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13
Translation-invariant Gibbs measures for a model with logarithmic potential
on a Cayley tree
Yu. Kh. Eshkabilov1, Sh. P. Bobonazarov2, R. I. Teshaboev3 1 National University of Uzbekistan, Tashkent, Uzbekistan 2 Tashkent Institute of Irrigation and Melioration, Tashkent, Uzbekistan 3Termez State University, Termez, Uzbekistan [email protected], [email protected], [email protected]
DOI 10.17586/2220-8054-2016-7-5-893-899
In this paper, we consider a model with logarithmical potential and with the set [0,1] of spin values, on a Cayley tree rk of the order k. In the case k = 2, 3, we shall prove that, there is a unique translation-invariant splitting Gibbs measure for this model. For the case k = 4, we show that there are three translation-invariant Gibbs measures for this model.
Keywords: Cayley tree, configuration, translation-invariant Gibbs measure, fixed point, nonlinear operator.
Received: 15 April 2016. Revised: 25 May 2016.
1. Introduction
One of the central problems in the theory of Gibbs measures is to describe infinite-volume (or limiting) Gibbs measures corresponding to a Hamiltonian. The existence of such measures for a wide class of Hamiltonians was established in the ground-breaking work of Dobrushin. However, complete analysis of a set of limiting GMs for a specific Hamiltonian is quite often a difficult problem.
In [1,2,6,9-11,14-16] for several models on Cayley tree rk with the order k, using the Markov random field theory, Gibbs measures are described. These papers are devoted to models with a finite set of spin values. In [8], the Potts model with a countable set of spin values on a Cayley tree rk is considered and it was shown that the set of translation-invariant splitting Gibbs measures of the model contains at most one point, independently on parameters of the Potts model with countable set of spin values on the Cayley tree. This is a crucial difference from the models with a finite set of spin values, since those may have more than one translation-invariant Gibbs measure.
In [12], a Hamiltonian with an uncountable set (a set with continuum cardinality) of spin values (with the set [0,1] of spin values) on a Cayley tree rk is considered and it was shown that: the existence translation-invariant splitting Gibbs measure of the Hamiltonian is equivalent to the existence a positive fixed point of some nonlinear integral operator. For k = 1, the model with the continuous potential function was shown to have a unique translation-invariant splitting Gibbs measure. In the case k > 2, some models which have the unique splitting Gibbs measure were constructed. In the paper [4], sufficient conditions were found for the potential function of the model on a Cayley tree rk with an uncountable set of spin values under which the model had unique translation-invariant splitting Gibbs measure. In [3,5], several models were constructed, of which these models had at least two translational-invariant Gibbs measures, i.e the existence of phase transition for some models on a Cayley tree rk (k > 2) was proven.
This paper is a continuation of previous investigations [3-5,12]. We shall construct model with a logarithmic potential on a Cayley tree rk. We reduced the studying of translation-invariant Gibbs measures to a description of the fixed points for some nonlinear operator on R2. In the case k = 2, 3, we shall prove that, for the Hamiltonian on a Cayley tree rk with logarithmic potential, there is a unique translation-invariant splitting Gibbs measure. In the case k = 4, we show that, for the model on r4 with the logarithmic potential there are three translation-invariant Gibbs measures, i.e. there is a phase transition.
2. Preliminaries
A Cayley tree rk = (V, L) of order k g N is an infinite homogeneous tree, i.e., a graph without cycles, with exactly k + 1 edges incident to each vertex. Here, V is the set of vertices and L that of edges (arcs).
Consider models where the spin takes values in the set [0,1], and is assigned to the vertices of the tree. For A c V, a configuration aA on A is an arbitrary function aA : A ^ [0,1]. We denote QA = [0,1]A the set of all
configurations on A. A configuration a on V is then defined as a function x G V ^ a(x) G [0,1]; the set of all configurations is [0,1]V. The Hamiltonian of the model is :
H(a) = -J a G , (2.1)
(x,y)£L
where J G R \ {0} and £ : (u, v) g [0,1]2 ^ g R is a given bounded, measurable function. As usual, (x, y) represents the nearest neighbor vertices.
Let A be the Lebesgue measure on [0,1]. On the set of all configurations on A, the a priori measure Aa is introduced as the |A| fold product of the measure A. Here and subsequently, |A| denotes the cardinality of A. Below, Wm represents a 'sphere' and Vm for a 'ball' on the tree, of radius m = 1, 2,..., centered at a fixed vertex x0 (an origin):
Wm = {x G V : d(x, x0) = m}, Vm = {x G V : d(x, x0) < m};
and
Lm = {(x, y) G L : x, y G Vm}.
