shows the limit existence of the free energy of the height for the interface vector (Γ^𝑢,Γ^𝑙).

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)=lim_𝐿→∞1𝐿ln_{Γ𝑢,Γ𝑙∈𝒮𝐿}𝑒^{𝜇(𝜉ℎ(Γ𝑢)+𝜁ℎ(Γ𝑙))}𝑊Γ^𝑢,Γ^𝑙,(2.11) 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 (Γ^𝑢,Γ^𝑙). Further, we show that the asymptotical behavior of the backbone 𝜋(Γ^𝑢) can represent that of the corresponding interface Γ^𝑢 when 𝜇 is large enough; this means that the statistical properties of the interface Γ^𝑢 are similar to those of its backbone 𝜋(Γ^𝑢). From the definition of 𝜋(Γ^𝑢), 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 (Γ^𝑢,Γ^𝑙)∈𝒮_𝐿, let (0,𝑏₁) and (𝐿,𝑘_𝑢) be the starting and the ending points of Γ^𝑢; let 𝜋(Γ^𝑢) be the backbone of Γ^𝑢 connecting (0,𝑏₁) and (𝐿,𝑘_𝑢). Similarly, let (0,−𝑏₁) and (𝐿,𝑘_𝑙) be the starting and the ending points of Γ^𝑙; let 𝜋(Γ^𝑙) be the backbone of Γ^𝑙 connecting (0,−𝑏₁) and (𝐿,𝑘_𝑙). We decompose Γ^𝑢⧵𝜋(Γ^𝑢) into connected components {𝐶_𝑗}^𝑠_𝑗=1 and decompose Γ^𝑙⧵𝜋(Γ^𝑙) into connected components {𝐷_𝑗}^𝑟_𝑗=1. Then, from (2.6) and (2.9), we have_{Γ𝑢,Γ𝑙∈𝒮𝐿}𝑒^{𝜇(𝜉ℎ(Γ𝑢)+𝜁ℎ(Γ𝑙))}𝑊Γ^𝑢,Γ^𝑙=_{𝑘𝑢−𝑘𝑙>1−∞≤𝑘𝑙,𝑘𝑢≤+∞𝜋;(0,𝑏1)→(𝐿,𝑘𝑢)𝜋′;(0,−𝑏1)→(𝐿,𝑘𝑙)𝜋liesabove𝜋′𝐶1,…,𝐶𝑠;compatible𝐶𝜈𝑖𝜋,𝐶𝜈∩𝜋=∅𝜋;backboneof𝜋∪𝐶1∪⋯∪𝐶𝑠𝐷1,…,𝐷𝑟;compatible𝐷𝜈𝑖𝜋′,𝐷𝜈∩𝜋=∅𝜋′;backboneof𝜋′∪𝐷1∪⋯∪𝐷𝑠}×exp𝜇𝜉𝑘_𝑢−𝜇|𝜋|+𝑁𝜋,𝐶₁,…,𝐶_𝑠ln2−𝜇^𝑠_𝜈=1𝐶_𝜈×exp𝜇𝜁𝑘_𝑙−𝜇𝜋^′+𝑁𝜋^′,𝐷₁,…,𝐷_𝑟ln2−𝜇^𝑟_𝜈=1𝐷_{𝜈Λ1,…,Λ𝑡;connectedΛ𝛼𝑖𝜋∪𝐶1∪⋯∪𝐶𝑠orΛ𝛼𝑖𝜋′∪𝐷1∪⋯∪𝐷𝑟}^𝑡_𝛼=1𝑒^{−Φ(Λ𝛼)}−1,(3.1) where 𝑁(𝜋,𝐶₁,…,𝐶_𝑠) denotes the number of holes of 𝜋∪∪^𝑠_𝜈=1𝐶_𝜈, 𝑁(𝜋′,𝐷₁,…,𝐷_𝑟) denotes the number of holes of 𝜋′∪∪^𝑟_𝜈=1𝐷_𝜈, 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=𝛾^𝑢,𝛾^𝑙,𝐶₁,…,𝐶_𝑝,𝐷₁,…,𝐷_𝑞,Λ₁,…,Λ_𝑣,(3.2) is called a polymer with base [𝑎,𝑐] if 𝜂 satisfies the following conditions (1)–(4).(1)𝛾^𝑢,𝛾^𝑙 are self-avoiding paths in {𝑎≤𝑥¹≤𝑐}, 𝛾^𝑢 starts from (𝑎,𝑎^𝑢) and ending at a point (𝑐,𝑐^𝑢) in {𝑥¹=𝑐}, 𝛾^𝑙 starts from (𝑎,𝑎^𝑙) and ending at a point (𝑐,𝑐^𝑙) in {𝑥¹=𝑐}, where 𝑎^𝑢, 𝑎^𝑙, 𝑐^𝑢, 𝑐^𝑙 are fixed integers satisfying 𝑎^𝑢−𝑎^𝑙>1 and 𝑐^𝑢−𝑐^𝑙>1, and 𝛾^𝑢 lies above 𝛾^𝑙.(2){𝐶_𝜈}^𝑝_𝜈=1 is a compatible family of connected subsets of {𝑥∈Λ_𝐿,∞;𝑎≤𝑥¹≤𝑐} such that (i) 𝐶_𝜈∩𝑉(𝛾^𝑢)=∅, where 𝑉(𝛾^𝑢) is the set of vertices in 𝛾; (ii) 𝐶_𝜈∪𝑉(𝛾^𝑢) is connected; (iii) 𝛾^𝑢 is the backbone of 𝛾^𝑢∪𝐶₁∪⋯∪𝐶_𝑝 with starting point (𝑎,𝑎^𝑢) and ending point (𝑐,𝑐^𝑢). Similarly, {𝐷_𝜈}^𝑞_𝜈=1 and 𝛾^𝑙 have the same properties.(3){Λ_𝛼}^𝑣_𝛼=1 is a collection of connected subsets of {𝑥∈Λ_𝐿,∞;𝑎≤𝑥¹≤𝑐} such that Λ_𝛼𝑖𝛾^𝑢∪∪^𝑝_𝜈=1𝐶_𝜈orΛ_𝛼𝑖𝛾^𝑙∪∪^𝑞_𝜈=1𝐷_𝜈.(3.3)(4)For 𝑎≤𝑗<𝑐, 𝑗∈ℕ, the line ℓ_𝑗={𝑥¹=𝑗+1/2} intersects at least two edges of 𝛾^𝑢∪ℰ(∪^𝑝_𝜈=1𝐶_𝜈∪∪^𝑣_𝛼=1Λ_𝛼)∪ℰ(𝛾^𝑢,∪^𝑝_𝜈=1𝐶_𝜈∪∪^𝑣_𝛼=1Λ_𝛼) or at least two edges of 𝛾^𝑙∪ℰ(∪^𝑞_𝜈=1𝐷_𝜈∪∪^𝑣_𝛼=1Λ_𝛼)∪ℰ(𝛾^𝑙,∪^𝑞_𝜈=1𝐷_𝜈∪∪^𝑣_𝛼=1Λ_𝛼). 