The uniqueness of the translation-invariant Gibbs measure for four state HC-models on a Cayley tree Текст научной статьи по специальности «Математика»

УДК 517.98

The Uniqueness of the Translation-invariant Gibbs Measure for Four State HC-models on a Cayley Tree

Rustam M. Khakimov*

Institute of Mathematics Do'rmon Yo'li str., 29, Tashkent, 100125

Uzbekistan

Received 04.01.2015, received in revised form 17.02.2015, accepted 06.03.2015

We consider fertile Hard-Core (HC) models with activity parameter Л > 0 and four states on the Cayley tree of order two. It is known that there are three types of such models. In this paper for each of these models the uniqueness of the translation-invariant Gibbs measure is proved.

Keywords: Cayley tree, configuration, HC-model, Gibbs measure, translation-invariant measures.

Introduction

The study of limiting Gibbs measures play an important role in many fields of science. The hard core (HC) model arises in the study of random independent sets of a graph ( [1]), the study of gas molecules on a lattice [2], and in the analysis of multi-casting in telecommunication networks [3].

A HC model on d-dimensional lattice Zd, was introduced and studied by Mazel and Suhov in [4].

The description of all limiting Gibbs measures for a given Hamiltonian is one of main problems in the theory of Gibbs measures. (see f.e. [5-7]).

In [8] a HC (Hard Core) model with two states on a Cayley tree was studied and it was proved that the translation-invariant Gibbs measure for this model is unique. Moreover, it was proved non uniqueness of periodic Gibbs measures of period two for some conditions on parameters. In [9] weekly periodic Gibbs measure for the two state HC-model is investigated and it is shown that the weekly periodic measure is unique.

Works [10,11] are devoted to Gibbs measures for three state HC-models on a Cayley tree of order k ^ 1. In [12] the fertile three-state HC-models corresponding to graphs "the hinge", "the pipe", "the wand", "the key" and four-state HC-models corresponding to graphs "the stick", "the key", "the gun" are introduce. In [10] translation-invariant and periodic Gibbs measures for HC-model in the case "the key" on a Cayley tree is studied and it was proved that the translation-invariant measure is unique for any positive activity A. In [11] translation-invariant and periodic Gibbs measures for HC-model in cases "the pipe", "the hinge", "the wand" are studied. In [13] translation-invariant Gibbs measures for three state HC-models on a Cayley tree of order three are considered and the exact critical values of the parameter A are found such that for activities larger than these critical values the measure is not unique.

In this paper we consider fertile four states HC-models corresponding to graphs "the stick", "the key" and "the gun" on a Cayley tree of order two. In each case it is proved that the translation-invariant Gibbs measure is unique.

* [email protected] © Siberian Federal University. All rights reserved

1. Definitions and known facts

The Cayley tree 9fc of order k > 1 is an infinite tree, i.e., a graph without cycles, such that exactly k + 1 edges originate from each vertex. Let 9fc = (V, L, i), where V is the set of vertices 9fc, L the set of edges and i is the incidence function setting each edge l G L into correspondence with its endpoints x, y G V. If i(l) = {x, y}, then the vertices x and y are called the nearest neighbors, denoted by l = (x,y). The distance d(x,y),x,y G V on the Cayley tree is the number of edges of the shortest path from x to y:

d(x, y) = min {d|3x = xo, xi,..., x^_i, x^ = y G V such that (xo, xi),..., (x^_i, x^)}.

For a fixed x0 G V we set Wn = {x G V | d(x, x0) = n}, Vn = {x G V | d(x, x0) < n}, Ln = {l = (x, y) G L | x, y G Vn}.

We consider the four-state nearest-neighbor HC-model on a Cayley tree. In this model each vertex x is assigned one of the values a(x) G {0,1,2,3}. The values a(x) = 1, 2, 3 mean that the vertex x is "occupied" and the value a(x) = 0 means that x is "vacant".

