Ganikhodjaev, N N

Rozikov, U A

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Available at: http://www.ictp.it/~pub−off IC/2004/117United Nations Educational Scientific and Cultural OrganizationandInternational Atomic Energy AgencyTHE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSUNIQUENESS OF GIBBS MEASURE FOR POTTSMODEL WITH COUNTABLE SET OF SPIN VALUESN.N. Ganikhodjaev1International Islamic University Malaysia, 53100 Kuala Lumpur, Malaysia,National University of Uzbekistan, Vuzgorodok, 700095, Tashkent, Uzbekistan,andThe Abdus Salam International Centre for Theoretical Physics, Triesty, ItalyandU.A. Rozikov2Institute of Mathematics, 29, F. Hodjaev str., Tashkent, 700125, UzbekistanandThe Abdus Salam International Centre for Theoretical Physics, Triesty, Italy.AbstractWe consider a nearest-neighbor Potts model, with countable spin values 0, 1, . . ., and nonzero external field, on a Cayley tree of order k (with k + 1 neighbors). We study translation-invariant ‘splitting’ Gibbs measures. We reduce the problem to the description of the solutionsof some infinite system of equations. For any k ≥ 1 and any fixed probability measure ν withν(i) > 0 on the set of all non negative integer numbers Φ = {0, 1, ...} we show that the setof translation-invariant splitting Gibbs measures contains at most one point, independently onparameters of the Potts model with countable set of spin values on Cayley tree. Also we give afull description of the class of measures ν on Φ such that with respect to each element of thisclass our infinite system of equations has unique solution {ai, i = 1, 2, ...}, where a ∈ (0, 1).MIRAMARE – TRIESTENovember 20041nasirgani@hotmail.com2rozikovu@yandex.ru1 IntroductionThe Potts model (with q ≥ 2 spin values) was introduced as a generalization of the Ising model.The idea came from the representation of the Ising model as interacting spins which can beeither parallel or antiparallel. An obvious generalization was to extend the number of directionsof the spins. Such a model was proposed by C.Domb as a PhD thesis for his student R.Pottsin 1952. At present the Potts model encompasses a number of problems in statistical physicsand lattice theory (see, e.g. [10]). It has been a subject of increasing intense research interestin recent years.One of the central problems in the theory of Gibbs measures is to describe infinite-volume (orlimiting) Gibbs measures corresponding to a given Hamiltonian. The existence of such measuresfor a wide class of Hamiltonians was established in the ground-breaking work of Dobrushin (see,e.g. [5]). However, a complete analysis of the set of limiting Gibbs measures for a specificHamiltonian is often a difficult problem.The Potts model can be studied on both Zd and Cayley tree Γk. In [2] for the Potts modelwith q ≥ 2 spin values on the Cayley tree it was proven that there are q translation-invariant anduncountable many non translation-invariant extreme Gibbs measures. In [3] the Potts modelwith countable set Φ of spin values on Zd was considered and it was proven that with respectto Poisson distribution on Φ the set of limiting Gibbs measure is not empty.In this paper we consider Potts model with a nearest neighbor interaction and countable setof spin values on a Cayley tree.A Cayley tree Γk = (V,L) of order k ≥ 1 is an infinite homogeneous tree, i.e., a graphwithout cycles, with exactly k + 1 edges incident to each vertex. Here V is the set of verticesand L that of edges (arcs).We consider a model where the spin takes values in the set of all non negative integer numbersΦ := {0, 1, . . .