CC BY
10
УДК 517.98+530.1
Positive Fixed Points of Cubic Operators on R2 and Gibbs Measures
Yusup Kh. Eshkabilov* Shohruh D. Nodirov"
Karshi State University 17, Kuchabag st., Karshi, 180100 Uzbekistan
Received 13.03.2019, received in revised form 16.04.2019, accepted 10.07.2019 One model with nearest neighbour interactions of spins with values from the set [0,1] on the Cayley tree of order three is considered in the paper. Translation-invariant Gibbs measures for the model are studied. Results are proved by using properties of the positive fixed points of a cubic operator in the cone R+.
Keywords: Cayley tree, Gibbs measure, translation-invariant Gibbs measure, fixed point, cubic operator,
Hammerstein's integral operator.
DOI: 10.17516/1997-1397-2019-12-6-663-673.
Introduction
A lot of nonlinear operators are connected with problems in statistical physics, biology, thermodynamics, statistical mechanics and so on. One of the central problem in statistical physics is the existence of phase transitions. Phase transitions are connected with the theory of Gibbs measures [1]. In the theory of Gibbs measures there are a lot of papers devoted to Gibbs measures on a Cayley tree [2]. Splitting Gibbs measure is studied for models on a Cayley that can be divided into three classes: 1) models with a finite set of spin values; 2) models with a countable set of spin values; 3) models with a continuum set of spin values. Let us note that problems of studying Gibbs measures for models with a finite and countable set of spin values on a Cayley tree are reduced to the study of systems of algebraical or functional equations [3-13]. One of the main factors is that studying translation-invariant Gibbs measures for models with a continuum set of spin values is reduced to the study of positive fixed points of non-linear integral operator [14-20].
In the case of continuum set of spin values (i.e., [0,1]) various models with the nearest neighbour interactions on a Cayley tree were considered [14-20]. It was found that the existence of translation-invariant Gibbs measure for the models is equivalent to the existence of a positive fixed point of Hammerstein's nonlinear integral operator [17,20]. It was proved that the existence of translation-invariant Gibbs measures for the models on a Cayley tree of an arbitrary order and uniqueness of translation-invariant Gibbs measures on the Cayley tree of order one was shown (see [16,18]).
It is found that a sufficient condition for the model has the unique translation-invariant splitting Gibbs measure and each constructed model has at least two periodic Gibbs measures (see [15,16]).
* yusup62@mail.ru t shoh0809@ mail.ru © Siberian Federal University. All rights reserved
Models on the Cayley tree of order two were considered [20]. The study of translationinvariant Gibbs measures was reduced to the study of positive fixed points of some quadratic operator on R2. Sufficient conditions were also given such that the model has one, two or three translation-invariant Gibbs measure by using quadratic operators.
In this paper we consider the translation-invariant Gibbs measures for models on the Cayley tree of order three given in [20] .
1. Preliminaries
A Cayley tree Tfc = (V, L) of order k > 1 is an infinite homogeneous tree, i.e., a graph without cycles with exactly k + 1 edges. Here V is the set of vertices and L is the set of edges.
Let us consider models where the spin takes values from the set [0,1], and it 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]. Let us denote Ha = [0,1]A It is the set of all configurations on A. A configuration a on V is defined as a function x G V ^ a(x) G [0,1], and the set of all configurations is [0,1]V. The Hamiltonian of the model is
H(a) = — J a € (1)
(x,y)£L
where J € R \ {0} and 3 = T, T > 0 is temperature, £ : (u,v) € [0, l]2 ^ £uv € R is a given bounded measurable function. As usual, (x, y) stands for the nearest neighbour vertices.