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 (𝛾^𝑢,𝛾^𝑙) the backbone of 𝜂. For two disjoint self-avoiding paths 𝛾^𝑢₁,𝛾^𝑢₂ (similarly for 𝛾^𝑙₁,𝛾^𝑙₂) such that the starting point of 𝛾^𝑢₂ is the nearest neighbor of the endpoint of 𝛾^𝑢₁, we can define the concatenation 𝛾^𝑢₁∘𝛾^𝑢₂ of these paths by simply connecting them. Let 𝜂₁=𝛾^𝑢₁,𝛾^𝑙₁,𝐶₁,…,𝐶_𝑝,𝐷₁,…,𝐷_𝑞,Λ₁,…,Λ_𝑣,𝜂₂=𝛾^𝑢₂,𝛾^𝑙₂,𝐶^′₁,…,𝐶^′_𝑢,𝐷^′₁,…,𝐷^′_𝑤,Λ^′₁,…,Λ^′_𝑧(3.4) 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:(a)𝑐+1<𝑎′,(b)𝑎′=𝑐+1, the backbone of Γ=𝛾^𝑢₁∪𝐶₁∪⋯∪𝐶_𝑝∪𝛾^𝑢₂+0,ℎ𝛾^𝑢₁∪𝐶^′₁+0,ℎ𝛾^𝑢₁∪⋯∪𝐶^′_𝑢+0,ℎ𝛾^𝑢₁𝛾^𝑙₁∪𝐷₁∪⋯∪𝐷_𝑞∪𝛾^𝑙₂+0,ℎ𝛾^𝑙₁∪𝐷^′₁+0,ℎ𝛾^𝑙₁∪⋯∪𝐷^′_𝑤+0,ℎ𝛾^𝑙₁,(3.5) is the concatenation (𝛾^𝑢₁∘(𝛾^𝑢₂+(0,ℎ(𝛾^𝑢₁))),𝛾^𝑙₁∘(𝛾^𝑙₂+(0,ℎ(𝛾^𝑙₁)))), and connected components of the set Γ⧵(𝛾^𝑢₁∘(𝛾^𝑢₂+(0,ℎ(𝛾^𝑢₁)))∪𝛾^𝑙₁∘(𝛾^𝑙₂+(0,ℎ(𝛾^𝑙₁)))) are {𝐶₁,…,𝐶_𝑝,𝐶^′₁,…,𝐶^′_𝑢} and {𝐷₁,…,𝐷_𝑞,𝐷^′₁,…,𝐷^′_𝑤}. 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 {𝜂_𝑝}^𝑛+1_𝑝=0 is compatible if 𝜂_𝑝 and 𝜂_𝑝′ (𝑝≠𝑝′) are compatible. Let 𝜂=𝜋Γ^𝑢,𝜋Γ^𝑙,𝐶₁,…,𝐶_𝑠,𝐷₁,…,𝐷_𝑟,Λ₁,…,Λ_𝑡.(3.6) An edge 𝑒={𝑥,𝑦} of 𝜂 is not admissible if it is a horizontal edge in ℰ(𝜋(Γ^𝑢),∪^𝑠_𝜈=1𝐶_𝜈∪∪^𝑡_𝛼=1Λ_𝛼)∪ℰ(𝜋(Γ^𝑙),∪^𝑟_𝜈=1𝐷_𝜈∪∪^𝑡_𝛼=1Λ_𝛼), such that(i)the left vertex 𝑥 is in a connected 𝐸 of ∪^𝑠_𝜈=1𝐶_𝜈∪∪^𝑟_𝜈=1𝐷_𝜈∪∪^𝑡_𝛼=1Λ_𝛼, and the right vertex 𝑦 is in 𝑉(𝜋(Γ^𝑢))∪𝑉(𝜋(Γ^𝑢));(ii)further, there exists a horizontal edge 𝑒′={𝑥^′,𝑦^′} of 𝜂 such that 𝑥′∈𝑉(𝜋(Γ^𝑢))∪𝑉(𝜋(Γ^𝑢)) and 𝑦′∈𝐸, where 𝑥′ is the left vertex of 𝑒′.

Other edges of 𝜂 are admissible.

We say that the line ℓ_𝑗={𝑥¹=𝑗+1/2} (0≤𝑗≤𝐿−1) is the cutting line of 𝜂 if ℓ_𝑗 intersects only two admissible edges of 𝜂=(𝜋(Γ^𝑢),𝜋(Γ^𝑙),{𝐶_𝜈}^𝑠_𝜈=1,{𝐷_𝜈}^𝑟_𝜈=1,{Λ_𝛼}^𝑡_𝛼=1). Here, one of two admissible edges is connected to 𝜋(Γ^𝑢); the other is connected to 𝜋(Γ^𝑙).

Let ℓ₀<ℓ_𝑗1<⋯<ℓ_𝑗𝑛<ℓ_𝑗𝑛+1=ℓ_𝐿−1 be all the cutting lines of (𝜋(Γ^𝑢),𝜋(Γ^𝑙), {𝐶_𝜈}^𝑠_𝜈=1,{𝐷_𝜈}^𝑟_𝜈=1, {Λ_𝛼}^𝑡_𝛼=1). For each 𝑚∈{0,1,…,𝑛+1}, there are only two edges 𝑒^𝑢_𝑚={𝐵^𝑢_𝑚,𝐴^𝑢_𝑚+1} and 𝑒^𝑙_𝑚={𝐵^𝑙_𝑚,𝐴^𝑙_𝑚+1} of 𝜋(Γ^𝑢), and 𝜋(Γ^𝑙), respectively, which intersect ℓ_𝑗𝑚. Let 𝛾^𝑢_𝑚 be the portion of 𝜋(Γ^𝑢) starting from 𝐴^𝑢_𝑚 and ending at 𝐵^𝑢_𝑚; let 𝛾^𝑙_𝑚 be the portion of 𝜋(Γ^𝑙) starting from 𝐴^𝑙_𝑚 and ending at 𝐵^𝑙_𝑚. Also let {𝐶_𝜈^(𝑚)}^𝑠(𝑚)_𝜈=1, {𝐷_𝜈^(𝑚)}^𝑟(𝑚)_𝜈=1, and {Λ_𝛼^(𝑚)}^𝑡(𝑚)_𝛼=1 be the set of elements of {𝐶_𝜈}^𝑠_𝜈=1, {𝐷_𝜈}^𝑟_𝜈=1, and {Λ_𝛼}^𝑡_𝛼=1, respectively, such that they are subsets of [𝑗_𝑚−1+1,𝑗_𝑚]×(−∞,∞)∩ℤ². Then 𝐴^𝑢_𝑚=(𝑗_𝑚−1+1,𝑝^𝑢), 𝐴^𝑙_𝑚=(𝑗_𝑚−1+1,𝑝^𝑙) for some 𝑝^𝑢,𝑝^𝑙∈ℤ, where 𝑝^𝑢>𝑝^𝑙. Thus, we obtain the 𝑚th polymer 𝜂_𝑚 by setting 𝜂_𝑚=𝛾^𝑢_𝑚−0,𝑝^𝑢,𝛾^𝑙_𝑚−0,𝑝^𝑢,𝐶_𝜈^(𝑚)−0,𝑝^{𝑢𝑠(𝑚)}_𝜈=1,𝐷_𝜈^(𝑚)−0,𝑝^{𝑢𝑟(𝑚)}_𝜈=1,Λ_𝛼^(𝑚)−0,𝑝^{𝑢𝑡(𝑚)}_𝛼=1.(3.7) By the above definitions, {𝜂₀,𝜂₁,…,𝜂_𝑛+1} are compatible.

