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Positive fixed points of Lyapunov operator
R. N. Ganikhodjaev, R. R. Kucharov, K. A. Aralova National University of Uzbekistan, 100174, Tashkent, Uzbekistan ramz3364647@yahoo.com
DOI 10.17586/2220-8054-2020-11-4-373-378
In this paper, fixed points of Lyapunov integral equation are found and considered the connections between Gibbs measures for four competing
interactions of models with uncountable (i.e. [0,1]) set of spin values on the Cayley tree of order two.
Keywords: Lyapunov integral operator, fixed points, Cayley tree, Gibbs measure.
Received: 13 January 2020
Revised: 9 August 2020
1. Introduction
Spin models on a graph or in continuous spaces form a large class of systems considered in mechanics, biology, nanoscience, etc. Some of them have a real physical meaning, others have been proposed as suitably simplified models of more complicated systems. The geometric structure of the graph or a physical space plays an important role in such investigations. For example, in order to study the phase transition problem on a cubic lattice Zd or in space one uses, essentially, the Pirogov-Sinai theory; see [1-3]. A general methodology of phase transitions in Zd or Rd was developed in [4]; some recent results in this direction have been established in [5,6] (see also the bibliography therein).
During last years, an increasing attention was given to models with a uncountable many spin values on a Cayley tree. Until now, one considered nearest-neighbor interactions (J3 = J = a = 0, J1 = 0) with the set of spin values [0,1] (for example, [7-12]).
In [13] it is described that splitting Gibbs measures on r2 by solutions to a nonlinear integral equation for the case J| + J2 + J2 + a2 = 0 which a generalization of the case J3 = J = a = 0, J1 = 0. Also, it is proven that periodic Gibbs measure for Hamiltonian (1) with four competing interactions is either translation-invariant or Gfc2) — periodic.
In this paper, we consider Lyapunov's operator with degenerate kernel. In [11], Fixed points of Lyapunov's operator with special degenerate kernel are studied. The present paper is a continuation of the paper [11], i.e., we give full description of fixed points of Lyapunov's operator with another special degenerate kernel.
A Cayley tree rk = (V, L) of order k e N is an infinite homogeneous tree, i.e., a graph without cycles, with exactly k + 1 edges incident to each vertices. Here V is the set of vertices and L that of edges (arcs). The distance d(x, y),x,y e V is the number of edges of the path from x to y. Let x0 e V be a fixed and we set
Wn = {x e V | d(x, x0) = n}, Vn = {x e V | d(x, x0) < n},
Ln = {l =<x,y >e L I x,y e Vn}, If the distance from x to y equals one then we say x and y are nearest neighbors and use the notation l = (x, y). The set of the direct successors of x is denoted by S(x), i.e.
S(x) = {y e Wn+iI d(x,y) = 1}, x e Wn.
We observe that for any vertex x = x0, x has k direct successors and x0 has k + 1. The vertices x and y are called second neighbor which is denoted by > x,y <, if there exist a vertex z e V such that x, z and y, z are nearest neighbors. We will consider only second neighbors > x,y <, for which there exist n such that x,y e Wn. Three vertices x, y and z are called a triplet of neighbors and they are denoted by < x,y,z >, if < x,y >, < y,z > are nearest neighbors and x, z e Wn, y e Wn-1, for some n e N.
Now, we consider models with four competing interactions where the spin takes values in the set [0,1]. For some set A c V an arbitrary function aA : A ^ [0,1] is called a configuration and the set of all configurations on A we denote by QA = [0,1]A. Let &(•) belong to QV = ^ and ^ : (t,u,v) e [0,1]3 ^ £1(t,u,v) e R, : (u, v) e [0,1]2 ^ £,i(u, v) e R, i e {2, 3} are given bounded, measurable functions.
We consider models with four competing interactions where the spin takes values in the unit interval [0,1]. Given a set A c V a configuration on A is an arbitrary function aA : A ^ [0,1], with values a(x), x e A. The set of all
configurations on A is denoted by QA = [0,1]A = Q and denote by B the sigma-algebra generated by measurable cylinder subsets of Q.
Fix bounded, measurable functions £1 : (t, u, v) G [0,1]3 ^ &(t, u, v) G R and £ : (u, v) G [0,1]2 ^ £î(w, v) G R, i = 2,3. We consider a model with four competing interactions on the Cayley tree which is defined by a formal Hamiltonian
H(a) = -J3 Y, £1 (a(x),a(y),a(z)) - J £ £2 (a(x),a(z))
<x,y,z) )x,y(
-J1 £3 (a(x),a(y)) - aX^a(x), (1)
<x,y) x
where the sum in the first term ranges all triples of neighbors, the second sum ranges all second neighbors, the third sum ranges all nearest neighbors, and J, J1, J3, a g R \ {0}.