We write x < y if the path from x0 to y goes through x. Vertex y is a direct successor of x if y > x and x,y are nearest neighbours. Let us denote the set of direct successors of x by S(x). Any vertex x = x0 has k direct successors, and x0 has k + l direct successors. Let h : x € V ^ hx = (htxx,t € [0, l]) € R[0'1] be mapping of x € V \ {x0}. Now, we consider the following equation
1
/exp( J3£tu)f (u,y)du f (t,x)= J] 1-• (2)
yes(x) J exp( J3£0u)f (u, y)du
0
Here and below f (t,x) = exp(ht,x — h0,x), t € [0, l] and du = X(du) is the Lebesgue measure.
It is known that necessary and sufficient condition of the existence of the splitting Gibbs measure for model (1) is the existence of a solution of equation (2) for any x € V \ {x0}. Thus, we know that splitting Gibbs measure n for model (1) depends on the function f (t,x) and each splitting Gibbs measure corresponds to a solution f (t,x) of equation (2). Let us note that number of the Gibbs measures for model (1) is equal to the number of positive solutions of integral equation (2).
A detailed definition of the splitting Gibbs measure for models with nearest neighbour interactions and continuum set of spin values on the Cayley tree can be found in [14-20]. In what follows the splitting Gibbs measure will be called the Gibbs measure.
Let us note that the analysis of solutions to (2) is not easy. It is difficult to give a full description of the given potential function £t,u. We study Gibbs measures of model (2) in the case f (t,x) = f (t) for all x € S(x). Such Gibbs measure is called translation-invariant measure. We introduce
C+[0, l] = {f € C[0, l] : f (x) > 0}, C>[0, l] = C+[0, l] \{0 = 0}
Let £tu be a continuous function. For every k € N we consider an integral operator Hh acting in the cone C+ [0,1]
ri
rh i K (t,u)f
10
where K(t,u) = exp(jp£tu).
The operator Hh is called the Hammerstein integral operator of order k.
(Hkf)(t)= i K(t,u)fk(u)du, k G N, Jo
Lemma 1 ([16]). Let k ^ 2. Hamiltonian H (1) has a translation-invariant Gibbs measure iff the Hammerstein integral operator Hh has a positive eigenvalue, i.e., the Hammerstein integral equation
Hhf = Xf, f € C+[0,1] (3)
has a non-zero positive solution for some X > 0.
Moreover, if X0 > 0 is an eigenvalue of the operator Hh,k > 2 then an arbitrary positive number is the eigenvalue of the operator Hh. A number of positive eigenfunctions that correspond to positive eigenvalues Xi > 0 and X2 > 0 of the operator Hh are equal (see [16]). Then we have the following lemma.
Lemma 2. Let k ^ 2. A number Ntigm(H) of translation-invariant Gibbs measures for model (1) is
Ntigm(H) = N+iX(Hh), where N+iX(B) is a number of non-trivial positive fixed points of the operator B.
2. Main results
Let ^i(t), p2(t) and fai(t), fa2(t) are strictly positive functions that belong to C+ [0,1]. We consider Hamiltonian (1) on the Cayley tree r3 with the potential
In (¿1(t)<f1(u)+ fa2(t)<f2(u)) ^U = -Jp-• (4)
We consider the Hammerstein integral operator H3 on C+[0,1] in the following form
(Hsf)(t) = t (Mt)fi(u)+ fa(t)Mu))f(u)du
0
Let us introduce the following designations
i [ i [ i
3/„.\j„. „, _ _ I fu)d>i(u)6o(u)du, a2i = wi(u)6i(u)62(u)du,
an = / p1(u)4>l(u)du, ai2 = pi(u)4>l(u)02(u)du, a.i\ = pi(u)$i(u)4>2 (u)d
o o o
a22 = pi(u)4>l(u)du, ßii = p2(u)ft! (u)du, ßi2 = / ¥2(u)4>\(u)$2(u)du,
ooo
ß2i = / P2(u)$i(u)4>2(u)du, ß22 = / tp2(u)j>2(u)du.
oo
00 It is easy to verify that aij > 0 and ¡3ij > 0 for all i,j € {1,2}• We introduce a fourth degree polynomial P4 (£)
Pa(0 = Hoi4 + Hie + 3^2Î2 + - Hi, (5)
where
M0 = «22, Ml = 3«21 — ¡22, M2 = «12 — ¡21, M3 = «11 — 3^12, M4 = ¡11-We use the following designations
o=(P)3+(2 )2,
ût = 2fl+ 2n<k - 2>) , k = ,
where
and
= + 3^2 = Ml + M3
^ ^ 2 r !
cos a =----, a G [0, n I.
