The study of random functions σ _x from the vertices of some graph G (usually Z ^d or a Cayley tree) into a space with measure (E, E ) is a central problem in ergodic theory and statistical physics.

In many classical models of physics (e. g., the Ising model, the Potts model, the HardCore model), E is a finite set and σ _x has a physical interpretation as the spin of a particle at vertex x in a crystal lattice (graph G ). For such models, the theory of Gibbs measures is well developed. It is known that each limiting Gibbs measure is associated with one phase of

(0 2026 Makhammadaliev, M. and Khakimov, R.

a physical system. Therefore, one of the important problems in the theory of Gibbs measures is the existence of a phase transition, i. e. when a physical system changes its state with a change in temperature. This occurs when the number of measures changes with a change in temperature. In this case, the temperature at which the state of a physical system changes is usually called critical (see [1–4]).

For many classical models with a finite set of spin values (for example, the Ising model, the Potts model, the SOS model, the λ -model, the Hard-Core model), the theory of Gibbs measures is well developed (see, e. g., [5–12]). However, in the case of a countable set of spin values, the existence of Gibbs measures is not guaranteed. There are several papers devoted to the study of such models (see, e. g., [13–19]).

In [20], A. Mazel and Yu. Suhov introduced the HC (Hard-Core) model on a d -dimensional lattice Z ^d . This model is interesting from the point of view of statistical mechanics, as well as combinatorics and the theory of neural networks [2, 21, 22]. There are a lot of works devoted to the study of limiting Gibbs measures for HC models with a finite number of states on the Cayley tree (see [3, 23–27]).

In [28], for the first time, the HC model with a countable set of spin values was studied. For this model, on a Cayley tree of arbitrary order, some conditions for the existence of TIGM, and also the uniqueness of such a measure under the existence conditions are found. In addition, the conditions for the existence and non-uniqueness of periodic (not translationinvariant) Gibbs measures with period two are found. In [29] it is proved that for this model each H -periodic Gibbs measure is either translation-invariant or two-periodic. In [30] it is proved that in the case of an arbitrary normal divisor of index two the weakly periodic Gibbs measure is translation-invariant and unique, and in the case of a normal divisor of index four, under certain conditions the translation invariance of weakly periodic Gibbs measures on a Cayley tree of arbitrary order is shown. And also under certain conditions the exact critical value is found, at which weakly periodic (non-periodic) Gibbs measures exist. The present paper can be considered as a continuation of the mentioned works.

In this paper, we consider an HC model with a countable set of spin values on a Cayley tree. This model is defined by a countable set of parameters specified by the activity function A i > 0 , where i G Z . It is shown that each boundary law constructed for the model under consideration defines the Gibbs measure. The cases _{j ∈ Z 0} A 2 j = A , j A 2 j +1 = Ф are considered. In the case A = Ф on a Cayley tree of orders k = 1, 2, 3,4, 5 and in the case A = Ф on a Cayley tree of orders k = 1, 2, 3 , a complete description of the TIGM is obtained. Also, at A = Ф , the lower bound for the number of TIGMs is indicated for k ^ 2 .

• 2.    Preliminaries

The Cayley tree Q k 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 Q k = (V, L, i) , where V is the set of vertices of Q k , L is the set of its 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 nearest neighbors, denoted by l = ( x,y ) .

For fixed x ⁰ G V , we set

Wn = {x G V : djx,x°j = nJ, Vn = {x G V : djx,x°j < nJ, where d(x, y) is the distance between the vertices x and y on the Cayley tree, i. e., the number of edges of the shortest path connecting vertices x and y.

Write x ^ y , if the path from x ⁰ to y passes through x . A vertex y is called a direct successor of x , if y ^ x and x , y are nearest neighbors. Note that in Q k every vertex x = x ⁰ has k direct successors, and x ⁰ has k + 1 successors. The set of direct successors of x is denoted by S(x) , i. e., if x G W _n , then

S(x) = {iJ i G W n +i : d(x,y i ) = 1, i = 1, 2,... ,k}.

We consider the Hard-Core (HC) nearest neighbor model with countable number of states on the Cayley tree, where the spin variables are take values in the set of integers Z and are located at the tree vertices. A configuration on the Cayley tree is defined as a function a = { a(x) G Z : x G V } , i. e., in this model each vertex x is assigned one of the values a(x) G Z . The value a(x) = 0 means that the vertex x is “occupied”, and a(x) = 0 means that it is “vacant”.