Let h : [0,1] x V \ {xo} y R and |h(t, x) | = |ht,x | < C where xo is a root of Cayley tree and C is a constant which does not depend on t. For some n G N, an : x G Vn ^ a(x) and Zn is the corresponding partition function we consider the probability distribution ^(n) on QVn defined by:
M(nVn) = Z-1 exp ( -£H(an) + ^ hff(x),J , (2)
V xeWn /
exp -£H(ân) + ^ h?(x),J A<p—1 (dân), (3)
o(p) xeW"
1
— 1
where
QWn x x ... x QWn = ^W), AWn x AWn x ... x AWn = aW'1, n,p G N,
V y n V ' n
Let a„_i G _1 and a„_i V w„ G is the concatenation of a„_i and w„. For n G N we say that the probability distributions M(n) are compatible if M(n) satisfies the following condition:
V wn)(Aw„ x Aw„)(dwn) = M(n-1)(^n-i). (4)
By Kolmogorov's extension theorem, there exists a unique measure m on QV such that, for any n and an G , M (iCT|Vn = an}) = M(n) (CTn). The measure m is called splitting Gibbs measure corresponding to Hamiltonian (1) and function x ^ hx, x = x0 (see [7,8,14,15]). We denote:
K(u, t, v) = exp { J3(t, u, v) + (u, v) + Ji£ (£3 (t, u) + £3 (t, v)) + a£(u + v)} , (5)
and
f (t, x) = exp(ht,x - ho,x), (t, u, v) G [0,1]3, x G V \ {x0}.
The following statement describes conditions on hx guaranteeing the compatibility of the corresponding distributions M(n)(^n).
Proposition 1 [16] The measure M(n)(a„), n = 1, 2,... satisfies the consistency condition (4) iff for any x g V \ {x0} the following equation holds:
f (t x) = H Jo1 Jo1 K(t, ^ y)f (6)
>y,z<es(x) Jo1 Jo1 K(0, U v)f y)f z)dudv '
where S(x) = {y, z}, < y, x, z > is a ternary neighbor.
Zn
2. Lyapunov operator with degenerate kernel
Let yi(t), ^>2(s) and <p3(u) are positive functions from C+[0,1]. We consider Lyapunov's operator A (see [9,17]):
(Af )(t)= i I (<pi(t)+ ^(s) + ^s(u)) f (s)f (u)dsdu. Jo Jo
and quadratic operator P on R3 by the rule
P(x, y, z) = (aiix2 + xy + xz, «21 x2 + «22xy + a22xz, a3ix2 + a^xy + «33x2).
Here,
aii = / ^i(s)ds > 0;
o
r-i c i
«22 = / ¥2{s)ds > 0, «21 = / > 0;
Jo Jo
«33 = <f3(s)ds > 0, «31 = <pi(s)<p3(s)ds > 0. oo
oo
r-i i
, As)y>3(s)
oo
The existence of fixed points of Lyapunov's operator A is proved in [16]. A sufficient condition of uniqueness of fixed points ofLyapunov operator A s given (see [8]).
Lemma 2.1 Lyapunov's operator A has a nontrivial positive fixed point iff the quadratic operator P has a nontrivial positive fixed point, moreover, N+ix(A) = N+ix(P). Proof (a) Put
R+ = {(x,y,z) e R3 : x > 0, y > 0,z > 0} , R> = {(x,y,z) e R3 : x > 0, y > 0,z> 0} .
Let Lyapunov's operator A has a nontrivial positive fixed point f (t) e C++ [0,1]. Let
i
xi = f (u)du, (7)
Jo
/ 0
1
/0 and
X2 = / ^2 (u)f (u)du, Jo
(8)
X3 = / ^3 (u)f (u)du, (9)
Jo
Clearly, x1 > 0, x2 > 0,x3 > 0, i.e. (x1, x2, x3) G R>. Then, for the function f (t), the equality
f (t) = y>1 (t)x1 + x1x2 + x1x3 (10)
holds.
Consequently, for parametrs c1, c2, c3 from the equality (7), (8) and (9), we have the three identities:
x1 = x1(a11x1 + x2 + x3), x2 = x1(«21x1 + «22x2 + «22x3), x3 = x1(«31x1 + «33x2 + «33x3).
Therefore, the point (c1, c2) is fixed point of the quadratic operator P.
(b) Assume, that the fixed point x0,y0, z0 is a nontrivial positive fixed point of the quadratic operator P, i.e.
(x0, y0, z0) g R> and number x0, y0, z0 satisfies the following equalities
xo(anxo + y0 + Z0) = x0, x0(«21x0 + «22y0 + «22Z0) = y0,
x0(«31x0 + «33 y0 + «33Z0) = Z0. 20
Similary, we can prove that the function f0(t) = ^i(t)x^ + x0y0 + x0z0 is fixed point of Lyapunov's operator A
and f0(t) g C+[0,1]. This completes the proof.
3. Positive fixed points of the quadratic operators in cone R+
We define quadratic operator (QO) Q in cone R3 by the rule
Q(x, y, z) = (aux2 + xy + xz, a2ix2 + a22xy + a^xz, a3ix2 + a33xy + a^xz).
3.1-lemma If the point w = (x0, y0, z0) G R+ is fixed point of QO Q, then x0 is a root of the quadratic algebraic equation
(a2i + a3i - aiia22 - aiia33)x2 + (aii + a22 + a33)x - 1 = 0 (11)
Proof Let the point w = (x0, y0, z0) G R+ be a fixed point of QO Q. Then
22 aiix0 + x0 y0 + x0z0, a2ix0 + a22x0y0 + a22x0z0,
a3ix^ + a33x0y0 + a33x0z0.