For a polymer 𝜂_𝑚=𝛾^𝑢_𝑚,𝛾^𝑙_𝑚,𝐶_𝜈^(𝑚),𝐷_𝜈^(𝑚),Λ_𝛼^(𝑚),(3.8) let ℎ^𝑢_𝑚=ℎ(𝜂^𝑢_𝑚)=ℎ(𝛾^𝑢_𝑚) be the height of the endpoint of the self-avoiding path 𝛾^𝑢_𝑚; similarly let ℎ^𝑙_𝑚=ℎ(𝜂^𝑙_𝑚)=ℎ(𝛾^𝑙_𝑚). Then the heights ℎ(𝜋(Γ^𝑢)) and ℎ(𝜋(Γ^𝑙)) are given by ℎ𝜋Γ^𝑢=^𝑛+1_𝑚=0ℎ𝛾^𝑢_𝑚,ℎ𝜋Γ^𝑙=^𝑛+1_𝑚=0ℎ𝛾^𝑙_𝑚.(3.9) Now we introduce a statistical weight of a polymer 𝜂′=𝛾^𝑢,𝛾^𝑙,𝐶_𝜈^𝑝_𝜈=1,𝐷_𝜈^𝑞_𝜈=1,Λ_𝛼^𝑣_𝛼=1,(3.10) by setting Ψ𝜂^′=exp−𝜇𝛾^𝑢−𝜇𝛾^𝑙+𝑁^∗𝛾^𝑢,𝐶₁,…,𝐶_𝑝ln2+𝑁^∗𝛾^𝑙,𝐷₁,…,𝐷_𝑞ln2−𝜇^𝑝_𝜈=1𝐶_𝜈−𝜇^𝑞_𝜈=1𝐷_𝜈×^𝑣_𝛼=1𝑒^{−Φ(Λ𝛼)}−1,(3.11) where 𝑁^∗𝛾^𝑢,𝐶₁,…,𝐶_𝑝=𝑁𝛾^𝑢,𝐶₁,…,𝐶_𝑝̂𝑙+𝑁𝛾^𝑢,𝐶₁,…,𝐶_𝑝+𝑁𝑟̂𝛾^𝑢,𝐶₁,…,𝐶_𝑝,(3.12) and 𝑁̂𝑙(𝛾^𝑢,𝐶₁,…,𝐶_𝑝) is the number of new holes created by 𝑉(𝛾^𝑢)∪∪^𝑝_𝜈=1𝐶_𝜈 and the line {𝑥¹=𝑙̂−1}, where 𝑏𝑎𝑠𝑒(𝜂′)=[𝑙,̂𝑟]̂; 𝑁𝑟(̂𝛾^𝑢,𝐶₁,…,𝐶_𝑝) is the number of new holes created by 𝑉(𝛾^𝑢)∪∪^𝑝_𝜈=1𝐶_𝜈 and the line {𝑥¹=𝑟̂+1}. Similarly, we can give the definition of 𝑁^∗(𝛾^𝑙,𝐷₁,…,𝐷_𝑞).

A polymer 𝜂 is called simple if 𝑏𝑎𝑠𝑒(𝜂) is one point and 𝜂=(𝛾^𝑢,𝛾^𝑙,∅,∅,∅). Thus, the weight Ψ(𝜂) is given by Ψ𝜂=𝑒^{−𝜇|𝛾𝑢|−𝜇|𝛾𝑙|}.(3.13) A polymer 𝜂 is called decorated if it is not simple. A decorated polymer 𝜂=(𝛾^𝑢,𝛾^𝑙,{𝐶_𝜈},{𝐷_𝜈},{Λ_𝛼}) with 𝑏𝑎𝑠𝑒(𝜂)=[𝑙,̂𝑟]̂ is said to be 𝑟-active if there exists a simple polymer 𝜂₁=(𝛾^𝑢₁,𝛾^𝑙₁,∅,∅,∅) with 𝑏𝑎𝑠𝑒(𝜂₁)={𝑟̂+1} such that 𝜂₁ is incompatible with 𝜂, or the concatenation of 𝛾^𝑢 and 𝛾^𝑢₁ together with ∪_𝜈𝐶_𝜈 produces a new hole, or the concatenation of 𝛾^𝑙 and 𝛾^𝑙₁ together with ∪_𝜈𝐷_𝜈 produces a new hole. 𝜂 is said to be 𝑙-active if there exists a simple polymer 𝜂₂=(𝛾^𝑢₂,𝛾^𝑙₂,∅,∅,∅) with 𝑏𝑎𝑠𝑒(𝜂₂)={𝑟̂−1} such that 𝜂₂ is incompatible with 𝜂, the concatenation of 𝛾^𝑢 and 𝛾^𝑢₂ together with ∪_𝜈𝐶_𝜈 produces a new hole, or the concatenation of 𝛾^𝑙 and 𝛾^𝑙₂ together with ∪_𝜈𝐷_𝜈 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,…,𝜂_𝑚} such that(1){𝜂₁,…,𝜂_𝑚} are compatible;(2)if 𝑏𝑎𝑠𝑒(𝜂_𝑢)=[𝑙_𝑢̂,𝑟_𝑢̂], 1≤𝑢≤𝑚, then 𝑙̂_𝑢+1=𝑟_𝑢̂+1 or 𝑟_𝑢̂+2 for each 𝑢;(3)if 𝑙̂_𝑢+1=𝑟_𝑢̂+2 for some 𝑢, then 𝜂_𝑢 is 𝑟-active and 𝜂_𝑢+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)=[𝑙_𝑢̂,𝑟_𝑢̂] and 𝑏𝑎𝑠𝑒(𝒞₂)=[𝑙^′_𝑢̂,𝑟^′_𝑢̂] have 𝑙^′_𝑢̂−𝑟_𝑢̂>2, then 𝒞₁∪𝒞₂ is compatible polymers, but not a polymer chain.