We consider the set Z as the set of vertex set of some infinite graph G . Using the graph G , we define a G -admissible configuration as follows. A configuration σ is called a G - admissible configuration on the Cayley tree (in V n ), if { a(x), a(y) } is an edge of the graph G for any ( x, y ) from V (from V _n ). Denote the set of G -admissible configurations by Q G ( Q G ).

The activity set [24] for a graph G is a bounded function A : G ^ R + ( R + is the set of positive real numbers). The value A i of function A at the vertex i G Z is called its “activity”.

• 2. 1. Boundary law equation for Hard-Core model. For given G and λ we define the Hamiltonian of the G -HC model as

  H G (a) =

  
  

  - E ln A ^ ₍ x ) ,

  <    x E V

  + TO ,

  
  

  if a G Q G , if a G Q ^G .

  
  
  
  

Let L(G) be the set of edges of a graph G. We let A = AG = [сц^. denote the adjacency matrix of the graph G, i. e., aij    aij    ^

1,

0,

if { i,j }G L(G) if { i,j }G L(G).

Definition 1 (see [2, Chapter 12]). 1) A family of vectors l = {lxy^xy^L with lxy = {lxy(i) : i G Z} G (0, ro)Z is called the boundary law for the Hamiltonian (1), if for each (x,y) G L there exists a constant cxy > 0, such that the consistency equation lxy (i)= Cxy Ai   П   E“ij lzx(j)

zEdx\{y} jEZ holds for every i G Z, where dx is the set of nearest neighbors of a vertex x.

• 2)    A boundary law l is said to be normalisable if and only if

E Ai n Е°.izx(j)) < ™ iEZ     zEdx jEZ at any x G V.

• 3)    A boundary law is called translation-invariant , if it does not depend on edges of the tree, i. e., l xy (i) = l(i) for every oriented edge ( x,y ) and each i G Z .

Now we give a specially established correspondence between the boundary laws and Gibbs measures for the HC model (1). From our Hamiltonian we have the nearest-neighboring interaction potential A = (A b ) b , where b = ( a(x), a(y) ) .

For a given configuration ш , graph G with A = (a ij ) , edges b = { х,у} , and i = ш(х) , j = ш(у) , we define the symmetric transition matrix Q b as

^Q b ^(i,j) • — ^i a ij ^ j •

Let ш ь — { ш(х),ш(у) } , when b — ( x,y) .

For a finite subset Л С V define the Markov (Gibbsian) specification as

Ya(ш • ш|л — ад) — (Za)(w)-1 Д Qb(wb), ьпл=0

where Z Л (w) is a normalization constant (partition function).

Now we give the main theorem in [31] for the model (1).

Theorem 1. For any Gibbsian specification γ with associated family of transfer matrices (Q b ) b e L we have

•    Each normalizable boundary law (lxy )xy for (Qb)b^L defines a unique Gibbs measure у (corresponding to y) via the equation given for any connected set Л С V,

ц(а л и д л — ш д и д л ) — (Z Л ) ¹ П 1уу л (ш(у)) П Q b (^w b ),              (5)

у е д л           ь п л= 0

where for any у G дЛ, ул denotes the unique nearest-neighbor of у in Л.

•    Conversely, every Gibbs measure у admits a representation of the form (5) in terms of a normalizable boundary law (unique up to a constant positive factor).

Denote Zq — Z \{0} and assume lxy(0) — 1 (normalization at 0), for each {х,у} G L, then dividing (2) to the equality obtained for i — 0 we get lxy (i) — П

0 zedx\{y} aiQ + E aijlzx(j) j о____________

We consider the graph G with a i Q — 1 and a Q i — 1 for any i G Z and

{ a ij — 1, if i + j even , a ij — 0, otherwise .

Given a boundary law lxy(i), we define zi,x — lxy(i) where x are a direct successors of у, i. e., x G S(у). Then in case of G, (6) can be written as z2i,x

¹ + E ^z 2 j,y

\ П       ^{j eZ} o

~'              E ’ yeS(x) 1 +     zjy jeZo

i ^G Z Q ,

^z 2 i +1 ,x

^— ^ 2 i +1 П y e S ( x )

¹ + E ^z 2 j +1 ,y j e Z

1+ E jy ’ j eZ o

i G Z .

