On free energies of the Ising model on the Cayley tree
OON FREE ENERGIES OF THE ISING MODEL ON THE CAYLEYTREE
D. GANDOLFO, M.M. RAKHMATULLAEV, U. A. ROZIKOV, J. RUIZ
Abstract.
We present, for the Ising model on the Cayley tree, some explicit formulaeof the free energies (and entropies) according to boundary conditions (b.c.). They in-clude translation-invariant, periodic, Dobrushin-like b.c., as well as those correspondingto (recently discovered) weakly periodic Gibbs states. The later are defined through apartition of the tree that induces a 4-edge-coloring. We compute the density of eachcolor.
Mathematics Subject Classifications (2010).
Key words.
Cayley tree, Ising model, boundary condition, Gibbs measure, freeenergy, entropy. 1.
Introduction and definitions
On non-amenable graphs, not only Gibbs measures but also the free energy (and theentropy) depend on the boundary conditions.The purpose of this paper is to study this dependence for one of the simplest suchgraph, the Cayley tree (Bethe lattice). Our analysis is restricted to the Ising model.Let Γ k = ( V, L ) be the uniform Cayley tree, where each vertex has k + 1 neighborswith V being the set of vertices and L the set of edges.On this tree, there is a natural distance to be denoted d ( x, y ), being the number ofnearest neighbor pairs of the minimal path between the vertices x and y (by path onemeans a collection of nearest neighbor pairs, two consecutive pairs sharing at least agiven vertex).The Ising model is defined by the formal Hamiltonian H ( σ ) = − J (cid:88) (cid:104) x,y (cid:105)⊂ V σ ( x ) σ ( y ) , (1.1)where the sum runs over nearest neighbor vertices (cid:104) x, y (cid:105) and the spins σ ( x ) take valuesin the set Φ = { +1 , − } .For a fixed x ∈ V , the root, let W n = { x ∈ V : d ( x, x ) = n } , V n = { x ∈ V : d ( x, x ) ≤ n } be respectively the sphere and the ball of radius n with center at x , and for x ∈ W n let S ( x ) = { y ∈ W n +1 : d ( y, x ) = 1 } , be the set of direct successors of x . a r X i v : . [ m a t h - ph ] N ov D. GANDOLFO, M.M. RAKHMATULLAEV, U. A. ROZIKOV, J. RUIZ
The (finite-dimensional) Gibbs distributions at inverse temperature β = 1 /T are de-fined by µ n ( σ n ) = Z − n exp (cid:110) βJ (cid:88) (cid:104) x,y (cid:105)⊂ V n σ ( x ) σ ( y ) + (cid:88) x ∈ W n h x σ ( x ) (cid:111) , (1.2)with partition functions given by Z n ≡ Z n ( β, h ) = (cid:88) σ n ∈ Φ Vn exp (cid:110) βJ (cid:88) (cid:104) x,y (cid:105)⊂ V n σ ( x ) σ ( y ) + (cid:88) x ∈ W n h x σ ( x ) (cid:111) . (1.3)Here h = { h x ∈ R, x ∈ V } is a collection of real numbers that stands for (generalized) boundary condition.The probability distributions (1.2) are said compatible if for all σ n − (cid:88) ω n ∈ Φ Wn µ n ( σ n − , ω n ) = µ n − ( σ n − ) . (1.4)It is well known that this compatibility condition is satisfied if and only if for any x ∈ V the following equation holds h x = (cid:88) y ∈ S ( x ) f ( h y , θ ) , (1.5)where θ = tanh( βJ ) , f ( h, θ ) = arctanh( θ tanh h ) . (1.6)Namely, for any boundary condition satisfying the functional equation (1.5) thereexists a unique Gibbs measure, the correspondence being one-to-one.We will be interested in the dependence with respect to boundary conditions of thefree energy defined as the limit F ( β, h ) = − lim n →∞ β | V n | ln Z n ( β, h ) , (1.7)where | · | denotes hereafter the cardinality of a set.A boundary condition satisfying (1.5) will be in the sequel called compatible .The paper is organized as follows. Section 2 provides the first part of results: a gen-eral formula applied then to various known boundary conditions (translation-invariant,Bleher-Ganikhodjaev, Zachary, ART), and those about entropy. Periodic and weaklyperiodic cases are the subject of Section 3. A first appendix concerns the density ofcolors mentioned in the abstract. A second appendix provides a sufficient condition forthe existence of the free energy in case of compatible boundary conditions.2. General formula and first results
For compatible boundary conditions, the free energy is given by the formula F ( β, h ) = − lim n →∞ | V n | (cid:88) x ∈ V n a ( x ) , (2.1) N FREE ENERGIES OF ISING MODEL 3 where a ( x ) = 12 β ln[4 cosh( h x − βJ ) cosh( h x + βJ )] . To see it, first notice that Z n ( β, h ) = A n − Z n − ( β, h ) , (2.2)where A n = (cid:81) x ∈ W n b ( x ) with b ( x ) satisfying (cid:89) y ∈ S ( x ) (cid:88) u = ± exp( βJ εu + uh y ) = b ( x ) exp( εh x ) , ε = ± . (2.3)This formula used with both values of ε implies b ( x ) = (cid:89) y ∈ S ( x ) (4 cosh( h y − βJ ) cosh( h y + βJ )) / = exp (cid:16) β (cid:88) y ∈ S ( x ) a ( y ) (cid:17) . It is then enough to insert this formula into the recursive equation (2.2) to get by iteration Z n ( β, h ) = (cid:89) x ∈ V n − b ( x )which gives (2.1).Notice that F ( β, h ) = F ( β, − h ) , (2.4)where − h = {− h x , x ∈ V } .2.1. Translation-invariant boundary conditions.
