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

. 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.


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 ii Let b 1 b 1 L > 0, let b 2 b 2 L > 0, and assume that M > b 1 , M > b 2 .We define a boundary condition ω b as 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 b 1 L c 1 L, b 2 L c 2 L, for some 0 < c 1 , c 2 < 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 S 0 σ denote the set of points in Λ L,M such that the configuration σ takes 0 value.The connected components of S 0 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, b 1 and the ending point L, b 2 ; Γ l σ is the lower interface with the starting point 0, −b where |S| denotes the cardinality of a set S, and Z b L,M is the corresponding partition function.

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, b 1 with the ending point L, b 2 in Γ u , we select a selfavoiding 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 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 π x {x 1 , . . ., x n } and π y {y 1 , . . ., 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 U 1 , . . ., 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 Z 2 , we say that a set D ⊂ Z 2 is * -connected if, for every x, y ∈ D, there exist a sequence x z 0 , z 1 , . . ., z n y in D such that |z 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: 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 μ 0 > 0 and μ > μ 0 , according to the theory of the cluster expansions see 19 , we have 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: Moreover, if μ is sufficiently large, we have lim where 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 In the definition of boundary condition ω b see 1.2 , for a fixed b 1 ≥ 0, let S L ∪ b S b,u L , ∪ b S b,l 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 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 ∈ C 2 , 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: 11

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 {x 1 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 twolayered interface model.
For the interface vector Γ u , Γ l ∈ S L , let 0, b 1 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, b 1 and L, k u .Similarly, let 0, −b 1 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, −b 1 and L, k l .We decompose and decompose Γ l \ π Γ l into connected components {D j } r j 1 .Then, from 2.6 and 2.9 , we have . ., D r denotes the number of holes of π ∪ ∪ r ν 1 D 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, 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 ≤ x 1 ≤ c}, γ u starts from a, a u and ending at a point c, c u in {x 1 c}, γ l starts from a, a l and ending at a point c, c l in {x 1 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 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.
4 For a ≤ j < c, j ∈ N, the line j {x 1 j 1/2} intersects at least two edges of Here, for B ⊂ Z 2 , 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 Z 2 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 u 1 , γ u 2 similarly for γ l 1 , γ l 2 such that the starting point of γ u 2 is the nearest neighbor of the endpoint of γ u 1 , we can define the concatenation γ u 1 • γ u 2 of these paths by simply connecting them.Let be two polymers with bases a, c and a , c a ≤ a , respectively.We say that η 1 and η 2 are compatible if either of the following conditions holds: , and connected components of the set are {C 1 , . . ., C p , C 1 , . . ., C u } and {D 1 , . . ., 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

and the right vertex
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 j {x 1 j 1/2} 0 ≤ j ≤ L − 1 is the cutting line of η if j intersects only two admissible edges of η Here, one of two admissible edges is connected to π Γ u ; the other is connected to π Γ l . 3.9 Now we introduce a statistical weight of a polymer

3.11
where 2 if base η u l u , r u , 1 ≤ u ≤ m, then l u 1 r u 1 or r u 2 for each u; 3 if l u 1 r u 2 for some u, then η u is r-active and η u 1 is l-active.
Let C 1 and C 2 be two polymer chains.We say that C 1 and C 2 are compatible if C 1 ∪C 2 is a compatible family of polymers, but now it is not a polymer chain.For example, if base C 1 l u , r u and base C 2 l u , r u have l u − r u > 2, then C 1 ∪ C 2 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
e μξk u e −|k u |μ e μζk l e −|k l |μ , 3.14 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 {η 1 , . . ., η m }, let base C base η 1 ∪ • • • ∪ base η m .Further, for a polymer η, from 3.9 and 3.11 , we define where |base η | r − l 1 for base η l, r .By the formula 3.16 , for a polymer chain C {η 1 , . . ., η m }, we put 3.17 where J l , J r , J are defined in the following way.For base η l, r and base η c, d with c > r, we define where l−1 η c η means over polymers η γ u , γ l , ∅, ∅, ∅ with base{ l − 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 ∪ ν C ν or by the concatenation of γ l and γ l together with ∪ ν D ν .Similarly, and J η, η is defined in the following two cases.
1 If c r 2, η is r-active and η is l-active, then ..,C p −N r γ l ,D 1 ,...,D q −N l γ u ,C 1 ,...,C p −N l γ l ,D 1 ,...,D q . 3.20 2 If c r 1, η and η are compatible, then 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.

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 μ 1 is large enough, then for some δ > 0, one has 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 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 {x 0 , x 1 , . . ., x n } be a self-avoiding path with base γ u I.For j 0 0 and i ≥ 1, we let j i min j > j i−1 ; x j−1 , x j is a horizontal edge .