Let 𝒦=𝒦_𝐿 be the set of all decorated polymers with base in [0,𝐿], and let 𝒞𝒫_𝐿 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≤+∞}𝑒^{𝜇𝜉𝑘𝑢}𝑒^{−|𝑘𝑢|𝜇}𝑒^{𝜇𝜁𝑘𝑙}𝑒^{−|𝑘𝑙|𝜇},(3.14) then one has 1𝑄(𝜉,𝜁)^𝐿_{Γ𝑢,Γ𝑙∈𝒮𝐿}𝑒^{𝜇(𝜉ℎ(Γ𝑢)+𝜁ℎ(Γ𝑙))}𝑊Γ^𝑢,Γ^𝑙=_{𝒞1,…,𝒞𝑧∈𝒞𝒫𝐿;compatible}^𝑧_𝑖=1Ψ𝒞_𝑖;𝜉,𝜁,(3.15) where Ψ(𝒞_𝑖,𝜉,𝜁) is the weight function of polymer chains which is given (3.17).

Proof of Lemma 3.1. Considering a polymer chain 𝒞={𝜂₁,…,𝜂_𝑚}, let 𝑏𝑎𝑠𝑒(𝒞)=𝑏𝑎𝑠𝑒(𝜂₁)∪⋯∪𝑏𝑎𝑠𝑒(𝜂_𝑚). Further, for a polymer 𝜂, from (3.9) and (3.11), we define Ψ𝜂;𝜉,𝜁=𝑒^{𝜇𝜉ℎ(𝜂𝑢)}𝑒^{𝜇𝜁ℎ(𝜂𝑙)}Ψ𝜂𝑄(𝜉,𝜁)^{−|𝑏𝑎𝑠𝑒(𝜂)|},(3.16) where |𝑏𝑎𝑠𝑒(𝜂)|=𝑟̂−𝑙̂+1 for 𝑏𝑎𝑠𝑒(𝜂)=[𝑙,̂𝑟]̂. By the formula (3.16), for a polymer chain 𝒞={𝜂₁,…,𝜂_𝑚}, we put Ψ𝒞_𝑖;𝜉,𝜁=^𝑚_𝑢=1Ψ𝜂_𝑢̂𝑙;𝜉,𝜁×𝒥𝜂₁𝒥𝑟̂𝜂_𝑚^𝑚−1_𝑢=1𝒥𝜂_𝑢,𝜂_𝑢+1,(3.17) where 𝒥̂𝑙, 𝒥𝑟̂, 𝒥 are defined in the following way. For 𝑏𝑎𝑠𝑒(𝜂)=[𝑙,̂𝑟]̂ and 𝑏𝑎𝑠𝑒(𝜂)=[𝑐,𝑑] with 𝑐>𝑟̂, we define 𝒥̂𝑙𝜂=𝑙̂_{−1𝜂′𝑐𝜂}Ψ𝜂^′;𝜉,𝜁2^{𝑁(𝜂′,𝜂)−𝑁}𝑙̂^{(𝛾𝑢,𝐶1,…,𝐶𝑝)−𝑁}𝑙̂^{(𝛾𝑙,𝐷1,…,𝐷𝑞)},if𝜂is𝑙-active,1,otherwise,(3.18) where ∑𝑙̂_{−1𝜂′𝑐𝜂} 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 𝛾^𝑢 together with ∪_𝜈𝐶_𝜈 or by the concatenation of 𝛾′^𝑙 and 𝛾^𝑙 together with ∪_𝜈𝐷_𝜈. Similarly, 𝒥𝑟̂𝜂=𝑟̂_{+1𝜂′𝑐𝜂}Ψ𝜂^′;𝜉,𝜁2^{𝑁(𝜂,𝜂′)−𝑁}𝑟̂^{(𝛾𝑢,𝐶1,…,𝐶𝑝)−𝑁}𝑟̂^{(𝛾𝑙,𝐷1,…,𝐷𝑞)},if𝜂is𝑟-active,1,otherwise,(3.19) and 𝒥(𝜂,𝜂) is defined in the following two cases.(1)If 𝑐=𝑟̂+2, 𝜂 is 𝑟-active and 𝜂 is 𝑙-active, then 𝒥𝜂,𝜂=𝑟̂_{+1𝜂′𝑐𝜂,𝜂}Ψ𝜂^′;𝜉,𝜁2^{𝑁(𝜂,𝜂′)+𝑁(𝜂′,𝜂)−𝑁}𝑟̂^{(𝛾𝑢,𝐶1,…,𝐶𝑝)−𝑁}𝑟̂^{(𝛾𝑙,𝐷1,…,𝐷𝑞)−𝑁}𝑙̂^{(𝛾𝑢,𝐶1,…,𝐶𝑝)−𝑁}𝑙̂^{(𝛾𝑙,𝐷1,…,𝐷𝑞)}.(3.20)(2)If 𝑐=𝑟̂+1, 𝜂 and 𝜂 are compatible, then 𝒥𝜂,𝜂=2^{𝑁(𝜂,𝜂)−𝑁}𝑟̂^{(𝛾𝑢,𝐶1,…,𝐶𝑝)−𝑁}𝑟̂^{(𝛾𝑙,𝐷1,…,𝐷𝑞)−𝑁}𝑙̂^{(𝛾𝑢,𝐶1,…,𝐶𝑝)−𝑁}𝑙̂^{(𝛾𝑙,𝐷1,…,𝐷𝑞)}.(3.21)
From the formulas (3.16) and (3.18)–(3.21), we show the weight expression of Ψ(𝒞_𝑖;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 _{𝛾𝑏𝑎𝑠𝑒𝑢=𝐼}𝑒^{−(𝛿/3)𝑁𝑣(𝛾𝑢)−(𝜇1−2𝛿/3)(𝑁ℎ(𝛾𝑢)+1)}≤𝑅𝜇₁,𝛿^|𝐼|1−𝑅𝜇₁,,𝛿(4.1) where 𝛾^𝑢 is a upper backbone, 𝑁_𝑣(𝛾^𝑢) is the number of vertical edges in 𝛾^𝑢, 𝑁_ℎ(𝛾^𝑢) is the number of horizontal edges in 𝛾^𝑢, and 𝑅𝜇₁,𝛿=2𝑒^{−(𝜇1−2𝛿/3)}1+𝑒^−𝛿/31−𝑒^{−𝛿/3−1}.(4.2) The upper bound of (4.1) also exists for the lower backbone 𝛾^𝑙.