}, and is assigned to the vertices of the tree. A configuration σ on V is then definedas a function x ∈ V 7→ σ(x) ∈ Φ; the set of all configurations is ΦV . The (formal) Hamiltonianof the Potts model is :H(σ) = −J∑〈x,y〉∈Lδσ(x)σ(y) − α∑x∈Vδ0σ(x), (1)where J, α ∈ R are constants. As usual, 〈x, y〉 stands for nearest neighbor vertices and δ is theKroneker’s symbol.We consider a standard sigma-algebra B of subsets of ΦV generated by the cylinder subsets.A probability measure µ on (ΦV ,B) is called a Gibbs measure (with Hamiltonian H) if it satisfiesthe DLR equation, namely for any n = 1, 2, . . . and σn ∈ ΦVn :µ({σ ∈ ΦV : σ|Vn = σn})=∫ΦVµ(dω)νVnω|Wn+1(σn),2where νVnω|Wn+1is the conditional probabilityνVnω|Wn+1(σn) =1Zn(ω|Wn+1) exp (−βH (σn ||ω|Wn+1)) ,and β = 1T, T > 0 is temperature. Here and below, Wl stands for a ‘sphere’ and Vl for a ‘ball’on the tree, of radius l = 1, 2, . . ., centered at a fixed vertex x0 (an origin):Wl = {x ∈ V : d(x, x0) = l}, Vl = {x ∈ V : d(x, x0) ≤ l};andLn = {〈x, y〉 ∈ L : x, y ∈ Vn};distance d(x, y), x, y ∈ V , is the length of (i.e. the number of edges in) the shortest pathconnecting x with y. ΦVn is the set of configurations in Vn (and ΦWn that in Wn; see below).Furthermore, σ|Vn and ω|Wn+1 denote the restrictions of configurations σ, ω ∈ ΦV to Vn andWn+1, respectively. Next, σn : x ∈ Vn 7→ σn(x) is a configuration in Vn and H(σn ||ω|Wn+1)isdefined as the sum H (σn) + U(σn, ω|Wn+1)whereH (σn) = −J∑〈x,y〉∈Lnδσn(x)σn(y) − α∑x∈Vnδ0σn(x),U(σn, ω|Wn+1)= −J∑〈x,y〉: x∈Vn,y∈Wn+1δσn(x)ω(y).Finally, Zn(ω|Wn+1)stands for the partition function in Vn, with the boundary condition ω|Wn+1 :Zn(ω|Wn+1)=∑σ˜n∈ΦVnexp(−βH(σ˜n ||ω|Wn+1)).According to the nearest-neighbour character of the interaction, the Gibbs measures possessa natural Markov property: for given a configuration ωn on Wn, random configurations in Vn−1(i.e., ‘inside’ Wn) and in V \ Vn+1 (i.e., ‘outside’ Wn) are conditionally independent.We use a standard definition of a translation-invariant measure (see, e.g., [5]). The mainobject of study in this paper are translation-invariant Gibbs measures for Potts model withcountable set of spin values on Cayley tree .2 An infinite system of functional equationsFollowing [4], [6] we consider a special class of Gibbs measures. These measures are called in[4], [6] Markov chains, in [8], [9] - entrance laws and in [7] - splitting Gibbs measures. In thispaper we also call them splitting Gibbs measures.Write x < y if the path from x0 to y goes through x. Call vertex y a direct successor of x ify > x and x, y are nearest neighbors. Denote by S(x) the set of direct successors of x. Observethat any vertex x 6= x0 has k direct successors and x0 has k + 1.3For A ⊂ V denote by ΦA the configuration space on A. Let h : x 7→ hx = (h0,x, h1,x, ...) ∈R∞ be a real sequence-valued function of x ∈ V \ {x0}. Fix a probability measure ν = {ν(i) >0, i ∈ Φ}. Given n = 1, 2, . . ., consider the probability distribution µn on ΦVn defined byµ(n)(σn) = Z−1n exp−βH(σn) + ∑x∈Wnhσ(x),x ∏x∈Vnν(σ(x)), (2)Here, as before, σn : x ∈ Vn 7→ σ(x) and Zn is the corresponding partition function:Zn =∑σ˜n∈ΦVnexp−βH(σ˜n) + ∑x∈Wnhσ˜(x),x ∏x∈Vnν(σ˜(x)). (3)Remark. Note that Zn is finite, since ν is a probability measure and exp(−βH(σ˜n)+∑x∈Wn