2 V Pj
We also introduce
a M1 a M1 a M1
71 = #3 — ^—, 72 = #1 — ^—, 73 = 02 — ^— •
4^0 4^0 4^0
Theorem 2.1. Let Q ^ 0. Then model (1) on the Cayley tree of order three has the unique translation-invariant Gibbs measure, i.e., Ntigm(H) = l.
Theorem 2.2. Let Q < 0. If one of the following conditions
(a) 72 < 0,
(b) 72 > 0, P4(Y2) < 0,
(c) 72 > 0, P4(y3) > 0,
is satisfied then model (1) on the Cayley tree of order three has the unique translation-invariant Gibbs measure, i.e., Nti3m(H) = l.
Theorem 2.3. Let Q < 0. If one of the following conditions
(d) 72 > 0, P4(Y2) =0,
(e) 72 > 0, P4(Y3) =0,
is satisfied then model (1) on the Cayley tree of order three has two translation-invariant Gibbs measures, i.e., Nti3m(H) = 2.
Theorem 2.4. Let Q < 0. If the following condition
(f) 72 > 0, P4(72) > 0, P4(73) < 0,
is satisfied then model (1) on the Cayley tree of order three has three translation-invariant Gibbs measures, i.e., Ntigm(H) = 3.
3. Positive fixed points of cubic operators on R2
We introduce
R+ = {(x,y) G R2 : x > 0, y > 0}, R> = {(x, y) G R2 : x > 0, y > 0}. We consider the following cubic operator (CO) C on the cone R+
C(x, y) = (aux3 + 3ai2x2y + 3a2ixy2 + a22y3, bnx3 + 3bi2x2y + 3b2ixy2 + b22 y3), (6)
where aij > 0 and bij > 0 for all i,j G {1, 2}.
Clearly, an arbitrary non-trivial positive fixed point (xo,yo) G R+ of the CO C is strictly positive, i.e., xo > 0, yo > 0. We denote a number of fixed points of the CO C that belongs to
R2> by N>ix(C).
Lemma 3. If w = (xo, yo) G R> is a fixed point of the CO C then w G R> and Ço = — is a root
xo
of the algebraic equation
a22£4 + (3a2i - b22) Ç3 + 3 (ai2 - b2i) Ç2 + (an - 3bi2)Ç - bn = 0. (7)
Proof. Let the point w = (x0, y0) G R> be a fixed point of CO C. Then aiixO + 3ai2x0yo + 3a2ixoy2 + a22 y3 = xo,
biixO + 3bi2xoyo + 3b2ixoyo + b22y3 = yo.
Taking into account that — = Ço, we obtain
x0
3 3 3 2 3 3
aiix3 + 3ai2x0Ço + 3a2ix0Ç0 + a22x0Ç0 = xo,
Q Q Q O Q Q
biix° + 3bi2x3^o + 3b2lXoÇo + b22X0Ç0 = x oÇo.
x3 (an + 3ai2^o + 3a2i£° + a22Ç°) = xo, x3 (bu + 3bi2Ço + 3b2iÇ2 + b22Ç°) = Çox o.