• 3.    Constant Solutions. Translation Invariant Gibbs Measures

It is natural to begin with translation invariant solutions of (7), where zx = z G R“ is constant. In this case, the system of equations (7) reads z2i = A2i

¹ + 12 ^z 2 j j S Z o

1+ E z j j eZ o

i ^{G Z} 0 ,

^z 2 i +1 = ^A 2 i +1

1 +     ^z 2 j +1

j s Z

1+ E ^z j j sZ o

i G Z ,

where A i > 0 , z i >  0 .

• 3.1.    Normalisability of boundary law. The aim of this subsection is to prove the following

Proposition 1. Let k ^ 2 and Ejez Aj < от. Then any boundary law l = {lxy := zx}{x,y)SL for the Hamiltonian (1) with graph G is normalizable.

• <1 Let ^j _s z A j <  от . Recall Q(i; j) := A i a ij A j . The normalisablity of boundary laws can be reduced (see [29]) to show

z x,i Q(i; j)z y,j <  от ( V( x,y ) G L). i S Z j SZ

Now we check this condition for solution (8)

££ zx,i ^Q(i; j)^z y,j i S Z j S Z

= ££z x,i X i a ij A j z yj < £X i zx,£A j z yj .

i S Z j S Z

i S Z        j S Z

Since the solution is independent on the vertices of the Cayley tree, for the RHS of (9) have we

£ A i z x,i£ A j z yj =  £A i z x,i   .

i SZ        j SZ               i SZ

Therefore, the lemma follows from the following estimate (in the case of (8))

£ A i z x,i = A o + £A 2 i i SZ                  i SZ

1 +     z2j j sZo

1+ E z j j S Z o

+ £ A 2 i +1 i SZ

1 +     ^z 2 j +1

j S Z

1+ E z j j SZ o

<     λ i ² .

i SZ

It is easy to check that if Ej _SZ A j <  от , then Ej _SZ A j <  от . >

Remark 1. We assume k ^ 1 and Ej_SZ A j <  от . Therefore, each boundary law which we construct, by Theorem 1 and Proposition 1, defines a Gibbs measure.

Let series ^j _s z _o z j and Ej _SZ _o A j obtained, respectively, from { z i } i _S Z _o and { A j } j _SZ _o converge, i. e.,

£ z 2 j = A, £ z 2 j +1 = B, £ ^A 2 j = A,  £ ^A 2 j +1 = Ф.

j SZ o              j SZ                 j SZ o              j SZ

Then from (8) we obtain

A = A

B = Ф

(

1 + A

1 + A + B

(

1 + B

1 + A + B

^, ) ^k .

We investigate the system of equation (10) in the following subcases.

• 3.2.    Subcase A = Ф . In this case, the system of equations (10) reads

  A = y

  B = y

  
  

  {

  
  

  1 + A

  1 + A + B

  
  

  {

  
  

  1 + B

  1 + A + B

  
  

  ^,

  )'

  
  
  
  

where y := A = Ф .

In the system of equations (11), we first consider the case A = B . In this case, we get

A = ’( (12)

The following proposition is true

Proposition 2. Let k G N. Then for any y > 0 the system of equations (12) has exactly one positive solution.

• <1 The proof follows from the fact that the function f (x) = ( i^+ X ) k is strictly decreasing under the conditions of the proposition. >

Proposition 3. For k = 1 and any y > 0 the system of equations (11) has exactly one positive solution (A; A).

• < The proof is clear. >

Remark 2. In [10], the system of equations (11) for k = 2 is studied completely and it is proved that for y C 9/4 has a unique solution (A * , A * ) , and for y > 9/4 has exactly three solutions (A * , A * ) , (A i ; B i ) and (B i ; A i ) , where

(1 + 71-4? y, B 1 ₌/1-7Г-4? V a =----2    .

V 2a J        V    2a    7        ^ 7 + Vt +4

Remark 3. In [12], the system of equations (11) for k = 3 is studied completely and it is proved that for y C 32/27 has a unique solution, and for y > 32/27 it has exactly three solutions.

Remark 4. In [11], it is shown that in the case k ^ 2 the system of equations (11) for y C k -- i ( k +1 ) k has a unique solution, and for y >  j^ ( k +1 ) k it has at least three solutions.

The following statement is true.