Using the bellowing equalities, we obtain:
y0 + z0 = 1 - aiix0 y0 = x0(a2ix0 + a22(1 - aiix0))
z0 = x0(a3ix0 + a33(1 - aiix0)) y0 + z0 = x0(a2ix0 + a22(1 - aiix0)) + x0(a3ix0 + a33(1 - aiix0)) = = (a2i + a3i - aiia22 - aiia33)x;° + (a22 + a33)x0 = aiix0
By the last equality, we get:
(a2i + a3i - aiia22 - aiia33)x;° + (aii + a22 + a33)x0 - 1 = 0.
This completes the proof.
3.2-lemma If the positive number x0 is root of the quadratic algebraic Eq.(11), thenthepoint w0 = (x0, x0(a2ix0 + a22(1 - aiia is fixed point of QO Q.
Proof Let x0 be a root of the quadratic Eq.(11), i.e.,
(a2i + a3i - aiia22 - aiia33)x;° + (aii + a22 + a33)x0 - 1 = 0.
x0(aiix0 + y0 + z0) = = x0(aiix0 + x0(a2ix0 + a22(1 - aiix0)) + x0(a3ix0 + a33(1 - aiix0))) = = x0(anx0 + (a2i + a3i - aiia22 - ana33)x°; + (a22 + a33)x0) = = x0((a2i + a3i - aiia22 - aiia33)x0 + (aii + a22 + a33)x0 - 1 + 1) = x0(0 + 1) = x0
Then
y0 + z0 = 1 - aiix0.
From the last equality, we get:
a2ix;° + a22x0 y0 + a22x0z0 =
= xo(a2iXo + (yo + zo)) = xo(a2ixo + 022(1 - «11x0)),
a^ix2; + a33xo yo + a33xozo = = xo(a3ixo + 033(yo + zo)) = xo(a3ixo + 033(1 - aiixo)).
This completes the proof. We put
Mo = «2i + «3i — aiia22 — aiia33, Mi = aii + «22 + 033
and define polynomial P2 (x):
P2(x) = Mox2 + Mixi — 1. (12)
Theorem 3.3 QO Q has a unique nontrivial positive fixed point.
Proof To prove the Theorem, we use properties of the polynomial P2 (x). It is known that there are two roots of the polynomial. They are:
—Mi + \J Mi + 4mo
xi =
x2
2mo
—Mi — \J Mi + 4MO
2mo
I Let mo > 0. In this case, xi > 0 and x2 < 0.
-, -, -Mi + V Mi + 4^0
1 - aiixi = 1----an =
2M0
2mo + aiiMi - \J(Mi + 4mo) aii
2mo
>
> 2^0 + «iiMi - VMia2i + 4M0Miaii > 2^0
> 2^0 + aiiMi - VMiau + Q'M0Miaii + 4M0Miaii = 0 2^0
i.e., 1 — aiixi > 0. It means:
yi = xi(a2ixi + a22(1 — aiixi)) > 0, zi = xi(a3ixi + a33(1 — aiixi)) > 0.
II Let m0 < 0. In this case, xi > 0 and x2 > 0. Clearly,
(P2(x))' = 2M0x + Mi (13)
and P2 ( —— ) =0. Moreover, the function P2(x) is an increasing function on ( —to, —— ) and it is a
V 2M0 ) \ 2m0 )
decreasing function on ( ——, to
2M0
If we put x' = —Mi, then
2M0
III Let x' = —— < —.
2M0 aii
Xi < X < X2.
Then xi < and from 1 — aiixi > 0. Moreover,
aii
aiiMi < -2mo (14)
(Xi,yi,zi) G R+
By other hand, we have the following identity:
2m0 + aiiMi + aii M2 + 4M0
1 — anx2 = ---
2M0
By (14): _
2M0 + aiiMi + an\JM2 + 4M0 >
> 2M0 + (—2M0) + an\JMi + 4M0 = an^Mi + 4M0 > 0.
From the last inequality,
1 — aii x2 < 0
and
(x2,y2,z2) e R+.
II.IILetx' = —— > — .Wehave:
2M0 aii
2M0 + aiiMi — ai^ M2 + 4M0 1 — aiixi = -
Consequently,
2mo
a2i + a3i > 0, i
a2i + a3i — aiia22 — ацазз + an + ацац + ацазз > ац,
I ^ 2
Mo + aiiMi > ац,
4mo(MO + aiiMi) < 4Mo(aii), aMo + 4aiiMoMi + a^Mi < a^M2 + 4Moa:j!i,
(2MO + aiiMi)2 < (aii\JM2 +4MO)2
2MO + «11M1 < «11 \/M2 + 4Mo, 2mo + «11M1 - «11 \/M2 + 4MO < 0.
From the last identity:
1 - 011x1 > 0,
and
(x1 ,y1,z1) G R+.
By the other hand, x2 > x' > —. So, 1 - a11x2 < 0 and (x2, y2, z2) G R+.
«11
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