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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 858725, 21 pages
Analysis of Two-Layered Random Interfaces for Two Dimensional Widom-Rowlinson's Model
Department of Mathematics, Key Laboratory of Communication and Information System, Beijing Jiaotong University, Beijing 100044, China
Received 21 May 2011; Revised 31 August 2011; Accepted 12 September 2011
Academic Editor: Dumitru Baleanu
Copyright © 2011 Jun Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 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.
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 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 or ) in the lattice W-R system, and there is a strong repulsive interaction between particles of the different types. Namely, a particle cannot occupy a site within distance from a site where an particle has occupied and vice versa. This means that different types of particles are separated by the empty sites. In , 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 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 , the thickness of the random interface (or the intermediate “belt”) between the two coexisting phases of the model is expected to become thinner as 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 phase at an interface between the and the phases. Much research work has been devoted to the study of the wetting behavior for the -state Potts model. Derrida and Schick  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  exhibit the wetting of the interface between two ordered phases by the disordered one with being large. At the transition point, De Coninck et al.  display the similar wetting behavior by the correlation inequalities, and Messager et al.  present an analysis of the order-disorder transition for large 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 is small, which means that the layer of empty sites that separate and 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 , since we think that the layer between the two coexisting phases is expected to become thinner as becomes larger.
We consider the two-layered phase interfaces in the two-dimensional Widom-Rowlinson model on the rectangle , where . Suppose that the particles in are of two types and there is a strong repulsive interaction between particles of the different types. Let denote a configuration on , where denotes that the site is occupied by an particle, denotes that is occupied by a particle, and denotes that there is no particle at . We say that a configuration is feasible if for all pairs with , where is the Euclidean distance. Let denote the set of all feasible configurations in , so there is a finite diameter hard-core exclusion between particles and particles on . 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 for all , where (>0) denotes the chemical potential.
(ii) Let , let , and assume that , . We define a boundary condition as for each .
(iii) Suppose that there is a connected “−1” particles path from the left side of 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 , , for some ) ensure that we have the two-layered interface model. Let be the corresponding configuration space with the conditions (i)–(iii), such that the configuration is feasible, where for and for . For a fixed configuration , let denote the set of points in such that the configuration takes 0 value. The connected components of are called contours; among these contours, there are two contours and which are defined as the interfaces of the model. is the upper interface with the starting point and the ending point ; is the lower interface with the starting point and the ending point . Let be the set of upper interfaces and the set of lower interfaces, respectively. The conditional Gibbs distribution on with the boundary condition is given by where denotes the cardinality of a set , and 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.  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 and of interfaces and 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 of to represent ; that is, among self-avoiding paths connecting the starting point with the ending point in , we select a self-avoiding path (called backbone ) by an “order” given in the following definition (6). Since the backbone 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 as Among these paths, we select a self-avoiding path according to the following order; the order is defined with preference among four directions: More precisely, let and be the two self-avoiding paths in . Let be the first number such that . We define that if the direction of the ordered edge is preferred to the direction of the ordered edge . Let be the unique maximal element of with respect to this order, and call the backbone of . Similarly, let represent the backbone of . In this paper, our study is mainly focused on the backbone of the phase separation belt.
Let be the different connected subsets of , we say that the subsets are compatible if they are connected components of the set . We also say that are compatible with a connected set if are compatible for every . Next we define the hole of a connected set of , we say that a set is -connected if, for every , there exist a sequence in such that for every . A hole of a connected set is a finite -connected component of . 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 defined in (1.4) can be rewritten by the similar formulas in [1–3] as following: where the second summation is taken over compatible families , which are compatible with , is the number of points in , and is the number of holes in , similar to , , and , . Then for some large and , according to the theory of the cluster expansions (see ), we have where is the partition function with the plus boundary condition, denotes that the set is incompatible with the interfaces , and is a translation invariant function, which satisfies the following estimate: Moreover, if is sufficiently large, we have where and Let which is called the weight of the partition function expansion (2.6). From , the last part in (2.6) can be expanded as follows:
In the definition of boundary condition (see (1.2)), for a fixed , let , then, for the interfaces vector , the height of the last ending point of is defined as the vector The aim of this paper is to study the statistical behaviors of the free energy of the height vector . The following theorem shows the limit existence of the free energy of the height for the interface vector .
Theorem 2.1. For some and a complex vector , when the chemical potential is large enough, one has the existence of the free energy of the heights for the two-layered W-R model as follows: where the function is analytic in if the real parts and .
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 . Further, we show that the asymptotical behavior of the backbone can represent that of the corresponding interface when is large enough; this means that the statistical properties of the interface are similar to those of its backbone . From the definition of , 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 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.  introduce a new definition of polymers for one-layered interface of the W-R model; here we modify the definitions in  and define the polymer chains for the two-layered interface model.