Here, distance d(x,y), x,y g V, is the length of (i.e. the number of edges in) the shortest path connecting x with y. QVn is the set of configurations in Vn (and QWn that in Wn; see below). Furthermore, a|V and w|W denote the restrictions of configurations a, w G Q to Vn and Wn+1, respectively. Next, an : x G Vn ^ an(x) is a configuration in Vn. For each an G QVn, we define:
H (an) = -J ^
{x,y)£L„
We write x < y if the path from x0 to y goes through x. Call vertex y a direct successor of x if y > x and x, y are nearest neighbors. We denote by S(x) the set of direct successors of x. We observe that any vertex x = x0 has k direct successors and x0 has k + 1.
Let h : x G V ^ hx = (ht,x,t G [0,1]) G R[0,1] be mapping of x G V \ {x0}. Given n =1,2,..., consider the probability distribution ^(n) on QVn defined by
M(n)(a„) = Z—1 exp ( (a„) + £ , (2.2)
V IEW„ /
where ft = 1, T > 0 is temperature. Here, as before, an : x G Vn ^ a(x) and Zn is the corresponding partition function:
Zn = exp I -^H(?„) + ^ Avn(da„). (2.3)
oVn
The probability distributions ^(n) are compatible [12] if for any n > 1 and an-1 G :
J M(n)(a„-i V w„)Aw„(d(wn)) = M(n-1)(a„_i). (2.4)
0W„
Here, an-1 V wn G QVn is the concatenation of an-1 and wn. In this case, there exists [12] a unique measure ^ on such that, for any n and an G QVn, J a
x = x .
^ = a„ f I = M(n)(a„).
The measure ^ is called the splitting Gibbs measure corresponding to Hamiltonian (2.1) and function x ^ hx
o
Proposition 2.1. [12] The probability distributions ^(n)(^n), n = 1, 2,..., in (2.2) are compatible iff for any x G V \ {x0} the following equality holds:
f (t x) = n ^ exP(Jg&")f(u,y)du (2 5)
yes(x) fo exP(J0£o«)f (u,y)du
Here, and below f (t, x) = exp(ht,x - h0,x), t G [0,1] and du = A(du) is the Lebesgue measure.
From Proposition 2.1, it follows that for any h = {hx G R[0'1], x G V} satisfying (2.5) there exists a unique Gibbs measure ^ and vice versa. However, the analysis of solutions to (2.5) is not easy. Let £t„ be a continuous function. We put
C +[0,1] = {f G C[0,1] : f (x) > 0}, C0+[0,1] = C+[0,1] \ {Q = 0}. We define the operator Rk : C+ [0,1] ^ C+[0,1] by
( 1 \ k / K(t,u)f , k g N,
/01 K (0,u)f {u)duj
where K(t, u) = exp( J££t„), f (t) > 0, t, u G [0,1].
We will solve the equation (2.5) in the class of translational-invariant functions f (t, x), i.e f (t, x) = f (t) for any x g V. For such functions, equation (2.5) can be written as:
Rk(f )(t) = f (t). (2.6)
Note that equation (2.6) is not linear for any k G N. For every k G N we consider Hammerstein's integral operator Hk acting in the cone C+ [0,1] as
1
(Hkf )(t) = J K(t, u)fk(u)du, k G N.
0
We denote
Mo = {f G C +[0,1]: f (0) = 1}. Lemma 2.2. [4] Let k > 2. The equation
Rkf = f, f G C+[0, 1] (2.7)
has a nontrivial positive solution iff the Hammerstein's equation
Hkf = Af, f G C +[0,1] (2.8)
has a positive solution in M0 for some A > 0.
Let k > 2. Then, we can easily verify that: if the number A0 > 0 is eigenvalue of the operator Hk, then an arbitrary positive number is an eigenvalue of the operator Hk (see [4]). Consequently, we obtain:
Lemma 2.3. Let k > 2. The equation (2.7) has a nontrivial positive solution iff the Hammerstein's operator Hk has a nontrivial positive fixed point. Moreover, the number of nontrivial positive fixed points of the operator Rk is equal to the number of nontrivial positive fixed points of the Hammerstein's operator Hk.
Note, that if there is more than one nontrivial positive fixed point for the the Hammerstein's operator, Hk, then there is more than one translation-invariant Gibbs measure for the model (2.1) corresponding to these fixed points. We say that a phase transition occurs for the model (2.1), if the Hammerstein's operator Hk has more than one nontrivial positive fixed point. The number of the fixed points depends on the parameters of the model (2.1) and the order of Cayley tree rk.
3. A model on Cayley tree with logarithmic potential
We consider Hamiltonian H on the Cayley tree rk by rule:
HM = _ E m (i + -Q (.M -1 )(,(y) -1)), . G av, (3.1)
(x,y)eL P
where Q is a coupling constant and 0 < Q < 1, i.e. in the (2.1) function of potential is defined by the formula:
C =in (1 + 4Q (t - 2) (u - 2))
= j^ .