They correspond to constant func-tions, h x = h , in which case the condition (1.5) reads h = kf ( h, θ ) . (2.5)The equation (2.5) has a unique solution h = 0, if θ ≤ θ c = k and three distinct solutions h = 0 , ± h ∗ ( h ∗ > θ > θ c .Let us denote by µ , µ ± the corresponding Gibbs measures and recall the followingknown results for the ferromagnetic Ising model ( θ ≥ θ ≤ θ c , µ is unique and extreme.(2) If θ > θ c , µ − and µ + , are extreme.(3) µ is extreme if and only if θ < √ k .(see e.g. [8], [6], [3])According to formula (2.1), the free energies of translation-invariant (TI) b.c. aregiven by: F TI ( β,
0) = − β ln(2 cosh( βJ )) . (2.6) F TI ( β, h ∗ ) = F TI ( β, − h ∗ ) = − β ln[4 cosh( βJ − h ∗ ) cosh( βJ + h ∗ )] . (2.7)Some particular plots are shown in Fig. 1. D. GANDOLFO, M.M. RAKHMATULLAEV, U. A. ROZIKOV, J. RUIZ (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Β F k (cid:61) (cid:61) (cid:61) Figure 1.
The free energies F TI ( β,
0) (solid line) and F TI ( β, h ∗ ) (dottedlines) for θ ≥ θ c with J = 1 and k = 4 , , F T I ( β, h ) as a function of β , we notice that theequation (2.5) gives β ( h ) = 12 J ln e (1+ k )2 h − e h − e hk (2.8)and use the parametric representation defined by the following mapping: h → ( β ( h ) , F ( h )) , for h ≥ . The function F ( h ) is defined by inserting (2.8) into (2.6) and (2.7).2.2. Bleher-Ganikhodjaev construction.
Here one consider the half tree. Namelythe root x has k nearest neighbours. Consider an infinite path π = { x = x < x < . . . } (the notation x < y meaning that pathes from the root to y go through x ). Associate tothis path a collection h π of numbers given by the condition h πx = (cid:40) − h ∗ , if x ≺ x n , x ∈ W n ,h ∗ , if x n ≺ x, x ∈ W n , (2.9) N FREE ENERGIES OF ISING MODEL 5 n = 1 , , . . . where x ≺ x n (resp. x n ≺ x ) means that x is on the left (resp. right) fromthe path π .For any infinite path π , the collection of numbers h π satisfying relations (1.5) existsand is unique (see [2]).A real number t = t ( π ), 0 ≤ t ≤ h π ( t ) is uniquely defined. The set of numbers h π ( t ) being distinct for different t ∈ [0 , h π can differ from h ∗ or − h ∗ only on the path π , it is thus obviousthat the values h πx , x ∈ π do not contribute to the free energy. As a consequence, we getfrom the evenness (2.7), that the free energies of Bleher-Ganikhodjaev and translation-invariant boundary conditions coincide: F BG ( β, h π ) = F TI ( β, h ∗ ) . (2.10)2.3. Zachary construction.