A configuration a = {a(x), x G V} on a Cayley tree is given by a function from V to {0,1,2, 3}. The set of all configurations in V denote by Q. Configurations in Vn (Wn) is defined analogously and the set of all configurations in Vn (Wn) denote by QVn (QWn ).

Here we consider three types of fertile graphs (see Definition 2) with four vertices 0,1, 2, 3 (on the set of values a(x)) which have the forms:

stick : {0,1}{1, 2}{2, 3};

key : {0,1}{0, 2}{1, 2}{2, 3};

gun : {0,1}{0, 2}{1, 2}{2, 2}{2, 3}.

Graphs that are not fertile are said to be sterile (see. [12]).

Let O = {the stick, the key, the gun}, G G O. We say that a configuration a is G-admissible configuration on a Cayley tree (in Vn or Wn), if {a(x), a(y)} is an edge of G for any pair of nearest-neighbors x, y in V (in Vn). We let QG (QGn) denote the set of G—admissible configurations.

The set of activity of graph G is a function A : G ^ R+ (R+ denotes the positive real numbers) [12]. The value Aj of function A at a vertex i G {0,1,2,3} is called the activity of A.

Given G and A, we define the G—HC Hamiltonian as

, i E log Aff(x), if a G

HG(a) = < xev

\ if a G

We write x < y, if a path from x0 to y passes through x. A vertex y is called a direct descendant of x if y > x and x, y are neighbors. We let S(x) denote the set of direct descendants of x. We note that in 9fc, any vertex x = x0 has k direct descendants and the vertex x0 has k +1 descendants.

For an G QG we set

#an =^3 !(an(x) > 1)

xEVn

which is the number of occupied vertices in an.

Let z : x ^ zx = (z0,x, z1jX, G R+ be a vector-valued function on V. For n =

1,2,... and A > 0 we consider the probability measure ^(n) on Q^, defined as

M(n)(an) = -J"A#ffn H z.(x),x. (1)

n x£Wn

Here Zn is normalizing divider

Zn

E A#Tn n

?(x)

a(x),x •

<T„£ QÍ-

xew„

We say, that the probability measure is consistent if for any n > 1 and an-i G :

M(n)K-i V wn)1(CTn_i V G QG, ) = M(n-1)(^n-i).

(n-1)(

(2)

In this case, there exists a unique probability measure ^ on B) such that for any n and ct„ e

V n

M{*k = *n}) = M(n)(^n), where B is a-algebra generated by cylindrical subsets

Definition 1. The measure ^ defined by (1) with condition (2) is called (G-)HC-Gibbs measure with A > 0, associated to the function z : x e V \ {x0} ^ zx.

The set of such measures (for all sorts of z) denoted by SG.

A Gibbs measure is called translation-invariant if it corresponding to a constant function

Definition 2 ( [12]). If there is a set of activities A on G such that the corresponding G-HC Hamiltonian has at last two translation-invariant Gibbs measures, then the graph G is called a fertile graph.

G

aij = aij

Let L(G) be the set of edges of G. We let A = AG = ( a,,-). . „ „ „ denote the incidence matrix of G, i.e.

1, if {i,j}G L(G), 0, if |i,j}^L(G).

The next theorem states a condition on zx that guarantees that the measure is consistent.

Theorem 1 ( [14]). Probability measures , n =1, 2,..defined by (1), are consistent if and only if the equalities

z0,x

z2,x

A n

yes(x)

A n

yes(x)

A n

yes(x)

aiozO y + aiizi y + ai2z2 y + ai3

a3oz0

a20z0

a3oz0 a30z0

a30z0

+ a3izi

+ a2izi

+ a3izi + a3izi

+ a3izi

+ a32z2

+ a22^2

+ a32z2 + a32z2

+ a32z2

+ a33

+ a23

+ a33 + a33

+ a33

(where Zi x = AzijX/z3jX, i = 0,1, 2) hold for any x G V.