Proposition 4. Let k = 2, 3,4, 5 and Y _cr (k) = j-b i ( k +1 ) k • Then

•    if y C Ycr, the system of equations (11) has exactly one solution of the form (A*, A*), •    if y > Ycr, the system of equations (11) has exactly three solutions of the form (A*, A*), (A, B) and (B, A). < 1 Cases k = 2 and k = 3. The proof follows from Remark 2 and Remark 3, respectively.

Case k = 4 . In this case, from (11) we obtain

( a = Y(          )^4,                             ₍ ,

[ b = Y ( i+ A + b ) 4

For simplicity introduce the notation A = x, B = y. Then x = (i±x у.                          (14)

У V± v)

We introduce the notation 1—x = t , t >  0 . Then by virtue of (14) ( x = y • t ⁴ ), after some algebra, we get

(t - 1) (1 - y • (t ³ ± t ² ± t)) = 0.

Hence t = 1 or y • (t3 ± t2 ± t) = 1.

It is clear that for t = 1 we have the solution x = y = x * . By Propositon 2 it follows that for any Y > 0 a solution of this kind is unique.

Let t = 1. Then y^   t3 ± t2 ± t and the function y(t) is uniquely determined for each value of t, since

‘,.       3 t² ± 2t ±1

< 0.

y ^{( t )} = - (t ³ ± t ² ± t) ²

The values x(t) corresponding to each value t(t) are determined by the formula t4

x(t) = t ⁴ • y(t) = —----.

^{u u} t ³ ± t ² ± t

We substitute the expressions for x(t) = y(t) • t4 and y(t) into the first equation in (13). Then t4

t ³ ± t ² ± t ^Y ^

⁽ t ⁴ ± t ³ ± t ² ± t ^{) 4}

(t ⁴ ± t ³ ± t ² ± t ±1) 4 ’

Equation (15) has a solution t = t(Y) , but it is very hard to solve. Therefore, we consider equation (15) with respect to the variable γ , i. e. from (15) we find that

Y(t) =

⁽ t ⁴ ± t ³ ± t ² ± t ±1 ^{) 4}

(t ³ ± t ² ± t) (t ³ ± t ² ± t ±1) 4 ’

Let us prove that each value of γ corresponds to only one value of t . Note that if t is a solution to (15), then ¹ _t is also a solution to (15). Hence, it suffices to show that each value of Y corresponds to exactly one value t > 1 (or t < 1 ). To do this, consider the derivative of the function Y(t) :

Y ‘ ^{( t )} =

(t - 1) • (t ⁴ ±t ³ ±t ² ±t±1) ³ • (t ⁸ ±5t 7 ±15t 6 ±31t ⁵ ±36t ⁴ ±31t ³ ±15t ² ±5t±1)

Therefore, the function Y(t) decreases for t < 1 , increases for t >  1 , and reaches its minimum for t = 1 (see Fig. 1):

Y min = Y⁽¹⁾ = Y cr ⁽⁴⁾ =     •

Fig. 1. Graph of the function y (t) for k = 4 (on the left) and k = 5 (on the right).

Hence, each value y corresponds to only one value t >  1 (or t < 1 ) for y > Y cr (4) , the value t = 1 for y = Y cr (4) and the equation (13) has no solutions for 7 <  Y c_r (4) . Due to symmetry, the system of equations (13) has a unique solution of the form (x * , x * ) for 7 C Y cr (4) , and for 7 >  Y c_r (4) has exactly three solutions (x * ,x * ) , (x,y) and (y,x) , x = y , where x * is the only positive solution of (11).

Case k = 5 . In this case, similar to the case k = 4 , for Y(t) we get

(t ⁵ + t ⁴ + t ³ + t ² + t + 1) ⁵

Y(t) = (t4 + t3 + t2 + t)(t4 + t3 + t2 + t + 1)5 ’ and                 ,m = (t - 1)(t + 1)3(t2 + t + 1)4(t2 - t +1)4 • P(t)

^{Y ( t )}          t ² • (t ² + 1) ² (t ⁴ + t ³ + t ² + t + 1) 6

where

P (t) = t ¹⁰ + 4t 9 + 11t 8 + 24t ⁷ + 41t ⁶ + 38t ⁵ + 41t ⁴ + 24t ³ + 11t ² + 4t + 1.