Consequently we have
Hence, we have
1 a + 3ai2^0 + 3a2i^2 + 022^0
C0 bii + 36^0 + 362l£o + 622C0 ' Using the last equality, we obtain
«22^0 + (3a2i - 622) Co + 3 (ai2 - 621) Co + (aii - 36i2)Co - 611 = 0. This completes the proof. □
Lemma 4. If Co is a root of algebraic equation (7) then the point w0 = (x0,C0x0) G R> is a fixed point of the CO C, where
X0 = —, ■ (8)
V7 aii + 3ai2 C0 + 3a2iCo + a22Co
Proof. Let £0 > 0 and £0 is a root of equation (7). We assume that y0 = £0x0, where x0 is given by equality (8) and w0 = (x0, £0x0). From the equality y0 = £0x0 we have
anx0 + 3ai2x2y0 + 3a2iX0y°° + a22y3 = «iix3 + 3ai2x0 (£0X0) + 3a2iX0 (£0X0)2 + a<22 (£0X0^ =
= x3 • («11 + 3a12£0 + 3a21£0 + a22£o) = / a ^ o '
V aii + 3a12£0 + 3a21£5 + a22£3
i.e.
aiix3 + 3ai2X°y0 + 3a2ix 0y° + a22y3 = X0.
On the other hand
a22£0 + (3«21 - 622) £° + 3 («12 - 621) £° + («11 - 36i2)£0 - bii = 0. Then we obtain
OQ O Q A / OQ\
611+36i2£0+3&2i£o +622£o = aii£0+3ai2£0 +3a2i£o +a22£ci = £0(011 + 3«i2£ + 3«2i£ + a,22£ ) . From the last equality we have
£0 bii + 36i2£0 + 3621 £2 + 622 £o
•y/aii + 3ai2£0 + 3a2i£2 + a22£3 (V aii + 3«i2£0 + 3«2i ^ + a22£3 )3
= biix3 + 3bi2xly0 + 3b2iX0yl + b22yl = y0. This completes the proof. □
We introduce
M0 = a22, Mi = 3a2i - 622, M2 = ai2 - 621, M3 = aii - 3bi2, mo = bii
and define the fourth degree polynomial P4(£)
Po(£) = M0£4 + Mi£3 + 3m2£2 + M3£ - MO. (9)
Lemma 5. The CO C has at least one positive fixed point in R >, i.e., Nfix(C) > 1.
Proof. It is clear that P4(0) = -611 and P4(+to) = It means that there exists c > 0 such that P4(c) = 0. According to lemma 4, (x0, cx0) is a fixed point of CO C and
1
X0
%/ aii + 3ai2C + 3a2ic2 + a22c3
□
Lemma 6. A number of strictly positive fixed points of the CO C less than or equal to three, i.e., 1 < Nfix(C) < 3.
Proof. We have the following table for the number of sign changes of the coefficients of the polynomial P4(£) (Tab. 1).
Using this table and the Descartes rule, we can conclude that a number of positive solutions of the polynomial P4(£) is not more than three (see [22, pp. 27-29] ), i.e., 1 < N fix(C) < 3. □
Let us introduce
o=(P )3+(2 )2,
Table 1.
Pi(i) Ho Hi H2 H3 Hi the number of sign changes
1. + + + + - 1
2. + + + - - 1
3. + + - - - 1
4. + - - - - 1
5. + - - + - 3
6. + - + + - 3
7. + + - + - 3
8. + - + - - 3
6k = 2^-3cos(a + 2n(k , k = 1, 2, 3,
where
and
= 3H 1 + 3H2 = Hi 3H1H2 + H3 P 16h2 2H0 ' ^ 32H0 &HO 4H0
q f 3\ 2 r n
cos a =----, a G [0, ni.
2 \ Pj
We also introduce
a Hi a Hi a Hi
Yi = h - —, Y2 = 0i --—, Y3 = --.— • 4H0 4ho 4HO
Theorem 3.5. Let Q > 0 then the CO C has the unique fixed point in R>, i.e., N>iX(C) = 1.