The main aim of this paper is to study translation-invariant Gibbs measures for model (3.1) on the Cayley tree rk. We define Hammerstein's operator Hk on C[0,1] by the equality:
1
(Hkf )(t) = / ^1+4Q^t - (u - fk(u)du. (3.2)
We set:
and
k, if k is even,
k1 = <
k - 1, if k is odd,
k, if k is odd,
k2 =
k — 1, if k is even.
We define operator P on R by the rule:
P(x,y) ^ (x',y'),
where
= ^ (2^Ajxk-2jy2j, 2j + 1 k y ,
(k2 + 1)/2 (2fl)2j-1
y' = y -r A?j-1xfc-2j+1y2j-1.
y 2(2 j + 1) k y
Here
Am A
m!(n — m)!
Lemma 3.1. Let k > 2. The Hammerstein's operator Hk f3.2> has a nontrivial positive fixed point iff the operator P has a fixed point (x0,y0), such that x0 > 0 and f0(t) = x0 + 40y0 ^t — ^ > 0 for all t G [0,1], moreover
the function f0(t) = x0 + 40y0 ^t — ^ is a positive fixed point of the Hammerstein's operator Hk. Proof. Necessity. We set:
1
c1 = J fk(u)du (3.3)
0
1
/ — 1) fk(u)du. (3.4)
and
1
C2 = / ^— ik
0
It is clear, that c1 > 0. Let the Hammerstein's operator Hk (3.2) has a positive fixed point f (t). Then, for the function f(t), the equality:
f (t) = C1 +40c^t — 0 (3.5)
is holds.
Consequently, for the parameter c1, from the equality (3.3), we have:
1 k k 1 / x j
C1 = J ^C1 +40c^u — du = £Akck-j(40C2)^ (u — 0" du =
0 j=0 0
= £ Akkck-j(4^c2)j | ujdu = y jj Akjck-2jc2j.
j=0 -1/2 j=0
Analogously, for the parameter c2, by equality (3.4), we get:
c2 = j (u — + 40c^u — du = £ Akck-j (40c2)j J (u — j du
0 j=0 0
k 1/2 (k2 + 1)/2 (20)2j-1 = £Akck-j(4^/ uj+1du = £ j--A2kj-1ck-2j+1c2j-1.
j=0 -1/2 j=1
n!
Therefore, the point (01,02) is a fixed point of the operator P.
Sufficiency. We assume that x0 > 0 and the point (x0, y0) is a fixed point of the operator P, i.e. the following equalities for numbers x0 and y0 numbers are satisfied:
kl/2 (26)2j (k2 + 1)/2 (26)2j-1
E(26) a 2j k-2j 2j V^ ( ) /|2j-1 k —2j + 1 2j-1
2j+l Akj xk j yoj = ^ 2(2j + !) Akj xk j yoj = y0.
We can simply prove that the function f0(t) = x0 + 46y0 ^t — ^ is a fixed point of the Hammerstein's operator Hk, i.e. Hkf0 = f0. This completes the proof. □
Proposition 3.2. For each k G N, the function f0(t) = 1 is a fixed point of the Hammerstein's operator Hk.
Proof. One can clearly see that:
1 1 1/2
(Hk)f0(t) = J (l + w(t — (u — du == ^ du + 46 ^ — 0 J udu = 1 = f0(t).
0 0 -1/2
□
4. Uniqueness of translation-invariant Gibbs measures for the model (3.1)
In [12], a Hamiltonian with an uncountable set of spin values (with the set [0,1] of spin values) on the Cayley tree rk was considered for a continuous potential £t,„. For k = 1, it was shown that the model (2.1) with the continuous potential function has a unique translation-invariant splitting Gibbs measure. This statement holds for the model (3.1). We study translation-invariant splitting Gibbs measure for the model (3.1) for the case k > 2.
Theorem 4.1. The model H (3.1) on the Cayley tree of order two has a unique translation-invariant Gibbs measure.
Proof. Let be k = 2. Then, the operator P assumes the following simple form:
P(x,y)= + 4y2, 36xy^ .
For a fixed point (x, y) of the operator P, we have the following system of algebraic equations:
2 , 4 2 x + o y = x
2 3 3 6xy = y.
It follows that, the operator P has a unique nontrivial fixed point (1,0), as 6 g (0,1). By lemma 3.1, the Hammerstein's operator H2 has a unique nontrivial positive fixed point f0(t) = 1. Therefore, by lemma 2.3, the model H (3.1) on the Cayley tree of order two has a unique translation-invariant Gibbs measure. □
Theorem 4.2. The model H (3.1) on the Cayley tree of order three has the unique translation-invariant Gibbs measure.