For the interface vector , let and be the starting and the ending points of ; let be the backbone of connecting and . Similarly, let and be the starting and the ending points of ; let be the backbone of connecting and . We decompose into connected components and decompose into connected components . Then, from (2.6) and (2.9), we have where denotes the number of holes of , denotes the number of holes of , and the second summation taken over “ lies above ”, 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 be the positive integers, is called a polymer with base if satisfies the following conditions (1)–(4).(1) are self-avoiding paths in , starts from and ending at a point in , starts from and ending at a point in , where , , , are fixed integers satisfying and , and lies above .(2) is a compatible family of connected subsets of such that (i) , where is the set of vertices in ; (ii) is connected; (iii) is the backbone of with starting point and ending point . Similarly, and have the same properties.(3) is a collection of connected subsets of such that (4)For , , the line intersects at least two edges of or at least two edges of . Here, for , denotes the set of the nearest neighbor edges of ; is the set of edges that connect with the set . Also, we identify an edge of with the line segment connecting and .
We call the backbone of . For two disjoint self-avoiding paths (similarly for ) such that the starting point of is the nearest neighbor of the endpoint of , we can define the concatenation of these paths by simply connecting them. Let be two polymers with bases and , respectively. We say that and are compatible if either of the following conditions holds:(a),(b), the backbone of is the concatenation , and connected components of the set are and . Here, is the height of the endpoint of .
Note that the previous work (see ) 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 .
The family is compatible if and are compatible. Let An edge of is not admissible if it is a horizontal edge in , such that(i)the left vertex is in a connected of , and the right vertex is in ;(ii)further, there exists a horizontal edge of such that and , where is the left vertex of .
Other edges of are admissible.
We say that the line is the cutting line of if intersects only two admissible edges of . Here, one of two admissible edges is connected to ; the other is connected to .
Let be all the cutting lines of , , . For each , there are only two edges and of , and , respectively, which intersect . Let be the portion of starting from and ending at ; let be the portion of starting from and ending at . Also let , , and be the set of elements of , , and , respectively, such that they are subsets of . Then , for some , where . Thus, we obtain the th polymer by setting By the above definitions, are compatible.
For a polymer let be the height of the endpoint of the self-avoiding path ; similarly let . Then the heights and are given by Now we introduce a statistical weight of a polymer by setting where and is the number of new holes created by and the line , where ; is the number of new holes created by and the line . Similarly, we can give the definition of .
A polymer is called simple if is one point and . Thus, the weight is given by A polymer is called decorated if it is not simple. A decorated polymer with is said to be -active if there exists a simple polymer with such that is incompatible with , or the concatenation of and together with produces a new hole, or the concatenation of and together with produces a new hole. is said to be -active if there exists a simple polymer with such that is incompatible with , the concatenation of and together with produces a new hole, or the concatenation of and together with produces a new hole. If is both -active and -active, we call it -active.
A polymer chain is a family of decorated polymers such that(1) are compatible;(2)if , , then or for each ;(3)if for some , then is -active and is -active.
Let and be two polymer chains. We say that and are compatible if is a compatible family of polymers, but now it is not a polymer chain. For example, if and have , then is compatible polymers, but not a polymer chain.
Let be the set of all decorated polymers with base in , and let denote the set of polymer chains with base in , then we have the following Lemma 3.1.
Lemma 3.1. Let be the generating function of the heights of the endpoints of a simple polymer then one has where is the weight function of polymer chains which is given (3.17).
Proof of Lemma 3.1. Considering a polymer chain , let . Further, for a polymer , from (3.9) and (3.11), we define
where for . By the formula (3.16), for a polymer chain , we put
where , , are defined in the following way. For and with , we define
where means over polymers with which are compatible with , and is the number of new holes created by the concatenation of and together with or by the concatenation of and together with . Similarly,
and is defined in the following two cases.(1)If , is -active and is -active, then (2)If , and are compatible, then
From the formulas (3.16) and (3.18)–(3.21), we show the weight expression of . 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 be a fixed interval; if is large enough, then for some , one has where is a upper backbone, is the number of vertical edges in , is the number of horizontal edges in , and The upper bound of (4.1) also exists for the lower backbone .
Proof of Lemma 4.1. We separate into fragments by the following method. Let be a self-avoiding path with . For and , we let Each vertical part of with the direction of the exit vector is called a fragment. For a fragment with exit direction , we define Then the decomposition of into fragments leads to the identity Therefore, if is sufficiently large, we have where the last equality comes from . Following the same proving procedure, we can obtain the existence of the formula (4.1) for . This completes the proof of Lemma 4.1.
Lemma 4.2. There is a large chemical potential ; if , then for any with the conditions and , and for each polymer , one has where is the set of all decorated polymers with the base in , , and where 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
such that the above estimate (4.7) exists. First we consider the statistical properties of
and we have
By the definition of the decorated polymers, if is one point, then
since or or is nonempty if is one point. If , then we have
Let be the backbone of some decorated polymer with the base . Next we estimate the following function
From (2.4) and for large parameter , we have
(i) If , that is , then From the (3.11), (4.14), and (4.15), we have The summation over is estimated as follows: There exist constants such that the number of connected sets of points in which contain the origin is bounded as then we know that the function goes to zero exponentially fast as . Thus, from (4.17) and (4.18) we obtain where
(ii) If ; that is, , then we have From the formula (4.13), we have Further, according to the formula (4.18), we have where
We choose (>) sufficiently large, such that Assume that and . Since then from the formula (4.11), for , we have For , we have
Let , be defined in (4.7). Since and only depend on the backbone , let Then, for a fixed interval , we have where for any such that is the backbone of . Then By Lemma 4.1, if , and , where is sufficiently large, we have From (3.14), there is a such that then we have