Proof of Lemma 4.1. We separate 𝛾^𝑢 into fragments by the following method. Let 𝛾^𝑢={𝑥₀,𝑥₁,…,𝑥_𝑛} be a self-avoiding path with 𝑏𝑎𝑠𝑒(𝛾^𝑢)=𝐼. For 𝑗₀=0 and 𝑖≥1, we let 𝑗_𝑖=min𝑗>𝑗_𝑖−1;𝑥_𝑗−1,𝑥_𝑗isahorizontaledge.(4.3) Each vertical part {𝑥_𝑗𝑖−1,𝑥_{𝑗𝑖−1+1},…,𝑥_𝑗𝑖−1} of 𝛾^𝑢 with the direction of the exit vector {𝑥_𝑗𝑖−1,𝑥_𝑗𝑖} is called a fragment. For a fragment 𝑓={𝑥₀,𝑥₁,…,𝑥_𝑝} with exit direction 𝑒(𝑓), we define 𝑊𝑓=𝑒^{−(𝛿/3)𝑁𝑣(𝑓)−(𝜇1−2𝛿/3)}=𝑒^{−𝑝𝛿/3−(𝜇1−2𝛿/3)}.(4.4) Then the decomposition of 𝛾^𝑢 into fragments {𝑓₁,…,𝑓_𝑟} leads to the identity 𝑒^{−(𝛿/3)𝑁𝑣(𝛾𝑢)−(𝜇1−2𝛿/3)(𝑁ℎ(𝛾𝑢)+1)}=^𝑟_𝑗=1𝑊𝑓_𝑗.(4.5) Therefore, if 𝜇₁ is sufficiently large, we have _{𝛾𝑢𝛾;𝑏𝑎𝑠𝑒𝑢=𝐼}𝑒^{−(𝛿/3)𝑁𝑣(𝛾𝑢)−(𝜇1−2𝛿/3)(𝑁ℎ(𝛾𝑢)+1)}=^∞_{|𝐼|𝑟=𝑓1,…,𝑓𝑟}^𝑟_𝑗=1𝑊𝑓_𝑗≤^∞_|𝐼|𝑟=2^∞_𝑘=−∞𝑒^{−(𝛿/3)|𝑘|𝑟}×𝑒^{−(𝜇1−2𝛿/3)𝑟}=𝑅𝜇₁,𝛿^|𝐼|1−𝑅𝜇₁,,𝛿(4.6) where the last equality comes from [7]. Following the same proving procedure, we can obtain the existence of the formula (4.1) for 𝛾^𝑙. 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;𝜂𝑖𝜂0}𝑒^{𝑐(𝜂)+𝑑(𝜂)}Ψ𝜂;𝜉,𝜁≤𝑐𝜂₀,(4.7) where 𝒦_𝐿 is the set of all decorated polymers with the base in [0,𝐿], 𝑐(𝜂)=3|𝑏𝑎𝑠𝑒(𝜂)|, and 𝑑𝜂=𝜇−𝜇₂𝜂+𝛿𝑏𝑎𝑠𝑒6𝛾^𝑢+𝛿6𝛾^𝑙−𝜇−𝜇₁−1,if𝑏𝑎𝑠𝑒𝜂≥2,𝜇−𝜇₂𝜂+𝛿𝑏𝑎𝑠𝑒6𝛾^𝑢+𝛿6𝛾^𝑙,if𝑏𝑎𝑠𝑒𝜂=1,(4.8) where 𝜇₁ is given in Lemma 4.1.

Proof of Lemma 4.2. In order to verify the convergence and analyticity, we have to show that there exist two functions 𝑐,𝑑∶𝒦_𝐿=𝜉;decoratedpolymer⟶[0,∞),(4.9) such that the above estimate (4.7) exists. First we consider the statistical properties of 𝜂=𝛾^𝑢,𝛾^𝑙,𝐶₁,…,𝐶_𝑝,𝐷₁,…,𝐷_𝑞,Λ₁,…,Λ_𝑣,(4.10) and we have 𝛾^𝑢=𝑁_𝑣𝛾^𝑢+𝑁_ℎ𝛾^𝑢+1,𝛾^𝑙=𝑁_𝑣𝛾^𝑙+𝑁_ℎ𝛾^𝑙+1.(4.11) By the definition of the decorated polymers, if 𝑏𝑎𝑠𝑒(𝜂) is one point, then 𝑁_ℎ𝛾^𝑢+𝑁_ℎ𝛾^𝑙+^𝑝_𝜈=1𝐶_𝜈+^𝑞_𝜈=1𝐷_𝜈+^𝑣_𝛼=1Λ_𝛼≥1,(4.12) since {𝐶_𝜈} or {𝐷_𝜈} or {Λ_𝛼} is nonempty if 𝑏𝑎𝑠𝑒(𝜂) is one point. If |𝑏𝑎𝑠𝑒(𝜂)|≥2, then we have 𝑁_ℎ𝛾^𝑢+𝑁_ℎ𝛾^𝑙+^𝑝_𝜈=1𝐶_𝜈+^𝑞_𝜈=1𝐷_𝜈+^𝑣_𝛼=1Λ_𝛼≥3𝑏𝑎𝑠𝑒(𝜉)−1.(4.13) Let (𝛾^𝑢,𝛾^𝑙) be the backbone of some decorated polymer with the base 𝐼=[𝑙,̂𝑟]̂. Next we estimate the following function 𝐺𝛾^𝑢,𝛾^𝑙=_{𝛾𝜂;𝑢,𝛾𝑙isthebackboneof𝜂}𝑒^{𝜇𝜉ℎ(𝛾𝑢)}𝑒^{𝜇𝜁ℎ(𝛾𝑙)}Ψ𝜂.(4.14) 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|}.(4.15)