Proof.
a) Let Q > 0. Then the equation P4(£) = 0 has one real root. This root is stationary point of the function P4(£). Furthermore we have P4(0) = -bii < 0, P4(±x>) = Consequently, there exists the unique number £0 > 0 such that P4(£0) = 0.
b) Let Q = 0. Then the equation P4(£) = 0 has a multiple root and all of its roots are real. A simple root (of multiplicity 1) is stationary point of the function P4(£). Also P4(0) = -bii < 0, P4(±x>) = It indicates that there exist the unique number £0 > 0 such that P4(£0) = 0.
□
Theorem 3.6. Let Q < 0. If the CO C satisfies one of the following conditions
(a) Y2 < 0,
(b) Y2 > 0, P4(Y2) < 0,
(c) Y2 > 0, P4(ys) > 0,
then the CO C has the unique fixed point in R>, i.e., NfiX(C) = 1. Proof. Let Q < 0. We have
P4 (O = 4H0? + 3Hie + 6H2Z + H3- (10)
One can find roots of the equation P4(£) =0 by the Vieta method (see [23]). From Q < 0 it turns out that numbers d1, 62, 93 are real and 93 < 61 < 62. By the Vieta method numbers Y1,Y2,Y3 are roots of the polynomial P4(£). Then the polynomial P4(£) (10) has the following form
P4(£) = 4M0(£ - Yi)(£ - Y2)(£ - Y3).
It follows that function P4(£) is an increasing (decreasing) function on the set (y1,Y2) U (y3, +ro) ((-ro,Yi. ) U (y2, y3)). The function P4(£) has a local maximum value at the point y2 and local minimum values at the points y1 and y3.
(a) Let y2 < 0. It is clear that minie(72 +TO) P4(£) = P4(y3) and function P4(£) is an increasing function on the interval (y3, +ro). On the other hand, we have P4(0) < 0. Consequently, we obtain P4(y3) < 0. It means that polynomial P4(£) has the unique positive root.
(b) Let Y2 > 0 and Po(y2) < 0. Then max?e(7li73) Po(£) = PoY) < 0. Function Po(£) is an increasing function on the interval (y3, +ro). Then polynomial P4(£) has the unique positive root £1 and £1 G (y3, +ro).
(c) Let Y2 > 0 and P4(Y3) > 0. Then max?e(7l73) Po(£) = PoY) > 0 and min^e(72j+TO) P4(£) = P4(y3) > 0. Using inequality P4(0) < 0, we obtain that polynomial P4(£) has the unique positive root £1 and £1 G (0, y2). □
Theorem 3.7. Let Q < 0. If the CO C satisfies one of the following conditions
(d) Y2 > 0, Po(Y2) =0,
(e) Y2 > 0, Po(Y3) =0,
then the CO C has two fixed points in R>, i.e., Nfix(C)=2.
Proof. (d) Let Y2 > 0 and P4(Y2) = 0. Then max?e(7l 73) Po(£) = PoY) = 0 and £1 = Y2 is the root of the polynomial P4(£). Since P4(£) is an increasing function on the interval (y3, ro), the polynomial P4(£) has a root £2 G (y3, ro) for y3 > 0 and P4(y3) < 0. It is clear that polynomial P4(£) does not have any other roots in the (y3, ro).
(e) Let y2 > 0, P4(y3) = 0. Function P4(£) is an increasing function on the (-ro, y2). Then polynomial P4(£) has a positive root £1 G (0, y2). We have min^e(72jTO) P4(£) = P4(y3) = 0. Then £2 = y3 is the second positive root of the polynomial P4 (£). The polynomial P4(£) does not have another root. □
Theorem 3.8. Let Q < 0. If the CO C satisfies the following condition
(f) Y2 > 0, Po(Y2) > 0, Po(Y3) < 0,
then the CO C has three fixed points in R>, i.e., Nfix(C) =3.