hσ˜(x),x) is bounded on ΦVn .We say that the probability distributions µ(n) are compatible if for any n ≥ 1 and σn−1 ∈ΦVn−1 : ∑ωn∈ΦWnµ(n)(σn−1 ∨ ωn) = µ(n−1)(σn−1). (4)Here σn−1 ∨ ωn ∈ ΦVn is the concatenation of σn−1 and ωn. In this case there exists a uniquemeasure µ on ΦV such that, for any n and σn ∈ ΦVn , µ({σ∣∣∣Vn= σn})= µ(n)(σn). Sucha measure is called a splitting Gibbs measure corresponding to Hamiltonian (1) and functionx 7→ hx, x 6= x0.The following statement describes conditions on hx guaranteeing compatibility of distribu-tions µ(n)(σn).Proposition 1. Probability distributions µ(n)(σn), n = 1, 2, . . ., in (2) are compatible iff forany x ∈ V \ {x0} the following equation holds:h∗x =∑y∈S(x)F (h∗y, θ). (5)Here, and below θ = exp(Jβ), h∗x = (h1,x − h0,x −αβ + ln ν(1)ν(0) , h2,x − h0,x −αβ + ln ν(2)ν(0) , ...) andthe function F ( · , θ) : R∞ → R∞ is F (h, θ) = (F1(h, θ), F2(h, θ), ...), withFi(h, θ) = −αβ + ln ν(i)ν(0)+ ln(θ − 1) exp(hi) +∑∞j=1 exp(hj) + 1θ +∑∞j=1 exp(hj),h = (h1, h2, ..., ), i = 1, 2, . . .Proof. Necessity. Suppose that (4) holds; we want to prove (5). Substituting (2) in (4), weobtain that for any configurations σn−1: x ∈ Vn−1 7→ σn−1(x) ∈ Φ:Zn−1Zn∑ωn∈ΦWnexp ∑x∈Wn−1∑y∈S(x)(Jβδσn−1(x)ωn(y) + αδ0ωn(y) + hωn(y),y)×4∏y∈Wnν(ωn(y)) = exp ∑x∈Wn−1hσn−1(x),x , (6)where ωn: x ∈ Wn 7→ ωn(x).From (6) we get:Zn−1Zn∑ωn∈ΦWn∏x∈Wn−1∏y∈S(x)exp (Jβδσn−1(x)ωn(y) + αβδ0ωn(y)+hωn(y),y + ln ν(ωn(y))) =∏x∈Wn−1exp (hσn−1(x),x).Consequently, for any i ∈ Φ,∏y∈S(x)∑j∈Φ exp (Jβδij + αβδ0j + hj,y + ln ν(j))∑j∈Φ exp (Jβδ0j + αβδ0j + hj,y + ln ν(j))= exp (hi,x − h0,x),so that ∏y∈S(x)1 +∑∞j=1 exp (h∗j,y) + (θ − 1) exp (h∗i,y)θ +∑∞j=1 exp (h∗j,y)=exp (h∗i,x + αβ − lnν(i)ν(0)),where h∗i,x = hi,x − h0,x + ln ν(j)ν(0) − αβ, which implies (5).Sufficiency. Suppose that (5) holds. It is equivalent to the representations∏y∈S(x)∑j∈Φexp (Jβδij + αβδ0j + hj,y + ln ν(j)) = a(x) exp (hi,x), i = 0, 1, ... (7)for some function a(x) > 0, x ∈ V. We haveLHS of (4) =1Znexp(−βH(σn−1))∏x∈Vn−1ν(σ(x))×∏x∈Wn−1∏y∈S(x)∑j∈Φexp (Jβδσn−1(x)j + αβδ0j + hj,y + ln ν(j)). (8)Substituting (7) into (8) and denoting An(x) =∏x∈Wn−1 a(x), we getRHS of (8) =An−1Znexp(−βH(σn−1))∏x∈Vn−1ν(σ(x))∏x∈Wn−1hσn−1(x),x. (9)Since µ(n), n ≥ 1 is a probability, we should have∑σn−1∑σ(n)µ(n)(σn−1, σ(n))) = 1Hence from (9) we get Zn−1An−1 = Zn, and (4) holds.From Proposition 1 it follows that for any h = {hx, x ∈ V } satisfying (5) there exists aunique Gibbs measure µ and vice versa. However, the analysis of solutions to (5) is not easy.It is natural to begin with translation-invariant solutions where hx = h ∈ R∞ is constant.53 Translation-invariant solutions of (5)Assume hx = h = (h1, h2, ...) for any x ∈ V. In this case we obtain from (5):hi = −αβ + ln ν(i)ν(0)+ k ln(θ − 1) exp(hi) +∑∞j=1 exp(hj) + 1θ +∑∞j=1 exp(hj), i = 1, 2, ... (10)Set ui = exp(hi), i = 1, 2, ... From (10) we haveui =ν(i)ν(0)exp(−αβ)((θ − 1)ui +∑∞j=1 uj + 1θ +∑∞j=1 uj)k, i = 1, 2, ... (11)In this section we give full analysis of the system of equations (11).3.1 The set of solutions {ui} with ∑∞j=1 uj = ∞In this subsection we shall describe solutions of (11) with property∑∞j=1 uj = ∞. In this casefrom (11) we getui =ν(i)ν(0)exp(−αβ), i = 1, 2, ... (12)Since∑∞j=1 ν(j) = 1 by (12) we get∞∑j=1uj =1− ν(0)ν(0) exp(αβ)< +∞.Thus there is no solution of (11) with∑∞j=1 uj = ∞.3.2 The set of solutions with∑∞j=1 uj < +∞Now we want to describe solutions of (11) with∑∞j=1 uj = A < +∞, where A is some fixedpositive number. In this case from (11) we obtainηiui =((θ − 1)ui + A + 1θ + A)k, (13)where ηi =ν(0)ν(i) exp(αβ). Denote Bi = ηi(θ + A)k. Note thatui → 0 and Bi →∞ as i →∞. (14)From (13) we obtainBiui = ((θ − 1)ui + A + 1)k, i = 1, 2, ... (15)3.2.1 Case θ > 1. As function u → ((θ − 1)u + A + 1)k is concave increasing, we concludethat (15) has a unique positive solution, say u∗i if u∗i satisfies the following equations{Biui = ((θ − 1)ui + A + 1)k,Bi = k(θ − 1)((θ − 1)u∗i + A + 1)k−1.(16)In other words, if (16) is satisfied for u∗i > 0 then u∗i is a unique positive solution to (15).6Assume k ≥ 2. From (16) we have u∗i = u∗ where u∗ = A+1(k−1)(θ−1) and Bi = B∗ whereB∗ = (θ − 1)(A + 1k − 1)k−1kk.We conclude that (15) has two solutions 0 < u∗i,1 < u∗i,2 for Bi > B∗.Note that B∗ and u∗ do not depend on i and for Bi > B∗ we have0 < u∗i,1 < u∗ < u∗i,2, for any i = 1, 2, ... (17)By (17) we have∑∞i=1 u∗i,2 = ∞ thus u∗i,2 does not satisfy the convergence condition.Note that u∗i,1 depends on k, J, α, β, ν(i), ν(0). Assume (condition on ν(i), i = 0, 1, ...)∞∑i=1u∗i,1 = A. (18)Thus we have provedTheorem 2. If for any i = 1, 2, ...,Bi > B∗ and ν satisfies the condition (18) then thesystem of equations (11) has unique solution {u∗i,1} with∑∞i=1 u∗i,1 = A.In order to demonstrate the conditions of Theorem 2 we will consider cases k = 1 and k = 2separately.Assume k = 1. In this case from (15) we getui =(A + 1)ν(i)ν(0)eαβ(θ + A)− (θ − 1)ν(i) ,which is positive if ν(0)ν(i) eαβ(θ + A) > θ− 1. This condition corresponds to Bi > B∗ for k = 1. Inthis case the condition (18) can be written as∞∑i=1(A + 1)ν(i)ν(0)eαβ(θ + A)− (θ − 1)ν(i) = A. (19)Assume k = 2. In this case condition Bi > B∗ has the formBi =ν(0)ν(i)eαβ(θ + A)2 > 4(θ − 1)(A + 1). (20)The solutions u∗i,m, m = 1, 2 areu∗i,1 =Bi − 2(θ − 1)(A + 1)−√Bi[Bi − 4(θ − 1)(A + 1)]2(θ − 1)2 ,u∗i,2 =Bi − 2(θ − 1)(A + 1) +√Bi[Bi − 4(θ − 1)(A + 1)]2(θ − 1)2 .By (14) we have u∗i,2 →∞ if i →∞. The condition (18) (for k = 2) on ν can be rewritten as∞∑i=1(ν(0)(θ + A)2eαβ − 2(θ − 1)(A + 1)ν(i)−7(θ + A)√ν(0)eαβ [ν(0)eαβ(θ + A)− 4(θ − 1)(A + 1)ν(i)])(2(θ − 1)2ν(i))−1 = A. (21)3.2.2. Case θ ≤ 1. If θ = 1 then from (11) we obtain ui = ν(i)ν(0)eαβ . This is unique solutionof (11). Note that this case corresponds to zero interaction case and is not interesting.If θ < 1 then function ϕ(u) = ((θ − 1)u + A + 1)k is convex and decreasing for odd k andequation (15) has unique solution. For even k the function ϕ is decreasing for u < A+11−θ andincreasing for u > A+11−θ . Thus for θ < 1 equation (11) has unique solution u′i which may satisfycondition∑∞i=1 u′i = A. Note that for θ < 1 we do not need a condition like Bi > B∗. Here wejust need to condition (18).Remark. Summarizing we note that for a given measure ν = {ν(i), i ∈ Φ} the equation(11) has at most one solution for any α, J ∈ R and k ≥ 1. In the next section we shall describeexact values of a unique exponential solution of (11) and corresponding measure ν.4 Exponential solutions of (11)In this section we shall describe solutions of (11) such that ui = ai for some a ∈ (0, 1). In thiscase∑∞i=1 ui = A =a1−a .4.1. Case θ > 1.From (11) we haveν(i) ≡ ν(i, a) = ν(0)aieαβ(a + (1− a)θ(θ − 1)(1 − a)ai + 1)k. (22)Now we shall choose a such that∑∞i=1 ν(i) < +∞. Considerν(i + 1)ν(i)= a ·((θ − 1)(1 − a)ai + 1(θ − 1)(1 − a)ai+1 + 1)k. (23)Using d’Alembert’s convergence condition