This construction provides an (uncountable) set of dis-tinct functions h ( t ) satisfying condition (1.5) and parameterized by t ∈ ( − h ∗ , h ∗ ). It isassumed here that θ > θ c .Take t (cid:54) = 0 and define the sequence ( t n ) n ≥ recursively by t = t , t n = kf ( t n +1 , θ ) , n ≥ . (2.11)Since the function f is increasing and maps the interval ( − h ∗ , h ∗ ) into itself, the definitionof t n make sense. Moreover one can see that lim n →∞ t n = 0 for each t = t .Consider the function h ( t ) x = t n for all x ∈ W n . This function satisfies condition(1.5) for any t and by construction, distinct t assign distinct h ( t ) . The associated Gibbsmeasures are known to be extreme [11].The corresponding free energies can be written as F Zach ( β, h ( t ) ) = − β lim n →∞ | V n | n (cid:88) m =0 | W m | ˜ a ( t m ) , where ˜ a ( t ) = ln[4 cosh( βJ − t ) cosh( βJ + t )] . By Stolz-Ces´aro theorem (see e.g. [7]) applied to the sequences a n = n (cid:88) m =0 | W m | ˜ a ( t m ) , b n = | V n | (2.12)one haslim n →∞ | V n | n (cid:88) m =0 | W m | ˜ a ( t m ) = lim n →∞ n +1 (cid:88) m =0 | W m | ˜ a ( t m ) − n (cid:88) m =0 | W m | ˜ a ( t m ) | V n +1 | − | V n | = lim n →∞ | W n +1 | ˜ a ( t n +1 ) | W n +1 | = lim n →∞ ˜ a ( t n +1 ) = ˜ a (0) . (2.13) D. GANDOLFO, M.M. RAKHMATULLAEV, U. A. ROZIKOV, J. RUIZ
As a consequence F Zach ( β, h ( t ) ) = F TI ( β, . (2.14)2.4. ART construction.
Let h be a boundary condition satisfying (1.5) on Γ k . For k ≥ k + 1 define the following boundary condition on Γ k :˜ h x = (cid:40) h x , if x ∈ V k , if x ∈ V k \ V k , (2.15)where V k denote the set of vertices of Γ k . Namely, to each vertices of V k one adds k − k successors with vanishing value of the boundary condition. It is obvious the b.c. ˜ h satisfy the compatibility condition (1.5). In this way one constructs a new set of Gibbsmeasures that are extreme in the range 1 /k < θ < / √ k [1].For the corresponding free energy, we have F ART ( β, ˜ h ) = − β lim n →∞ | V kn | (cid:16) | V kn | − | V k n | (cid:17) ln[2 cosh( βJ )] + β (cid:88) x ∈ V k n a ( x ) . (2.16)Since lim n →∞ | V k n || V kn | = k − k − n →∞ ( k + 1) k n − k + 1) k n − , by taking into account 0 ≤ a ( x ) ≤ C b , we get0 ≤ (cid:88) x ∈ V k n a ( x ) ≤ | V k n | C β . As a consequence, lim n →∞ | V kn | (cid:88) x ∈ V k n a ( x ) = 0 , so that F ART ( β, ˜ h ) = − β ln[2 cosh( βJ )] = F TI ( β, . (2.17)2.5. Entropy.
To compute the entropy S ( β, h ) = − dF ( β,h ) dT , we first notice that equation(2.8) gives h (cid:48) ( β ) = 1 β (cid:48) ( h ) = J k cosh(2 h ) − cosh (cid:0) hk (cid:1) sinh(2 h ) − k sinh (cid:0) hk (cid:1) . (2.18)As a result of easy computations, we get the formula S ( β, h ) = 12 ln (cid:2) h ) + 2 cosh(2 βJ ) (cid:3) + βJ k sinh(2 h ) cosh(2 h ) − cosh ( hk ) sinh(2 h ) − k sinh ( hk ) + sinh(2 βJ )cosh(2 h ) + cosh(2 βJ ) (2.19) N FREE ENERGIES OF ISING MODEL 7 and S ( β,
0) = ln(2 cosh( βJ )) − βJ tanh( βJ ) . (2.20)Let us mention that in case k = 2, we get by solving equation (2.5) the following valueof h ∗ as a function of the inverse temperature: ± h ∗ = 12 ln (cid:20) − (cid:18) e βJ − e βJ − ± ( e βJ − (cid:113) ( e βJ + 1)( e βJ − (cid:19)(cid:21) . (2.21)The free energies and entropies then read F ( β, h ∗ ) = − β (cid:18) ln( e βJ −
1) + 12 ln( e − βJ + 1) (cid:19) , (2.22) S ( β, h ∗ ) = ln(2 sinh( βJ )) + 12 ln(2 cosh( βJ )) − βJ βJ ) + 12 sinh(2 βJ ) . (2.23)3. Periodic and weakly periodic Gibbs measures
A group representation of the Cayley tree.