Then the function y(t) decreases for t < 1, increases for t > 1, and reaches its minimum for t = 1 (see Fig. 1)

^Y min — ^{y (1)} — ^Y cr ⁽⁵⁾ —

3125 .

Hence, the system of equations (13) has a unique solution of the form (x * ,x * ) for y C Y cr (5) , and for Y > Y _cr (5) has exactly three solutions (x ^* , x ^* ) , (x, y) and (y, x) , x = y , where x ^* is the only positive solution of (11). >

By virtue of Proposition 3, Proposition 4 and Remark 4 we obtain the following theorem.

Theorem 2. For the HC model with a countable set of spin values ( corresponding to the graph G) , the following statements are true:

1.    Let k = 1 and y > 0. Then there is a unique TIGM. 2.    Let k = 2, 3,4, 5 and Ycr(k) = k~L (k+г)k• Then for Y C Ycr(k), there is exactly one TIGM ^o, and for y > Ycr(k) there are exactly three TIGMs ^0, ^1, ^2. 3.    Let k ^ 6. Then for y C Ycr(k) there is exactly one TIGM, and for y > Ycr(k) there are at least three TIGMs.

• 3.3.    Subcase A = Ф .

Case k = 1 and A = Ф . In this case, from (10) we obtain

A = a( ¹ + A ) , V1 + A + BJ

B =фГ 1+B Y

<1 + A + Bj

It is clear that A = B , A > A and Ф > B , since A = Ф . For convenience, we introduce the notations A = x , B = y , A = a and Ф = в . Then from (16), we get

'y=(1+x) " - 0 ■ x = (1 + y) (в - 1) .

Let us write (17) as follows:

I

y = ^f(x,a), ^x = ^f (y^-,

where f (x,a) := (1 + x) (x - 1).

From the last system of equations we obtain

( y = ^{f ( f (}y^,e)-a)- ( ^x = ^{f(f ( x,a ) ,e ) -}

or

(y ² + (a - в + 1)y — вЖ — y ² + в) y(1 + y)(в - y)

(x ² + (в - a + 1)x - a)(ax - x ² + a) x(1 + x)(a - x)

From (18) we find

в(y + 1) 2 (в - y) y(вy - y ² + в )

a(x + 1) 2 (a - x) x(ax - x ² + a)

Introduce the following

g(x,a)

a(x + 1) ² (a - x) x(ax - x ² + a)

and consider the derivative of the function g(x, a) with respect to the variable x :

‘            a(1 + x)(2x ³ - 2ax ² + a ² x + a ² )

g x ^{( x- a )} = ^-

x ² (xa - x ² + a) ²

Let us consider the equation g x (x, a) = 0 with respect to x , i. e., ^(x, a) := 2x 3 — 2ax 2 + a ² x + a ² = 0.

Let us show that the equation g(x, a) = 0 has no positive solution. To do this, it is sufficient to show that

2x 3 — 2ax ² + a ² x > 0.

Indeed, using the Cauchy inequality we obtain

2x 3 + a ² x >  2 ^ 2 ax ² > 2ax ² .

Therefore, the function g(x, a) decreases for x > 0 . Then each value of в corresponds to one value of x . Due to symmetry of (18), the above can also be said about the parameter α .

Summarising, the system of equations (16) has exactly one positive solution for A = Ф , i. e., the following theorem is true.

Theorem 3. Let k = 1, A > 0, Ф > 0 and A = Ф. Then for the HC model with a countable set of spin values (corresponding to the graph G) there is exactly one TIGM.

Case k = 2 and A = Ф . In this case, from (10) we obtain

A = A

B = Ф

(

(

1 + A

1 + A + B

1 + B

1 + A + B

,

) ■ ■

Since A = Ф, then A = B , A > A and Ф > B . For convenience, we introduce the notations A = x, B = y, A = a and Ф = в. After some algebras, from (19) we obtain y=(1+x’( g- - 1) ’ x=(1+y) (/I - 1) ■

We write the system of equations (20) as

Г y = f(x,a), ( x = f (yg), where

f (x, a) := (1 + x)

(f7 ')■

It follows that

( y = ^{f ( f (}y ^{,e ) ,a ) -}I^x = ^{f ( f ( x,a ) ,e )}.

or

y = Ш ⁺    - y) (] (1 + yO — g y) -  ’