4.3
Each vertical part {x j i−1 , x j i−1 1 , . . ., x j i −1 } of γ u with the direction of the exit vector {x j i −1 , x j i } is called a fragment.For a fragment f {x 0 , x 1 , . . ., x p } with exit direction e f , we define Then the decomposition of γ u into fragments {f 1 , . . ., f r } leads to the identity Therefore, if μ 1 is sufficiently large, we have 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 μ 2 > 0; if μ > μ 2 , then for any ξ, ζ ∈ C with the conditions Re ξ < 1 − δ/μ and Re ζ < 1 − δ/μ, and for each polymer η 0 , one has where K L is the set of all decorated polymers with the base in 0, L , c η 3|base η |, and where μ 1 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; decorated polymer −→ 0, ∞ , 4.9 such that the above estimate 4.7 exists.First we consider the statistical properties of and we have

4.11
By the definition of the decorated polymers, if base η is one point, then Let γ u , γ l be the backbone of some decorated polymer with the base I l, r .Next we estimate the following function G γ u , γ l η; γ u ,γ l is the backbone of η e μξh γ u e μζh γ l Ψ η .

4.17
The summation over {Λ α } is estimated as follows:

4.18
There exist constants K 1 , κ > 0 such that the number N m of connected sets of m points in Z 2 which contain the origin is bounded as then we know that the function goes to zero exponentially fast as μ 1 → ∞.Thus, from 4.17 and 4.18 we obtain

4.21
where ii If l < r; that is, |I| ≥ 2, then we have

4.23
From the formula 4.13 , we have

4.24
Further, according to the formula 4.18 , we have where

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 c η c γ u , γ l , d η d γ u , γ l .

4.31
Then, for a fixed interval I, we have η; γ u ,γ l is the backbone of η e c η d η e μξh γ u e μζh γ l Ψ η G γ u , γ l e c γ u ,γ l d γ u ,γ l where base γ u , γ l base η for any η such that γ u , γ l is the backbone of η.

4.34
From 3.14 , there is a

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 3.1-4.2are 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

4.38
In the following part, we will show the existence of the following limit:

4.46
Assuming that μ 2 > 3μ 1 μ 3 2μ 3 6, then, from the inequalities 4.45 and 4.46 and if μ 1 is large enough by following the similar estimate procedure of 4.37 in Lemma 4.2 , we obtain R μ 1 , δ 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 1 and the ending point L, −b 2 .Let S b,u L,M Γ u σ ; σ ∈ Ω b Lof 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

m 1 } 1 . 3 . 7
For each m ∈ {0, 1, . . ., n 1}, there are only two edges e u m of π Γ u , and π Γ l , respectively, which intersect j m .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 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 ofj m−1 1, j m × −∞, ∞ ∩ Z 2 .Then A u m j m−1 1, p u , A l m j m−11, p l for some p u , p l ∈ Z, where p u > p l .Thus, we obtain the mth polymer η m by settingη m γ u m − 0, p u , γ l m − 0, p u ,By the above definitions, {η 0 , η 1 , . . ., η n 1 } are compatible.m be the height of the endpoint of the self-avoiding path γ u m the heights h π Γ u and h π Γ l are given by
C ν and the line {x 1 r 1}.Similarly, we can give the definition of N * γ l , D 1 , . .., 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 1} such that η 1 is incompatible with η, or the concatenation of γ u and γ u 1 together with ∪ ν C ν produces a new hole, or the concatenation of γ l and γ l 1 together with ∪ ν D ν produces a new hole.η is said to be l-active if there exists a simple polymer η 2 γ u 2 , γ l 2 , ∅, ∅, ∅ with base η 2 { r − 1} such that η 2 is incompatible with η, the concatenation of γ u and γ u 2 together with ∪ ν C ν produces a new hole, or the concatenation of γ l and γ l 2 together with ∪ ν D ν produces a new hole.If η is both r-active and l-active, we call it bi-active.
p ν 1 C ν and the line {x 1 l−1}, where base η l, r ; N r γ u , C 1 , . . ., C p is the number of new holes created by V γ u ∪ ∪ p ν 1 A polymer chain is a family of decorated polymers C {η 1 , . . ., η m } such that 1 {η 1 , . . ., η m } are compatible; if r u < l u 1 < r u 2 for each 1 ≤ u ≤ n.So that the base of a polymer chain forms the linked intervals.For a fixed polymer chain C 0 , let base C 0 l 0 , r 0 be the smallest intervals in length including base C 0 .Let