Let G k be a free product of k + 1cyclic groups of the second order with generators a , a , . . . , a k +1 , respectively.It is known that there exists a one-to-one correspondence between the set of vertices V of the Cayley tree Γ k and the group G k .To give this correspondence we fix an arbitrary element x ∈ V and let it correspondto the unit element e of the group G k . Using a , . . . , a k +1 we numerate nearest-neighborsof element e , moving by positive direction (see Fig. 2). Now we shall give numerationof the nearest-neighbors of each a i , i = 1 , . . . , k + 1 by a i a j , j = 1 , . . . , k + 1. Since all a i have the common neighbor e we give to it a i a i = a i = e . Other neighbors are numeratedstarting from a i a i by the positive direction. We numerate the set of all nearest-neighborsof each a i a j by words a i a j a q , q = 1 , . . . , k + 1, starting from a i a j a j = a i by the positivedirection. Iterating this argument one gets a one-to-one correspondence between the setof vertices V of the Cayley tree Γ k and the group G k .In the group G k , let us consider the left (right) shift transformations defined as follows.For g ∈ G k , let us set T g ( h ) = gh, ( T g ( h ) = hg ) , for all h ∈ G k . (3.1)The set of all left (right) shifts in G k is isomorphic to the group G k .3.2. Periodic boundary conditions.
In this subsection, we consider periodic solutionsof (1.5) and use the above group structure of the Cayley tree.
Definition 1.
Let ˜ G be a normal subgroup of the group G k . The set h = { h x : x ∈ G k } is said to be ˜ G -periodic if h yx = h x for any x ∈ G k and y ∈ ˜ G . Let G (2) k = { x ∈ G k : the length of word x is even } . Note that G (2) k is the set of even vertices (i.e. with even distance to the root). Considerthe boundary conditions h ± and h ∓ : D. GANDOLFO, M.M. RAKHMATULLAEV, U. A. ROZIKOV, J. RUIZ b b b b b b b b b b b b b b b bb b b b b b b bb b bb bb bbb bb e a a a a a a a a a a a a a a a a a a Figure 2.
Some elements of group G on Cayley tree of order two. h ± x = − h ∓ x = h ∗ , if x ∈ G (2) k − h ∗ , if x ∈ G k \ G (2) k , (3.2)and denote by µ ( ∓ ) , µ ( ± ) ) the corresponding Gibbs measures.The ˜ G - periodic solutions of equation (1.5) are either translation-invariant ( G k -periodic)or G (2) k -periodic (see [5]), they are solutions to u = kf ( v, θ ) , v = kf ( u, θ ) . (3.3)In the ferromagnetic case only translation invariant b.c. can be found. In the antifer-romagnetic case ( θ ≤
0) the system (3.3) has a unique solution h = 0 if θ ≥ − /k , andthree distinct solutions h = 0, h ± and h ∓ if θ < − /k .Let us also recall that for the antiferromagnetic Ising model:(1) If θ ≥ − /k , µ is unique and extreme.(2) If θ < − /k , µ ( ± ) and µ ( ∓ ) , are extreme.see [6].For periodic measures, we have F Per ( β, h ( ± ) ∗ ) = − β ln[4 cosh( βJ − h ∗ ) cosh( βJ + h ∗ )] = F TI ( β, h ∗ ) . Weakly periodic Gibbs measures.
Let G k / (cid:98) G k = { H , ..., H r } be a factor group,where (cid:98) G k is a normal subgroup of index r ≥ N FREE ENERGIES OF ISING MODEL 9
Definition 2.