/ = (/x ^{+    —} x) (/(1+ x)( g x —^ ^— 1) ■

After some algebras, from (21), we obtain

в(1 + y) ³ (ув -Т У) Ту (у Т в - y T y + ^тв ) ^{2 ,} a(i + x) ³ (т» — та) Т Х(х т а — х т х + т а ) ²

Let us introduce the function

h(x, а) :=

а(1 + x) ³ (та — ТХ )

Т Х(х Т а — х т х + т а ) ²

and calculate its derivative with respect to the variable x :

h X (x, а)

ха(1 + x) ² («(x ² — 1) — T аx ^3/2 (x — 5) — 6x ² ) 2x 5 / 2 (х т а — x ^3/2 + t« ) ³

Let us consider equation h' _x (x, а) = 0 with respect to x . Then

ф(х, а) := а(х ² — 1) — T«x ^{3 / 2} ( x — 5) — 6x ² = 0.

We write ф(х, а) as follows:

ф(х, а) = —Т ах ⁵/² — (6 — а)х ² + 5 Т ах ^3/2 — а.

It is easy to see that the equation ф(х, а) = 0 , by Descartes’ theorem on the number of positive roots of a polynomial, has at most two positive solutions or has not positive solutions. On the other hand,

ф(0, а) = — а < 0,   lim ф(х, а) = —от , ф(1, а) = 4Та — 6.

х ^^

Therefore, there exists a value «о of the parameter а such that the equation ф(х, а) = 0 has two positive solutions. Let Xi ( «o ) and X2 ( «o ) be solutions of the equation ф(а,х) = 0 . Then the function h(x, а) decreases for 0 < х <  Xi ( «o ) and х >  X2 ( «o ) , and increases for xi ( «o ) <  x <  X2 ( «o ) . Consequently, for the function h(x, а) the point х = X2 ( «o ) is a local maximum, and the point х = Xi ( «o ) is a local minimum (see Fig. 2).

Fig. 2. Graph of the function h(x, a) for a = 5 (on the left) and a = 2 (on the right).

As a result, for the parameter β we have:

• 1)    if 0 <  в < в(x 1 (a o ), a o ) , then each value of в corresponds to one value of x ;

• 2)    if в = в(х 1 (а о ), a o ) , then corresponds to two values of x ;

• 3)    if в(х 1 (a o ),a o ) < в < в(х 2 (а о ),a o ) , then corresponds to three values of x ;

• 4)    if в = в (x 2 (a o ), a o ) , then corresponds to two values of x ;

• 5)    if в > в(x 2 (a o ), a o ) , then corresponds to one value of x .

Since there is symmetry in the system of equations (20), the above can also be said about the parameter α . Thus, the following theorem is true.

Theorem 4. Let k = 2. Then for the HC model with a countable set of spin values (corresponding to the graph G) the number of TIGMs is:

1,

N = < 2,

3,

if 0 < Ф < Л 1 (Д) or Ф > Л ₂ (Д), if Ф = Л 1 (Д) or Ф = Л ₂ (Д), if Л 1 (Д) < Ф < Л 2 (Д).

Here

Л 1 (Д) =

Д(1+ x i ) ³ (УД _X x)

y x (x 1 y x — x 1 y x i + У д ) ²

_{4 к}      Д(1 + x 2 ) ³ (УД — y x 2 )

Л2(Д) — -----------------------------2 , yx2 (x2УД — x2yx2 + УД)

where x1 and x2 are positive solutions of the equation

У д x ⁵ / ² + (6 — Д)x ² — б У д x ³ / ² + д = о.

Case k = 3 and Д = Ф . In this case, from (10) we obtain

A = Д

B = Ф

1 + A

1 + A + B

1 + B

1 + A + B

Since Д = Ф , then A = B , Д > A and Ф > B . Introducing the notations A = x , B = y , Д = a and Ф = в , similarly to the previous case from (22) we obtain

( У = f (f (У,в ),a), Ix = f(f (x,a),в), where

f (x, a) := (1 + x)

(37 ').