2. Translation-invariant Gibbs measures

We assume that z3 ,x = 1 and zj, x = zi , x > 0, i = 0,1, 2. Then for any functions x G V zx = (zo, x, zi, x, z2, x), satisfying the relation

x

y

y

y

y

zi ,x =

y

y

A n

yes(x)

+ + ai2Z2,y + «¿3

a30z0,y + a31z1,y a32z2,y + a33

0,1, 2,

(3)

there exist a unique G-HC Gibbs measure ^ and vice versa. We consider the translation-invariant solutions such that zx = z G R+, x = x0. In cases G = stick, G = key and G = gun from (3) we get following systems of equations

z0 = A (t) ' ,

Z1 Z2

= x(

Z2 )

(4)

A 21+1

Z2

Z0 = A (^\ ,

A / 20 + 22

Z1

(5)

, Z2 = A (

Z0 Z1

= A l 20+21+1 | 22

A( ^ )k-

/ \ k

A'Z0+2) :

(6)

, Z2 = A

respectively.

Lemma. If (z0, z1? z2) is a solution of the system of equations (5) or (6) then z0 = z1.

The proof is obtained by subtracting directly from the first equation of the second in (5) and (6), respectively.

The following statement gives estimates for any solution of the system of equations (4). Statement 1. If z = (z0,z1?z2) is the solution of (4), then

1)

k+1

< z0 <

2k2 k+1

fc+V Ak(A + 1)k2

fc+V Ak (2kA + 1)k2

2) A < z1 < 2k A;

3) A(A+1)k < z2 < fc+VA(2kA + 1)k.

Proof. 1. The first inequality follows from the first equation (4) using estimates for z1 and z2.

2. From the second equation of (4) we obtain that z1 > A. Dividing the first equation to the third we will have

z0 = i z1 z2 \z1 + 1

Using the last inequality from the second equation follows z1 < 2kA.

3. Using estimate for z1; inequality obtained from the third equation of (4) we obtain z2 = fc+y A(z1 + 1)k. □

< 1.

1. The case G = stick

z

¿.x

k

k

In the case stick we have the following

Theorem 2. In the case G = stick for k = 2 and A > 0 there is a unique HC translationinvariant Gibbs measure.

Proof. We find z2 from the third equation of system (4) for k = 2 and substitute into the first equation. Using obtained expressions for z2 and from the second equation we get

zi = A • (+ 1) = f (zi). (7)

Rewrite the equation (7) in the form

A = —-T2 = V-(zi)- (8)

((zr+i)2 + 1)

It is easy to see that the function is strictly increasing for z7 > 0, consequently for any

value A > 0 there exists a unique value z1, i.e. the equation z7 = f (z7) has a unique solution for A > 0. □

2. The case G = key

In the case denoting = x > 0, = y > 0, VA = a and using lemma, from the system of equations (5) we obtain the following system of equations

xk + yk /x\ k

x = a •-k— = a • — + a,

yk ^ k (9)

2xk + 1 fx\k a V '

y = a--k— = 2a • " + "fc •

yk v^ yk

k

t x \ x — a

From the first equation of (9) we find — = - and substitute into second equation we

ya

a

get y = 2x — 2a +—t or yk

ya

x =---r + a.

2 2yk

Using the last expression for x, from the second equation of (9) we obtain

k

yk+1 = a •

2<f — 2yk + «1 +1

This equation is equivalent to the equation

f (y) = 2y7 — ay6 — 4a2 y5 — (4a3 + 2a)y4 + 2a2y3 + 4a3y2 — a3 = 0,

for k = 2 which by the known theorem of Descartes up to three positive solutions. Moreover, f (0) = —a3 < 0 and f(y) ^ +<» for y ^ i.e. the equation f (y) = 0 has at least one

positive solution.

Thus, we have the following

Statement 2. The system of equations (9) has at least one and at most three solutions for k = 2.

We show that the system (9) has only one solution for any values a > 0. It is easy to see that the equation z7 = f (z7) has more than one solution if and only if the equation zf '(z7) = f (z7)

2

2

has more than one solution. We use this property twice. From the system of equations (5) for k = 2 we get the equation

= A ■ ( zi + ^W+IF V = A.f -1 +1^2 = f (zi).