A set h = { h x , x ∈ G k } is called (cid:98) G k - weakly periodic, if h x = h ij , forany x ∈ H i , x ↓ ∈ H j , where x ↓ denotes the ancestor of x . Weakly periodic b.c. h coincide with periodic ones if h x is independent of x ↓ .Here, we will restrict ourself to the cases of index two and recall that any such subgrouphas the form H A = (cid:40) x ∈ G k : (cid:88) i ∈ A ω x ( a i ) − even (cid:41) , where ∅ (cid:54) = A ⊆ N k = { , , . . . , k + 1 } , and ω x ( a i ) is the number of a i in a word x ∈ G k .We consider A (cid:54) = N k : when A = N k weak periodicity coincides with standard periodicity.Let G k /H A = { H , H } be the factor group, where H = H A , H = G k \ H A . Then,in view of (1.5), the H A -weakly periodic b.c. has the form h x = h , x ∈ H , x ↓ ∈ H ,h , x ∈ H , x ↓ ∈ H ,h , x ∈ H , x ↓ ∈ H ,h , x ∈ H , x ↓ ∈ H , (3.4)where the h i satisfy the following equations: h = | A | f ( h , θ ) + ( k − | A | ) f ( h , θ ) ,h = ( | A | − f ( h , θ ) + ( k + 1 − | A | ) f ( h , θ ) ,h = ( | A | − f ( h , θ ) + ( k + 1 − | A | ) f ( h , θ ) ,h = | A | f ( h , θ ) + ( k − | A | ) f ( h , θ ) . (3.5)For sake of simplicity, consider k = 4 and | A | = 1. In this case H = { x ∈ G k : ω x ( a ) is even } ,H = { x ∈ G k : ω x ( a ) is odd } . These sets are shown in Fig. 3.Let us recall the following results of [9]:There exists a critical value α cr ( ≈ , α = e − βJ such that:(1) If α > α cr , there exists a unique weakly periodic state µ ,(2) If α = α cr , there are three distinct weakly periodic states µ , µ − , µ +1 .(3) If 0 ≤ α < α cr , there are five distinct weakly periodic states µ , µ − , µ +1 , µ − , µ +2 .These measures correspond to solutions of system (3.5) on the invariant set h = − h , h = − h , that is to solutions of: h = 3 f ( h , θ ) − f ( h , θ ) ,h = 4 f ( h , θ ) . (3.6)More results about weakly periodic Gibbs measures can be found in [9], [10]. Figure 3.
The sets H (black vertices) and H (gray vertices).Denote A n = |{(cid:104) x, y (cid:105) ∈ L n : x ∈ H , y = x ↓ ∈ H }| , B n = |{(cid:104) x, y (cid:105) ∈ L n : x ∈ H , y = x ↓ ∈ H }| , (3.7) C n = |{(cid:104) x, y (cid:105) ∈ L n : x ∈ H , y = x ↓ ∈ H }| , D n = |{(cid:104) x, y (cid:105) ∈ L n : x ∈ H , y = x ↓ ∈ H }| , where L n is the set of edges in V n . N FREE ENERGIES OF ISING MODEL 11
For weakly periodic b.c. (3.6) we have F WP ( β, h ) = − β lim n →∞ | V n | { ( A n + D n ) ln[4 cosh( βJ − h ) cosh( βJ + h )]+( B n + C n ) ln[4 cosh( βJ − h ) cosh( βJ + h )] } . By Proposition 1 of the appendix we obtain F WP ( β, h ) = − β (cid:26)
45 ln[4 cosh(
J β − h ) cosh( J β + h )]+ 15 ln[4 cosh( J β − h ) cosh( J β + h )] (cid:27) (3.8)In the case under consideration the proposition is straightforward. Indeed one imme-diately see, in view of definitions and Fig. 3, that ( A n + D n ) / ( | V n | −
1) = 4 / B n + C n ) / ( | V n | −
1) = 1 / n = 1; the induction is trivial.The equation h = 3 f ( h , θ ) − f (4 f ( h , θ ) , θ ) (3.9)that solves the system (3.6) (with h = 4 f ( h , θ )) can then be reduced to: α ξ − αξ − α ξ + α + 1 = 0 , (3.10)which has two solutions ξ and ξ when 0 < α < α cr (see [9]).The free energy then reads: F WP ( β, h ) = − β (cid:26) (cid:20) βJ ) + 2 ξ cosh(2 βJ ) −
42 cosh(2 βJ ) − ξ (cid:21) + ln[2 cosh(2 βJ ) + ξ − ξ + 2] (cid:27) . The solution ξ is given by ξ = 13 (cid:32) e β + 2 / (cid:0) e β (cid:1) U + 2 − / U (cid:33) , where U = (cid:16) − e β − e β + 2 e β + 3 (cid:112) − − e β + 54 e β + 69 e β − e β (cid:17) / . The corresponding free energy reads F WP ( β ) = − β ln (cid:0) − e − β + e β (cid:1) (cid:16) e − β + e β − (cid:0) e β W (cid:1) + (cid:0) e β W (cid:1) (cid:17)(cid:0) e − β + e β − e β W (cid:1) , where W = (cid:18) e − β + VU + 2 / e − β U (cid:19) with V = 2 / e − β (cid:16) e β (cid:17) . The plot is given in Fig. 4 from which we observe the strict inequalities F TI ( β, h ∗ ) < F WP ( β, h ) < F TI ( β, . (3.11) in the range α ≤ α cr . (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Β F Figure 4.