(i) If 𝑙̂=𝑟̂, that is |𝐼|=1, then 𝑁^∗𝛾^𝑢,𝐶₁,…,𝐶_𝑝=𝑁^∗𝛾^𝑙,𝐷₁,…,𝐷_𝑞=0.(4.16) From the (3.11), (4.14), and (4.15), we have 𝐺𝛾^𝑢,𝛾^𝑙≤𝑒^{−𝜇|𝛾𝑢|}𝑒^{𝜇ℎ(𝛾𝑢)Re𝜉}_{𝐶𝜈;𝐶𝜈𝑖𝛾𝑢}𝑒^{−𝜇∑𝜈|𝐶𝜈|}×𝑒^{−𝜇|𝛾𝑙|}𝑒^{𝜇ℎ(𝛾𝑙)Re𝜁}_{𝐷𝜈;𝐷𝜈𝑖𝛾𝑙}𝑒^{−𝜇∑𝜈|𝐷𝜈|}×_{Λ𝛼;Λ𝛼𝑖𝛾𝑢∪𝐶1∪⋯∪𝐶𝑝orΛ𝛼𝑖𝛾𝑙∪𝐷1∪⋯∪𝐷𝑞}𝑒^{−(𝜇−𝜇0−1)∑𝛼|Λ𝛼|}≤𝑒^{−𝜇|𝛾𝑢|+𝜇ℎ(𝛾𝑢)Re𝜉−𝜇|𝛾𝑙|+𝜇ℎ(𝛾𝑙)Re𝜁−(𝜇−𝜇1−1)}_{𝐶𝜈;𝐶𝜈𝑖𝛾𝑢}𝑒^{−𝜇1∑𝜈|𝐶𝜈|}×_{𝐷𝜈;𝐷𝜈𝑖𝛾𝑙}𝑒^{−𝜇1∑𝜈|𝐷𝜈|}_{Λ𝛼;Λ𝛼𝑖𝛾𝑢∪𝐶1∪⋯∪𝐶𝑝orΛ𝛼𝑖𝛾𝑙∪𝐷1∪⋯∪𝐷𝑞}𝑒^{−(𝜇1−𝜇0)∑𝛼|Λ𝛼|}.(4.17) The summation over {Λ_𝛼} is estimated as follows: _{{Λ𝛼};Λ𝛼𝑖𝛾𝑢∪𝐶1∪⋯∪𝐶𝑝orΛ𝛼𝑖𝛾𝑙∪𝐷1∪⋯∪𝐷𝑞}𝑒^{−(𝜇1−𝜇0)∑𝛼|Λ𝛼|}≤^∞_𝑡=01𝑡!_{Λ1𝑖𝛾𝑢∪𝐶1∪⋯∪𝐶𝑝orΛ1𝑖𝛾𝑙∪𝐷1∪⋯∪𝐷𝑞}⋯_{Λ𝑡𝑖𝛾𝑢∪𝐶1∪⋯∪𝐶𝑝orΛ𝑡𝑖𝛾𝑙∪𝐷1∪⋯∪𝐷𝑞}𝑒^{−(𝜇1−𝜇0)∑𝛼|Λ𝛼|}≤exp4𝛾^𝑢∪𝐶₁∪⋯∪𝐶_𝑝∪𝛾^𝑙∪𝐷₁∪⋯∪𝐷_{𝑞Λ∋0;connected}𝑒^{−(𝜇1−𝜇0)|Λ|}=exp𝛾^𝑢+𝛾^𝑙+_𝜈𝐶_𝜈+_𝜈𝐷_𝜈𝑔₁𝜇₁,𝜇₀.(4.18) There exist constants 𝐾₁,𝜅>0 such that the number 𝑁_𝑚 of connected sets of 𝑚 points in ℤ² which contain the origin is bounded as 𝑁_𝑚≤𝐾₁𝜅^𝑚(𝑚≥1),(4.19) then we know that the function 𝑔₁𝜇₁,𝜇₀=4_{Λ∋0;connected}𝑒^{−(𝜇1−𝜇0)|Λ|},(4.20) goes to zero exponentially fast as 𝜇₁→∞. Thus, from (4.17) and (4.18) we obtain 𝐺𝛾^𝑢,𝛾^𝑙≤𝑒^{−(𝜇−𝑔1(𝜇1,𝜇0))|𝛾𝑢|+𝜇ℎ(𝛾𝑢)Re𝜉−(𝜇−𝑔1(𝜇1,𝜇0))|𝛾𝑙|+𝜇ℎ(𝛾𝑙)Re𝜁}×𝑒^{−(𝜇−𝜇1−1)}×_{𝐶𝜈;𝐶𝜈𝑖𝛾𝑢}𝑒^{−(𝜇1−𝑔1(𝜇1,𝜇0))∑𝜈|𝐶𝜈|}_{𝐷𝜈;𝐷𝜈𝑖𝛾𝑙}𝑒^{−(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)},(4.21) where 𝑔₂𝜇₁,𝜇₀=4_{𝐷∋0;connected}𝑒^{−(𝜇1−𝑔1(𝜇1,𝜇0))|𝐷|}.(4.22)
(ii) If 𝑙̂<𝑟̂; that is, |𝐼|≥2, then we have 𝑁^∗𝛾^𝑢,𝐶₁,…,𝐶_𝑝≤𝑁_ℎ𝛾^𝑢+^𝑝_𝜈=1𝐶_𝜈,𝑁^∗𝛾^𝑙,𝐷₁,…,𝐷_𝑞≤𝑁_ℎ𝛾^𝑙+^𝑞_𝜈=1𝐷_𝜈.(4.23) From the formula (4.13), we have 𝐺𝛾^𝑢,𝛾^𝑙≤𝑒^{−𝜇|𝛾𝑢|}𝑒^{𝜇ℎ(𝛾𝑢)Re𝜉}𝑒^{−𝜇|𝛾𝑙|}𝑒^{𝜇ℎ(𝛾𝑙)Re𝜁}_{𝐶𝜈;𝐶𝜈𝑖𝛾𝑢}𝑒^{−𝜇∑𝜈|𝐶𝜈|}2^{𝑁∗(𝛾𝑢,𝐶1,…,𝐶𝑝)}×_{𝐷𝜈;𝐷𝜈𝑖𝛾𝑙}𝑒^{−𝜇∑𝜈|𝐷𝜈|}2^{𝑁∗(𝛾𝑙,𝐷1,…,𝐷𝑞)}_{Λ𝛼;Λ𝛼𝑖𝛾𝑢∪𝐶1∪⋯∪𝐶𝑝orΛ𝛼𝑖𝛾𝑙∪𝐷1∪⋯∪𝐷𝑞}𝑒^{−(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)}2^{𝑁ℎ(𝛾𝑢)}2^{𝑁ℎ(𝛾𝑙)}×_{𝐶𝜈;𝐶𝜈𝑖𝛾𝑢}𝑒^{−(𝜇1−ln2)∑𝜈|𝐶𝜈|}_{𝐷𝜈;𝐷𝜈𝑖𝛾𝑙}𝑒^{−(𝜇1−ln2)∑𝜈|𝐷𝜈|}×_{Λ𝛼;Λ𝛼𝑖𝛾𝑢∪𝐶1∪⋯∪𝐶𝑝orΛ𝛼𝑖𝛾𝑙∪𝐷1∪⋯∪𝐷𝑞}𝑒^{−(𝜇1−𝜇0)∑𝛼|Λ𝛼|}.(4.24) Further, according to the formula (4.18), we have 𝐺𝛾^𝑢,𝛾^𝑙≤𝑒^{−(𝜇−𝑔1(𝜇1,𝜇0))|𝛾𝑢|+𝜇ℎ(𝛾𝑢)Re𝜉−(𝜇−𝑔1(𝜇1,𝜇0))|𝛾𝑙|+𝜇ℎ(𝛾𝑙)Re𝜁}𝑒^{−(𝜇−𝜇1−1)(3|𝐼|−𝑁ℎ(𝛾𝑢)−𝑁ℎ(𝛾𝑙)−3)}×_{𝐶𝜈;𝐶𝜈𝑖𝛾𝑢}𝑒^{−(𝜇1−𝑔1(𝜇1,𝜇0)−ln2)∑𝜈|𝐶𝜈|}2^{𝑁ℎ(𝛾𝑢)}_{𝐷𝜈;𝐷𝜈𝑖𝛾𝑙}𝑒^{−(𝜇1−𝑔1(𝜇1,𝜇0)−ln2)∑𝜈|𝐷𝜈|}2^{𝑁ℎ(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},(4.25) where 𝑔₃𝜇₁,𝜇₀=4_{𝐷∋0;connected}𝑒^{−(𝜇1−𝑔1(𝜇1,𝜇0)−ln2)|𝐷|}.(4.26)