As a result, solving the system of equations (22) leads to the following y=

(W⁺ y ^—y) (/i+yf—ii)^—■

x=

(1 + x)( 3/a — yx)

β ³ x

—

From (23) we get

a = в =

в(1+ y) 4 (X -у ) № ( y V P - y V y + V P ) ^{2 ,}

a(1 + x) ⁴ ( V a — v x )

V x (x v a — x V x + v a ) ²

Consider the function

(   )a(1 + xY 4 ( V a - V x )

^{x x,a} ’  V x (x 3/ a — x- V x + V a ) 3

and calculate its derivative. Then from the equation X x (x, a) = 0 we have

z(x, a) := - 2a ^1/3 x ^7/ 3 + 2a ² / ³ x ² - 12x ⁵ / ³ + 10a ^1/3 x ^4/3 + a ^2/3 x - a ² / ³ = 0.

By the well-known Descartes theorem on the number of positive roots of a polynomial, the equation Z (x, a) = 0 has four positive solutions, or two positive solutions, or does not have positive solutions.

On the other hand,

Z (0, a) = - a ² / ³ < 0,   lim Z (x, a) = -от ,    Z (1, a) = 8a ^1/3 + 2a ^2/3 - 12.

x ^^

Therefore, there exists a value a o of the parameter a such that the equation Z (x, a) = 0 has two or four positive solutions. Let us prove that the equation Z (x, a) = 0 for x G (0; a) and α > α 0 has two solutions or has no solution. To do this, we will show that the equation

Z ' ( x, a) := - ¹⁴ a ^1/3 x ^4/3 + 4a ^2/3 x - 20x ^2/3 + - a ^1/3 x ^1/3 + a ^2/3 = 0

x— -      3                              3

has a unique positive solution. We denote x = t ³ . Then

Z X (t, a) := - 14 a ^1/3 1 ⁴ + 4a ^2/3 1 ³ - 20t ² + 40 a ^1/3 1 + a ^2/3 = 0.

Using the Ferrari formula, we find the only positive solution t(a) of the equation Z X (t, a) = 0 . Then the function Z(x, a) increases for x < t ³ (a) and decreases for x > t ³ (a) . Consequently, for x G (0; a) and a > a o , the equation Z(x, a) = 0 has two positive solutions or no solutions. In the case where the equation Z(x, a) = 0 has no solutions, then each value of P corresponds to one value of x .

Let the equation Z(x, a) = 0 have two positive solutions of the form x i (a o ) and x 2 (a o ) . Then the function x(x, a) decreases for 0 < x < x i (a o ) and x > x 2 (a o ) and increases for x i (a o ) < x < x 2 (a o ) . Consequently, for the function x(x, a) the point x = x 2 (a o ) is a local maximum, and the point x = x i (a o ) is a local minimum (see Fig. 3).

Fig. 3. Graph of the function x(a,x) for a = 2 (on the left) and a = 1 (on the right).

From all that has been said, for the parameter β we have:

• 1)    if 0 <  в < в(х 1 (a o ), a o ) , then each value of в corresponds to one value of x ;

• 2)    if в = в (x i (a o ),a o ) , then corresponds to two values of x ;

• 3)    if в(х 1 (a o ),a o ) < в <  в (x 2 (a o ),a o ) , then corresponds to three values of x ;

• 4)    if в = в (x 2 (a o ), a o ) , then corresponds to two values of x ;

• 5)    if в >  в(x 2 (a o ), a o ) , then corresponds to one value of x .

Since there is symmetry in the system of equations, the above can also be said about the parameter α .

Thus, the following theorem is true.

Theorem 5. Let k = 3. Then for the HC model with a countable set of spin values (corresponding to the graph G) the number of TIGMs is:

1,

N = < 2,

3,

if 0 < Ф < Л 1 (Д) or Ф > Л ₂ (Д), if Ф = Л 1 (Д) or Ф = Л ₂ (Д), if Л 1 (Д) < Ф < Л 2 (Д).

Here

Л 1 (Д) =

д(1+£1№-,^5 Г )

^/x i (x 1 / Д — x 1 ^/x ! + / Д ) ³

Л 2 (Д) =

Д(1+ x 2 ) 4 (\ Д —^ ) //x 2 (x 2 / Д — x 2 //x 2 + / Д ) 3

where x1 and x2 are positive solutions equations

2Д ^1/3 x ^7/3 — 2Д ^2/3 x ² + 12x ⁵ / ³ — 10Д ^1/3 x ^4/3 — Д ^2/3 x + Д ^2/3 = 0.

Acknowledgement. The authors dedicate this work to the anniversary of the outstanding mathematician Evgeny Mikhailovich Semenov and wish him good health and further creative success.