The free energies F WP ( β ) (dotted line) for α ≤ α cr to-gether with the previous free energies F TI ( β,
0) (solid line), and F TI ( β, h ∗ )(dashed line). Here J = 1 and k = 4.The results for weakly periodic boundary conditions have to be compared with theinequalities given recently in [4]; the weakly periodic b.c. (3.6) corresponding to theso-called dimer covering in [4]. There, the inequalities are easier to catch with clusterexpansion method in mind, the condition on the temperature is more restrictive, andthe free energies cannot be expressed explicitly. Appendix: Density of edges in a ball
In this appendix we consider a group representation of a Cayley tree and its partitionwith respect to an arbitrary subgroup of index two. This partition gives a 2-vertex-coloring on Cayley tree, which then gives 4-edge-coloring, say, colors i = 1 , , ,
4. Wefix a root of the Cayley tree and give explicit formulas for number A n,i of edges withcolor i in a ball V n of radius n with the center at the root. Moreover, we compute thelim n →∞ ( A n,i / | V n | ) for each i = 1 , , , N FREE ENERGIES OF ISING MODEL 13
We will use the notation of Subsection 3.3 and let α n = |{(cid:104) x, y (cid:105) ∈ L n \ L n − : x ∈ H , y = x ↓ ∈ H }| .β n = |{(cid:104) x, y (cid:105) ∈ L n \ L n − : x ∈ H , y = x ↓ ∈ H }| .γ n = |{(cid:104) x, y (cid:105) ∈ L n \ L n − : x ∈ H , y = x ↓ ∈ H }| .δ n = |{(cid:104) x, y (cid:105) ∈ L n \ L n − : x ∈ H , y = x ↓ ∈ H }| , for A = { , , , ..., j } and 1 ≤ j ≤ k + 1.Let M be the set of all unit balls with vertices in V and let S ( x ) denotes the set ofall nearest neighbors of x . For b ∈ M the center of b is denoted by c b . Lemma 1. If c b ∈ H , then |{ x ∈ S ( c b ) : x ∈ H }| = j, |{ x ∈ S ( c b ) : x ∈ H }| = k − j + 1 . If c b ∈ H , then |{ x ∈ S ( c b ) : x ∈ H }| = k − j + 1 , |{ x ∈ S ( c b ) : x ∈ H }| = j. Proof.
We have S ( c b ) = { c b a p : p = 1 , , . . . , k + 1 } . If c b ∈ H , (the case c b ∈ H is similar) then c b a p ∈ H for p = 1 , , ..., j and c b a p ∈ H for p = j + 1 , j + 2 , ..., k + 1 , i.e. we have |{ x ∈ S ( c b ) : x ∈ H }| = j, |{ x ∈ S ( c b ) : x ∈ H }| = k − j + 1 . (cid:3) Consider b = V ∈ M with the center x = e ∈ H , then in W we have j verticeswhich belong to H , and k − j + 1 vertices which belong in H , consequently, α = k − j + 1 , β = 0 , γ = j, δ = 0 . Lemma 2.
For any n ∈ N the following recurrence system hold α n +1 = ( k − j ) α n + ( k − j + 1) β n β n +1 = ( j − γ n + jδ n γ n +1 = ( j − β n + jα n δ n +1 = ( k − j ) δ n + ( k − j + 1) γ n , (A.1) with initial values α = k − j + 1 , β = 0 , γ = j , δ = 0 .Proof. By Lemma 1, an edge (cid:104) x, y (cid:105) ∈ L n \ L n − with x ∈ H , y = x ↓ ∈ H has ( k − j )neighbor edges (cid:104) z, x (cid:105) ∈ L n +1 \ L n with z ∈ H , x = z ↓ ∈ H . An edge (cid:104) z, t (cid:105) ∈ L n \ L n − with z ∈ H , t = z ↓ ∈ H has ( k − j + 1) neighbor edges (cid:104) u, z (cid:105) ∈ L n +1 \ L n with u ∈ H , z = u ↓ ∈ H . Moreover, it is easy to see that only α n and β n have contributionto α n +1 . Hence we have α n +1 = ( k − j ) α n + ( k − j + 1) β n . Other equations of the system(A.1) can be obtained by a similar way. (cid:3) Remark 1.