we should getν(i + 1)ν(i)≤ q < 1. (24)If k = 1 from (23) we have a ∈ (0, 1). If k ≥ 2 using AM-GM inequality from (23) we haveRHS of (23) ≤(a[(θ − 1)(1 − a)ai+1 + 1] + k[(θ − 1)(1 − a)ai + 1](k + 1)[(θ − 1)(1 − a)ai+1 + 1])k+1≤(a + kk + 1· Ti)k+1,whereTi =(θ − 1)(1 − a)ai + 1(θ − 1)(1− a)ai+1 + 1 , i = 1, 2, ...It is easy to see that Ti+1 < Ti, i = 1, 2, ... Hence in order to obtain (24) it is enough tosolve a+kk+1 · T1 < 1 which is equivalent to(a− 1)(a2 − a + 1k(θ − 1) ) < 0. (25)8From (25) we get a ∈ (0, 1) if 1 < θ ≤ 1 + 4kand a ∈ (0, a∗−) ∪ (a∗+, 1) if θ > 1 + 4k , wherea∗± =1±√1−4((θ−1)k)−12 .4.2. Case θ ≤ 1. Denote bi = (θ − 1)(1 − a)ai + 1. It is easy to see that bi+1 ≥ bi > 0 forany i = 1, 2, ..., θ ≤ 1 and a ∈ (0, 1). Using bi+1 > bi from (23) we get (24) for any a ∈ (0, 1).Thus we haveTheorem 3. (i) If k = 1 or k ≥ 2 and 1 < θ ≤ 1+ 4k( resp. θ > 1+ 4k) then for any a ∈ (0, 1)(resp. a ∈ (0, a∗−)∪(a∗+, 1)) and ν(i) = ν(i, a) (see (22)) there exists unique translation-invariantGibbs measure µa which corresponds to solution {ui = ai} of (11).(ii) If θ ≤ 1 then for any a ∈ (0, 1) and ν(i) = ν(i, a) there exists unique translation-invariantGibbs measure µa .Denote by G(H) the set of all splitting translation-invariant Gibbs measures for Hamiltonian(1).Summarizing, we obtain the followingTheorem 4. For any parameters α, J ∈ R, k ≥ 1, β > 0 and any fixed probability measureν on Φ the set G(H) contains at most one point.Remark. Note that (see [2]) for the Potts model with q ≥ 2 spin values on Cayley tree existq distinct translation-invariant Gibbs measures. Theorem 4 shows that the result is not true ifq →∞.Acknowledgments. The final part of this work was done at the Abdus Salam InternationalCentre for Theoretical Physics (ICTP), Trieste, Italy and the authors thank ICTP for providingfinancial support and all facilities. They would also like to thank IMU/CDE-program for travelsupport. The first author was particularly supported by the IIUM short term research Grant andthe second author by Italian scheme of Borsa di Studio NATO-CNR. The second author thanksCNR for providing financial support and the Physics Department of “La Sapienza” Universityin Rome for all facilities.9References1. R.J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic, London, 1982).2. N.N. Ganikhodjaev, On pure phases of the ferromagnet Potts with three states on theBethe lattice of order two. Theor. Math. Phys. 85:163–175 (1990).3. N.N. Ganikhodjaev, The Potts model on Zd with countable set of spin values. J. Math.Phys. 45: 1121-1127 (2004)4. C. Preston, Gibbs states on countable sets (Cambridge University Press, London 1974);5. Ya.G. Sinai, Theory of phase transitions: Rigorous Results (Pergamon, Oxford, 1982).6. F. Spitzer, Markov random fields on an infinite tree, Ann. Prob. 3: 387–398 (1975).7. Y.M. Suhov, U.A. Rozikov, A hard - core model on a Cayley tree: an example of a lossnetwork, Queueing Syst. 46: 197–212 (2004)8. S. Zachary, Countable state space Markov random fields and Markov chains on trees,Ann. Prob. 11: 894–903 (1983).9. S. Zachary, Bounded, attractive and repulsive Markov specifications on trees and on theone-dimensional lattice, Stochastic Process. Appl. 20:247–256 (1985).10. F.Y. Wu, The Potts model, Rev. Mod. Phys. 54: 235–268 (1982).10

Uniqueness of Gibbs measure for Potts model with countable set of spin values

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