Proof. Let y2 > 0, P4(y2) > 0, P4(y3) < 0. Function P4(£) is an increasing function on the set (y1,Y2) U (y3, +ro) and a decreasing function on the interval (y2,Y3). Then polynomial P4(£) has three positive roots £1 G (0,y2), £2 G (y2,Y3) and £3 G (y3, ro), as P4(0) = -611 < 0, PO(Y2) > 0, PO(Y3) < 0, Po(+ro) = +ro. □
4. Proofs of the main results
Let ^1(t), y2(t) and ^1(t), $2(t) are strictly positive functions that belong to C+[0,1]. We consider the Hammerstein integral operator H3 on C+ [0,1] in the following form
(H3f)(t)= i (Mt)fi(u) + fa(t)v2(u)) f (u)d
JO
and cubic operator C on R2 has the form
0 O O Q Q O O Q.
C(x,y) = (aiix +3ai2x y + 3aoixy + aooy , ßiix + 3ßi2x y + 3ß2ixy + ß22V ). Here
i f i f i
3f„.\j„. „.,__! !u)é>1(u)é2(u)du, a2i = œi(u)Si(u)S2(u)du,
aii = / pi(u)4>\{u)du, ai2 = pi(u)4>1(u)^2(u)du, aoi = pi(u)$i(u)4>2 (u)d
o o o
a22 = / pi(u)^3(u)du, ßii = p2(u)4>\ (u)du, ßi2 = / p2(u)4>2(u)&(u)du,
ooo
ß2i = / P2(u)$i(u)4>2(u)du, ß22 = / tp2(u)4>0(u)du.
oo
>0 Jo
It is clear that a ij > 0 and 3ij > 0 for all i,j € {1,2}•
Lemma 7. The Hammerstein integral operator H3 has a non-trivial positive fixed point iff the CO C has a non-trivial positive fixed point and N+iX(H3) = NfiX(C).
Proof. Let the Hammerstein integral operator H3 has a nontrivial positive fixed point f (t) € C+[0,1]. Let us introduce
ci = / pi(u)f3(u)du (11)
o
and
C2 = ! V2(u)f3(u)du. (12)
0
It is clear that ci > 0, c2 > 0, i.e. (ci, c2) € R>. Then function f (t) satisfies the equality
f (t) = fa (t)ci + fo (t)c2 (13)
and f (t) € C> [0,1].
Consequently, from (11) and (12) we have the following two identities for parameters ci, c2 ci = aiicf + 3ai2cjc2 + 3a2icic2 + a22c\, c2 = /3iic?i + 3fii2c\c2 + 3^2icic'2 + ¡322c3. Therefore, the point (ci, c2) is the fixed point of the CO C.
(b) Let us assume that point (x0,y0) is a non-trivial positive fixed point of the CO C and x0, y0 satisfy the following equalities
aiixO + 3ai2x0yo + 3a2i xoyO + a.22y0 = xo, 3iix3o + 33i2x0yo + 332ixo y20 + ¡22y3 = yo•
It is easy to verify that function f0(t) = fai(t)x0+fa2(t)y0 is the fixed point of the Hammerstein integral operator H3 and f0(t) € C>[0,1] for (x0,y0) € R+. This completes the proof. □
Taking into account potential (4), Lemma 2 and Lemma 7, the following equality holds for model (1) on the r3
N tigm(H) = N+iX(H3) = NfiX(C). Using the last equality and Theorems 3.5-3.8, we obtain Theorems 2.1-2.4, respectively.
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Положительные неподвижные точки кубических операторов на R2 и меры Гиббса
Юсуп Х. Эшкабилов Шохрух Д. Нодиров
Каршинский государственный университет Кучабог, 17, Карши, 180100 Узбекистан
В этой статье мы 'рассматриваем модель с взаимодействиями ближайших соседей и с множеством [0,1] значений спина на дереве Кэли третьего порядка. Трансляци,онно-и,нвари,антные меры Гиббса для модели исследованы свойствами положительных неподвижных точек кубического оператора в конусе R+.
Ключевые слова: дерево Кэли, мера Гиббса, трансляци,онно-и,нвари,антные меры Гиббса, неподвижная точка, кубический оператор, интегральный оператор Гаммерштейна.