V 3 A(2zi + 1)2 ^ VV A(2zi + 1)2 J

We calculate the derivative

f'(zi) = 2A (^i' +1^ 2z' +3

VA(2zi + 1)2 J VA(2zi + 1)5 '

Then from z1f '(z1) = f (z1) we obtain

^A(2zi + 1)5 ( ) Z1 = 2zi +5 = ^(Zl)'

Taking the derivative

4A(2zi + 1)(2zi + 11)

V (zi) "

3(2zi + 5)2 ^A2(2zi + 1)' we consider the equation zi^/(zi) = ^(zi). It is equivalent to the following quadratic equation

4z2 - 8zi + 15 = 0,

that has no real roots. Hence, the equation zi = f (zi) has unique solution. Consequently by Statement 2, we have

Theorem 3. In the case G = key for k = 2 and A > 0 Hard-Core translation-invariant Gibbs measure is a unique.

3. The case G = gun

By the lemma from (6) we obtain the following system of equations

x = a- xk+kyk = a- f-) + a,

yk \y J

2xk + yk + 1 o , a ,

y = a • +y + =2a • (-) + yak + a,

(10)

where kzi = x > 0, = y > 0, VA = a.

It is clear that x > a. Similarly to the previous case from (10) we have

k

yk+1 = 2a ^ | - ¿k + f) + ayk + a.

For k = 2 we transform this equation to

f (y, a) = 2y7 - 3ay6 - 2a2y5 - (a3 + 2)y4 + 2a2y3 + 2a3y2 - a3 = 0.

Analyzing the last equation similar to the previous case we get the following Statement 3. The system of equations (10) has at least one and at most three solutions for k = 2.

We show that the system (10) has only one solution for any values a > 0. For this from (10) we obtain the equation

2x3 — ax2 + x — a

+ 1

f (x).

The derivative of function f (x) is f '(x)

2 ax5

(2x3 — ax2 + x — a)3

(—ax2 + 2x — 3a).

(11)

(12)

Transforming equation xf'(x) = f (x), we get

2x6(—ax2 + 2x — 3a) = (x6 + (2x3 — ax2 + x — a)2)(2x3 — ax2 + x — a),

which has no solutions for a > —. Hence the equation (11) by Statement 3 has only one solution

3

1

for a > —= •

1

We consider the case 0 < a < —=. From (12) it follows that the function f (x) has critical

_ 1 — —1 — 3a2 1 + —1 — 3a2 1a,

points x7 = - and x2 = - (a < x7 < x2 for 0 < a < —=), decreases

a a 3

for a < x < xi, x > x2 and increases for xi < x < x2. Hence xi = xmin and x2 = xmax. We note, that the value of the function f(x) in the point a lies above the bisector y = x, because f (a) = 2a > a. Moreover, values of the function f (x) in points xmin and xmax lies below the

bisector y = x, because f (xmin) < xmin and f (xmax) < xmax for 0 < a < ^^ (see Fig. 1).

3

-20-

-40-

-60-

-80-

-100J

Fig. 1. Graphs of functions g(a) = f (xmin) — xmin for 0 < a < (on the left) and g(a) = f (xmax) — xmax for 0 < a < (on the right)

From all this it follows

Theorem 4. In the case G = gun for k = 2 and A > 0 Hard-Core translation-invariant Gibbs measure is a unique.

The author is grateful to Professor U. A. Rozikov for useful discussions.

2

3

x

x=a

References

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Единственность трансляционно-инвариантной меры Гиббса для ЫС-моделей с четырьмя состояниями на дереве Кэли

Рустам М. Хакимов

Рассмотрены плодородные ИаЫ-Сотв (ИС-модели) с параметром активности Л> 0 и четырьмя состояниями на дереве Кэли порядка два. Известно, что существуют три типа таких моделей. В данной работе для каждой из этих моделей доказана единственность трансляционно-инвариантной меры Гиббс.

Ключевые слова: дерево Кэли, конфигурация, НС-модель, мера Гиббса, трансляционно-инвариан-тные меры.