For j = k + 1 by Lemmas 1 and 2 we get α n = δ n = 0 , for any n ≥ and β n = (cid:40) , if n = 2 m − k + 1) k m − , if n = 2 m , m = 1 , , . . .γ n = (cid:40) , if n = 2 m ( k + 1) k m − , if n = 2 m − , m = 1 , , . . . So in the sequel of this section we consider j as ≤ j ≤ k . Lemma 3.
For α n we have α n +2 = j ( k − j + 1) | W n | + ( k − j ) α n +1 − (2 j − α n − kα n − , n ≥ , (A.2) with initial values α = k − j + 1 , α = ( k − j )( k − j + 1) , α = (cid:0) ( k − + j ( j − (cid:1) ( k − j + 1) (A.3) Proof.
The initial values follow from Lemma 2.By definitions of α n , β n , γ n , δ n we have α n + β n + γ n + δ n = | W n | = k n − ( k + 1) , n ≥ . (A.4)From (A.1) we get β n = k − j +1 ( α n +1 − ( k − j ) α n ) ,γ n = j − k − j +1 ( α n − ( k − j ) α n − ) + jα n − ,δ n = j ( β n +1 − ( j − γ n ) . (A.5)Substituting these values in (A.4) and then simplifying we get (A.2). (cid:3) To find solution of (A.2) we denote α n = q n k n − . (A.6)From (A.2) we get k n q n +2 = k n − ( k + 1)( k − j + 1) j + ( k − j ) k n − q n +1 − (2 j − k n − q n − k n − q n − , dividing by k n we obtain q n +2 = ( k + 1)( k − j + 1) jk + k − jk q n +1 − j − k q n − k q n − , (A.7)with initial values q = k ( k − j + 1) , q = ( k − j )( k − j + 1) , q = (cid:0) ( k − + j ( j − (cid:1) ( k − j + 1) k . (A.8)In order to find solution to (A.7) first we rid ( k +1)( k − j +1) jk by denoting q n = p n + k ( k − j + 1)2 . (A.9) N FREE ENERGIES OF ISING MODEL 15
Substituting (A.9) to (A.7) we get p n +2 = k − jk p n +1 − j − k p n − k p n − , (A.10)with p = 12 k ( k − j + 1) , p = ( k − j )( k − j + 1)2 , p = ( k − k + 2 + 2 j − j )( k − j + 1)2 k . (A.11)The characteristic equation for (A.10) has the following form (setting p n = λ n ): λ − k − jk λ + 2 j − k λ − k = 0 , which has solutions λ = − k , λ , = k − j + 1 ± (cid:112) ( k − j ) − k + 2 j ) + 12 k . (A.12)Then the general solution to (A.10) is p n = A λ n + A λ n + A λ n , (A.13)where the coefficients A , A , A are determined by the initial conditions (A.11).Using (A.9) and (A.6) we get α n = k − j + 12 k n − + A · ( − n k + A k · ( kλ ) n + A k · ( kλ ) n . (A.14)Then using (A.5) and (A.14) one can find β n , γ n and δ n .We have A n = n (cid:88) m =1 α m = k − j + 12( k −
1) ( k n −
1) + A k (( − n −
1) + A λ kk ( λ k −
1) (( λ k ) n −
1) + A λ kk ( λ k −
1) (( λ k ) n − . (A.15) Proposition 1.
For any j = 1 , . . . , k and any fixed q = 0 , , , . . . we have lim n →∞ α n − q | V n | = lim n →∞ δ n − q | V n | = ( k − k − j + 1)2( k + 1) k q +1 . lim n →∞ β n − q | V n | = lim n →∞ γ n − q | V n | = ( k − j k + 1) k q +1 . lim n →∞ A n − q | V n | = lim n →∞ D n − q | V n | = k − j + 12( k + 1) k q . lim n →∞ B n − q | V n | = lim n →∞ C n − q | V n | = j k + 1) k q . Proof.
It is easy to check that | λ | < | λ | <
1, i.e. (cid:12)(cid:12)(cid:12) k − j + 1 ± (cid:112) ( k − j ) − k + 2 j ) + 1 (cid:12)(cid:12)(cid:12) < k, for any 1 ≤ j ≤ k. Using these inequalities, formula (A.14) and (A.16) we getlim n →∞ α n − q | V n | = ( k − k − j + 1)2( k + 1) k q +1 . Now using this formula together with (A.5) we obtainlim n →∞ β n − q | V n | = 1 k − j + 1 (cid:18) lim n →∞ α n +1 − q | V n | − ( k − j ) lim n →∞ α n − q | V n | (cid:19) = j ( k − k + 1) k q +1 . The formulae involving γ n and δ n are obtained in a similar way.By (A.15) and | V n | = ( k + 1) · k n − k − n →∞ A n − q | V n | = k − j + 12( k + 1) k q . From (A.5) we get B n = k − j +1 ( A n − α + α n +1 − ( k − j ) A n ) , C n = j − k − j +1 ( A n − ( k − j ) A n − ) + j A n − , D n = j ( B n − β + β n +1 − ( j − C n ) , (A.17)that allows to prove the remaining formulae. (cid:3) Remark 2.