Proof. Let k = 3. Then, the operator P assumes the following form:
P(x, y) = ^x3 + 462xy2, 6x2y + 463y3 For a fixed point (x, y) of the operator P, we have the following system of algebraic equations:
x3 + 462xy2 = x,
6x2y + 4 63y3 = y.
It follows that, the point (1,0) is a fixed point of the operator P. Consequently, by lemma 3.1, the function f0(t) = 1 is a fixed point of the Hammerstein's operator H3. Conversely, for the case x > 0, y = 0, the last system of algebraic equations is equivalent to the following system of algebraic equations:
x2 + 462y2 = 1,
6x2 + 4 63y2 = 1. 5 y
We find x2 = 1 — 402y2. Hence, for y, we have:
(1 — 402y2) 0 +503y2 = 1,
i.e.
y2 = 5(0 — 1) y 1603 .
This is impossible, as 0 G (0,1).
Thus, the operator P has a unique nontrivial fixed point (1,0). Therefore, by lemmas 3.1 and 2.3, the model H (3.1) on the Cayley tree of order three has a unique translation-invariant Gibbs measure. □
5. A phase transition for the model (3.1)
In this section, we consider the model (3.1) on the Cayley tree r4. In the case k = 4, the operator P is acting on R2 by the rule:
P(x, y) = (x4 + 802x2y2 + 1604y4, 40x3y + 1603xy3
Theorem 5.1. Let k = 4. Then:
(a) for all 0 G (0,3/4] the model H (3.1) on the Cayley tree rk has a unique translation-invariant Gibbs measure;
(b) for all 0 G (3/4,1) the model H (3.1) on the Cayley tree rk has three translation-invariant Gibbs measures.
Proof. Let k = 4. For a fixed point (x, y) of the operator P, we have the following system of algebraic equations:
x4 + 802x2y2 + — 04y4 = x,
4 3 16 3 5
30x y + -50 xy = y.
In the case x > 0, y = 0, the above system of algebraic equations has the solution (1,0). We assume that y = 0. Then, we have x = 0 and from the second equation of the last system of equations, we obtain:
2 5(3 — 40x3)
y = 4803x . (5.1)
This means that:
,f3~
0 <x< V 40. (5.2)
From the first equation of the system of equations, for a fixed point of the operator P, we obtain:
16 6 30 — 5 3 5
¥x6 + -3Tx3 — 1602 = °. (5.3)
We set z = x3. Then, z > 0 and for the unknown variable z, by the equality (5.3), we have the quadratic equation:
16 2 30 — 5 5
—z2 +--z--^ = 0. (5.4)
9 + 30 1602 V '
One clearly sees that equation (5.4) has two roots:
z1 = z1(0) =
—1 + 31 — ^
32
9
< 0, z2 = z2(0) =
—1 + + ^
32
9
> 0,
where
D = D(0) = ( 1 — —) .
v y 1 30 / 902
Therefore, for x, by the lemma 3.1 and the inequality (5.2), we obtain x = x1 = x1(0), where
T
x1 (0)= ^0) < V-.
The question then arises: does the inequality (5.2) hold for x1(6) for all values of the parameter 6 g (0,1)?
3 l~3~
To this end, we consider the inequality x1(6) < \ —. This is equivalent to the inequality:
46
-1 + + VP
32 < V 40
< VTTI. (5-5)
9
Hence, it follows that VP < 1 + 1. It means
0
5\ 2 20 / 1X 2
1 - W) +902 < ^ + 0 3 3
From the last inequality, we get 0 > -. Thus, for the case - < 0 < 1, by equality (5.1), the operator P has three fixed points:
(1, 0), (xi(0), yi(0)), (xi(0), -yi(0)),
where
(0) 1 /5(3 - 40x3(0)) > 0
yi(°) = 4^V 30xi(0) >
3
We note that if 0 <0 < -, then the operator P has a unique fixed point: (1,0). Consequently, by lemmas 2.3
and 3.1, for all 0 g ^0, 3 the model H (3.1) on the Cayley tree r4 has a unique translation-invariant Gibbs
3
measure. In the case - < 0 < 1 by the lemma 3.1, the Hammerstein's operator H4 has three positive fixed points:
/„(t) = 1, /i(t) = xi(0) + 40yi(0)(t - 0 , f2(t) = xi(0) - 40yi(0)(t - 2
Because /¿(t) > 0 for all t g [0,1], where i = 1, 2. Therefore, By lemma 2.3 for all 0 g (3/4,1) the model H (3.1) on the Cayley tree r4 has three translation-invariant Gibbs measures. This completes the proof. □
Finally we note that in the case 0 g , for the model H (3.1) on the Cayley tree r4 there is a phase transition.
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