We choose 𝜇₁(>𝜇₀) sufficiently large, such that 𝑔₁𝜇₁,𝜇₀<𝛿4,𝑔₂𝜇₁,𝜇₀<𝛿4,𝑔₃𝜇₁,𝜇₀<𝛿4.(4.27) Assume that Re𝜉<1−𝛿/𝜇 and Re𝜁<1−𝛿/𝜇. Since 𝑁_𝑣𝛾^𝑢≥ℎ𝛾^𝑢,𝑁_𝑣𝛾^𝑙≥ℎ𝛾^𝑙,(4.28) then from the formula (4.11), for |𝐼|≥2, we have 𝐺𝛾^𝑢,𝛾^𝑙≤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)}.(4.29) For |𝐼|=1, we have 𝐺𝛾^𝑢,𝛾^𝑙≤𝑒^{−(𝛿/2)𝑁𝑣(𝛾𝑢)−(𝛿/2)𝑁𝑣(𝛾𝑢)−2(𝜇1−𝛿/2)}𝑒^{−3(𝜇−𝜇1−1)}.(4.30)
Let 𝑐(𝜂), 𝑑(𝜂) be defined in (4.7). Since 𝑐(𝜂) and 𝑑(𝜂) only depend on the backbone (𝛾^𝑢,𝛾^𝑙), let 𝑐𝜂=𝑐𝛾^𝑢,𝛾^𝑙,𝑑𝜂=𝑑𝛾^𝑢,𝛾^𝑙.(4.31) Then, for a fixed interval 𝐼, we have _{𝛾𝜂;𝑢,𝛾𝑙isthebackboneof𝜂}𝑒^{𝑐(𝜂)+𝑑(𝜂)}𝑒^{𝜇𝜉ℎ(𝛾𝑢)}𝑒^{𝜇𝜁ℎ(𝛾𝑙)}Ψ𝜂=𝐺𝛾^𝑢,𝛾^𝑙𝑒^{𝑐(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)|},(4.32) where 𝑏𝑎𝑠𝑒(𝛾^𝑢,𝛾^𝑙)=𝑏𝑎𝑠𝑒(𝜂) for any 𝜂 such that (𝛾^𝑢,𝛾^𝑙) is the backbone of 𝜂. Then _{𝛾𝑏𝑎𝑠𝑒=𝐼}𝑒^{𝑐(𝛾𝑢,𝛾𝑙)+𝑑(𝛾𝑢,𝛾𝑙)}𝐺𝛾^𝑢,𝛾^𝑙≤𝑒^{−(2𝜇+𝜇2−3𝜇1−6)|𝐼|}_{𝛾𝑏𝑎𝑠𝑒=𝐼}𝑒^{−(𝛿/3)𝑁𝑣(𝛾𝑢)−(𝜇1−2𝛿/3)(𝑁ℎ(𝛾𝑢)+1)}𝑒^{−(𝛿/3)𝑁𝑣(𝛾𝑙)−(𝜇1−2𝛿/3)(𝑁ℎ(𝛾𝑙)+1)}.(4.33) By Lemma 4.1, if Re𝜉<1−𝛿/𝜇,Re𝜁<1−𝛿/𝜇, and 𝜇>𝜇₁, where 𝜇₁ is sufficiently large, we have _{(𝜉)𝑏𝑎𝑠𝑒=𝐼}𝑒^{𝑐(𝛾𝑢,𝛾𝑙)+𝑑(𝛾𝑢,𝛾𝑙)}𝐺𝛾^𝑢,𝛾^𝑙≤𝑒^{−(2𝜇+𝜇2−3𝜇1−6)|𝐼|}𝑅(𝜇₁,𝛿)^|𝐼|1−𝑅𝜇₁,𝛿².(4.34) From (3.14), there is a 𝜇₃>0 such that 𝑄(𝜉,𝜁)≥𝑒^{−2𝜇−𝜇3},(4.35) then we have _{(𝜉)𝑏𝑎𝑠𝑒=𝐼}Ψ𝜂;𝜉,𝜁𝑒^{𝑐(𝛾𝑢,𝛾𝑙)+𝑑(𝛾𝑢,𝛾𝑙)}≤𝑒^{−(2𝜇+𝜇2−3𝜇1−6)|𝐼|}𝑅𝜇₁,𝛿^|𝐼|1−𝑅𝜇₁,𝛿²𝑒^{(2𝜇+𝜇3)|𝐼|}=𝑒^{−(𝜇2−3𝜇1−𝜇3−6)|𝐼|}𝑅𝜇₁,𝛿^|𝐼|1−𝑅𝜇₁,𝛿².(4.36) Let 𝜇₁ be large enough, we take 𝜇₂>3𝜇₁+𝜇₃+6 and 𝜇>𝜇₂. For a fixed 𝜂₀ with the 𝑏𝑎𝑠𝑒(𝜂₀)=[𝑙,̂𝑟]̂, then we have _{𝜂𝑖𝜂0}Ψ𝜂;𝜉,𝜁𝑒^{𝑐(𝜂)+𝑑(𝜂)}≤_𝑥∈𝑙̂_−1,𝑟̂₊₁^𝐼∋𝑥𝑅𝜇₁,𝛿^|𝐼|1−𝑅𝜇₁,𝛿²=𝑟̂−𝑙̂+31−𝑅𝜇₁,𝛿^2∞_𝑘=1𝑘𝑅𝜇₁,𝛿^2𝑘≤3𝑏𝑎𝑠𝑒𝜂₀1−𝑅𝜇₁,𝛿²×𝑅𝜇₁,𝛿²1−𝑅𝜇₁,𝛿²²≤𝑐𝜂₀.(4.37) 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 2.1. From Lemma 3.1, we have 1𝑄(𝜉,𝜁)^𝐿_{Γ𝑢,Γ𝑙∈𝒮𝐿}𝑒^{𝜇(𝜉ℎ(Γ𝑢)+𝜁ℎ(Γ𝑙))}𝑊Γ^𝑢,Γ^𝑙=_{𝒞1,…,𝒞𝑧∈𝒞𝒫𝐿;compatible}^𝑧_𝑖=1Ψ𝒞_𝑖;𝜉,𝜁.(4.38) In the following part, we will show the existence of the following limit: 𝜑(𝜉,𝜁)=lim_𝐿→∞1𝐿ln_{𝒞1,…,𝒞𝑧∈𝒞𝒫𝐿;compatible}^𝑧_𝑖=1Ψ𝒞_𝑖;𝜉,𝜁,(4.39) where 𝜑(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)=𝜑(𝜉,𝜁)+ln𝑄(𝜉,𝜁),(4.40) 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=𝑙₁̂,𝑟₁̂,…,𝐼_𝑛=𝑙_𝑛̂,𝑟_𝑛̂,(4.41) the