By Proposition 1 it is clear that the values of A , A , A do not give anycontribution to the equalities of the proposition. This is why we did not compute A , A , A . But one can obtain the numbers by the initial conditions (A.11) for p n . Forexample, in the case k = 4 , j = 1 from (A.13) and (A.11) we have p n = A ( −
14 ) n + A (cid:32) − √ i (cid:33) n + A (cid:32) √ i (cid:33) n ,p = 8 , p = 4 , p = 1 . The initial conditions give − A + A −√ i + A √ i A + A − √ i + A √ i − A + A − − √ i + A − √ i ⇒ A = 0 A = 4 + √ i A = 4 − √ i . Consequently, (for k = 4 , j = 1 ) we have α n = 2 · n − + 14 (cid:32) − √ i (cid:33) (cid:32) √ i (cid:33) n + 14 (cid:32) √ i (cid:33) (cid:32) − √ i (cid:33) n . (A.18) N FREE ENERGIES OF ISING MODEL 17
Using (A.5) and (A.18) one can find β n , γ n and δ n . Moreover, we have A n = n (cid:88) m =1 α m = 2(4 n − √ i · (3 − √ i ) n − (3 + √ i ) n n . Note that α n and A n are natural numbers for any n ≥ . Remark 3.
In the case j = k + 1 by Remark 1 we get lim n →∞ α n | V n | = lim n →∞ δ n | V n | = lim n →∞ A n | V n | = lim n →∞ D n | V n | = 0 . lim m →∞ β m − | V m − | = lim m →∞ γ m | V m | = 0 . lim m →∞ β m | V m | = lim m →∞ γ m − | V m − | = k − k . lim m →∞ B m | V m | = lim m →∞ C m − | V m − | = kk + 1 . lim m →∞ B m − | V m − | = lim m →∞ C m | V m | = 1 k + 1 . Appendix: Existence of the free energy
As we have seen in the previous sections free energy exists for each known compatibleboundary condition. We note that a ( x ) is bounded: β − ln 2 ≤ a ( x ) ≤ C β . Hence limit(2.1) is also bounded. But the problem of convergence of (2.1) is still open. Here weshall give some conditions on h , under which the corresponding free energy exists.Let π = { x = x < x < . . . } be an infinite path. A function h x on the path π is called monotone non increasing (non decreasing) if h x i ≥ h x i +1 , ( h x i ≤ h x i +1 ), i = 0 , , , . . . . Proposition 2.
Let h = { h x , x ∈ V } be a compatible b.c. If on any infinite path startingat x ∈ W n , | h x | is monotone non increasing (non decreasing), then the correspondingfree energy F ( β, h ) exists.Proof. Using Stolz-Ces´aro theorem we getlim n →∞ | V n | (cid:88) x ∈ V n a ( x ) = lim n →∞ A n , with A n = 1 | W n | (cid:88) x ∈ W n a ( x ) . We shall show that A n is monotone for n > n . We have A n − − A n = 1 | W n | k (cid:88) x ∈ W n − a ( x ) − (cid:88) x ∈ W n a ( x ) =1 | W n | (cid:88) x ∈ W n − ka ( x ) − (cid:88) y ∈ S ( x ) a ( y ) = 1 | W n | (cid:88) x ∈ W n − (cid:88) y ∈ S ( x ) ( a ( x ) − a ( y )) . By monotonicity of | h x | and by evenness of a ( x ) we notice that a ( x ) − a ( y ) does notchange sign for all x, y with x < y . Thus A n is monotone and since it is a boundedsequence it has a limit. (cid:3) Acknowledgements
U. Rozikov thanks CNRS for support and The Centre de Physique Th´eorique DeMarseille, France for kind hospitality during his visit (September-December 2012).
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E-mail address : [email protected] [email protected] M. M. Rakhmatullaev, Namangan State University, Namangan, Uzbekistan.
E-mail address : [email protected] U. A. Rozikov, Institute of mathematics, 29, Do’rmon Yo’li str., 100125, Tashkent,Uzbekistan.
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