linked intervals if 𝑟_𝑢̂<𝑙̂_𝑢+1<𝑟_𝑢̂+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)]=[𝑙₀̂,𝑟₀̂] be the smallest intervals (in length) including 𝑏𝑎𝑠𝑒(𝒞₀). Let 𝑐^∗(𝒞)=_𝜂∈𝒞𝑐𝜂,𝑑^∗(𝒞)=_𝜂∈𝒞𝑑𝜂,(4.42) where 𝑐(𝜂) and 𝑑(𝜂) are defined in Lemma 4.2. Then, for the functions 𝑐^∗,𝑑^∗∶𝒞𝒫=𝒞;polymerchain⟶[0,∞),(4.43) we show that _{𝒞∈𝒞𝒫𝐿;𝒞𝑖𝒞0}Ψ(𝒞;𝜉,𝜁)𝑒^{𝑐∗(𝒞)+𝑑∗(𝒞)}≤𝑐^∗𝒞₀,(4.44) 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}Ψ(𝒞;𝜉,𝜁)𝑒^{𝑐∗(𝒞)+𝑑∗(𝒞)}=_{𝒞;𝒞𝑖𝒞0}Ψ(𝒞;𝜉,𝜁)exp_𝜂∈𝒞𝑐𝜂+_𝜂∈𝒞𝑑𝜂≤_𝑥∈𝑙̂_0−2,𝑟₀^∞₊₂^{𝑛=1𝐼1,…,𝐼𝑛⊂[]0,𝐿;∪𝐼𝑢∋𝑥}_{linkedintervals}^{𝜂1,…,𝜂𝑛∈𝒦𝐿;}_{𝜂𝑏𝑎𝑠𝑒𝑢=𝐼𝑢𝑛,1≤𝑢≤𝑛𝑢=1}̂Ψ𝜂_𝑢;𝜉,𝜁𝑒^{𝑐(𝜂𝑢)+𝑑(𝜂𝑢)}𝒥_𝑙𝜂₁𝒥_𝑟𝜂_𝑛^𝑛−1_𝑢=1𝒥𝜂_𝑢,𝜂_𝑢+1.(4.45) From definitions (3.18)–(3.21) and the formula (4.35), there exists 𝜇^′₃>0 such that |𝒥_𝑙|, |𝒥_𝑟|, and |𝒥| are all bounded from above by 𝑒^𝜇′3 if Re𝜉<1−𝛿/𝜇 and Re𝜁<1−𝛿/𝜇. Therefore, from (4.36) or Lemma 4.2, we have that _{𝜂1,…,𝜂𝑛∈𝒦𝐿;𝜂𝑏𝑎𝑠𝑒𝑢=𝐼𝑢}^𝑛_{,1≤𝑢≤𝑛𝑢=1}Ψ𝜂_𝑢;𝜉,𝜁𝑒^{𝑐(𝜂𝑢)+𝑑(𝜂𝑢)}𝒥_𝑙𝜂₁𝒥_𝑟𝜂_𝑛^𝑛−1_𝑢=1𝒥𝜂_𝑢,𝜂_𝑢+1≤^𝑛_𝑢=1𝑒^{−(𝜇2−3𝜇1−𝜇3−2𝜇′3−6)|𝐼𝑢|}𝑅𝜇₁,𝛿^|𝐼𝑢|1−𝑅𝜇₁,𝛿².(4.46) Assuming that 𝜇₂>3𝜇₁+𝜇₃+2𝜇^′₃+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}Ψ(𝒞;𝜉,𝜁)𝑒^{𝑐∗(𝒞)+𝑑∗(𝒞)}≤𝑟₀̂−𝑙₀̂+4^∞𝑛_{𝑛=1𝑢=1𝐼1,…,𝐼𝑛⊂[]0,𝐿;∪𝐼𝑢∋𝑥linkedintervals}^𝑛_𝑢=1𝑅𝜇₁,𝛿^|𝐼𝑢|1−𝑅𝜇₁,𝛿²≤𝑟₀̂−𝑙₀̂+4𝑅𝜇₁,𝛿²1−𝑅𝜇₁,𝛿²1−𝑅𝜇₁,𝛿^22∞𝑛=1𝑛2𝑅𝜇₁,𝛿²1−𝑅𝜇₁,𝛿^22𝑛−1≤𝑟₀̂−𝑙₀̂+42≤𝑐^∗𝒞₀.(4.47) 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 𝒫_𝑓(𝒞𝒫) be the collection of all finite subsets of 𝒞𝒫, so that there exists a function Ψ^𝑇∶𝒫_𝑓(𝒞𝒫)×ℂ²⟶ℂ²,(4.48) such that Ψ^𝑇 is analytic for Re𝜉<1−𝛿/𝜇 and Re𝜁<1−𝛿/𝜇, and it satisfies _{𝒞1,…,𝒞𝑧∈𝒞𝒫𝐿;compatible}^𝑧_𝑖=1Ψ𝒞_𝑖;𝜉,𝜁=exp_{Δ∈𝒫𝑓𝒞𝒫𝐿}Ψ^𝑇(Δ;𝜉,𝜁_Δ𝑖𝒞0),Ψ^𝑇(Δ;𝜉,𝜁)𝑒^𝑑∗(Δ)≤𝑐^∗𝒞₀,(4.49) 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 Ψ^𝑇(Δ;𝜉,𝜁)=0. We call Δ∈𝒫_𝑓(𝒞𝒫_𝐿) a cluster if there are no such decomposition Δ=Δ₁∪Δ₂. Note that Ψ^𝑇(Δ;𝜉,𝜁) is invariant under horizontal translation of Δ. For Δ∈𝒫_𝑓(𝒞𝒫), we set 𝑏𝑎𝑠𝑒(Δ)=∪_𝒞∈Δ𝑏𝑎𝑠𝑒(𝒞). Then, by the cluster expansion theory of [19] and (4.49), we have that the limit 𝜑(𝜉,𝜁)=lim_𝐿→∞1𝐿ln_{𝒞1,…,𝒞𝑧∈𝒞𝒫𝐿;compatible}^𝑧_𝑖=1Ψ𝒞_𝑖;𝜉,𝜁=_{Δ∈𝒫𝑓();[(Δ=[]𝒞𝒫𝑏𝑎𝑠𝑒)]0,𝑘forsome𝑘≥0}Ψ^𝑇(Δ;𝜉,𝜁),(4.50) 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.