Ўзбекистон миллий университети ҳузуридаги илмий даражалар берувчи dsc. 03/30. 12. 2019. Fm. 01. 01 Рақамли илмий кенгаш математика институти
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Автореферат Ботиров
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- “Nontranslation invariant Gibbs measures for models with uncountable set of spin values
- Proposition 18.
- “Ground states for the Potts model with competing interactions
- НАУЧНЫЙ СОВЕТ DSc.03/30.12.2019.FM.01.01 ПО ПРИСУЖДЕНИЮ УЧЕНЫХ СТЕПЕНЕЙ ПРИ НАЦИОНАЛЬНОМ УНИВЕРСИТЕТЕ УЗБЕКИСТАНА
- МНОЖЕСТВОМ ЗНАЧЕНИЙ СПИНА 01.01.01 – Математический анализ (физико-математические науки)
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Theorem 10. Let 𝑘𝑘 = 2. Then for the model (6) defined on Cayley tree of order k the following statements hold: i) If ) 1 2 ( 2 3 2 0 + + ≤ ≤ n n θ
measure; ii) If 1 ) 1 2 ( 2 3 2 < < + + θ n n , then there are three translation-invariant Gibbs measures.
In the third chapter, titled “Nontranslation invariant Gibbs measures for models with uncountable set of spin values” we consider models with nearest- neighbour interactions and with set [0,1] of spin values, on a Cayley tree of order 1 ≥
. In the first section of Chapter 3, we give main definitions and ART construction. By H.Akin, U.A.Rozikov and S.Temir for the Ising model (with the set {-1,1} of spin values) the authors constructed a class of new Gibbs measures by extending the known Gibbs measures defined on a Cayley tree of order k
to a Cayley tree of higher order k>k
. Their construction is called the ART–construction. For a given ) ( σ H of the model (6), denote by ) (H G k the set of all Gibbs measures on the Cayley tree of order 2 ≥ k . By
M we denote the cardinality of a set M. Theorem 11. Let ,...}
3 , 2 { , 0 ∈ k k be numbers such that 0
k > . If 2 )
0 ≥
G k and ) , ( u t K satisfies the following equality 40
( ) ] 1 , 0 [ , 0 ) , 0 ( ) , ( 1 0 ∈ ∀ = − ∫ t du u K u t K , (13) then for each ) ( 0 H G k ∈ µ there exists a measure ( )
) (H G k ∈ = µ ν ν . Example 1. Let 2 = k . Suppose that the function 𝜉𝜉 𝑡𝑡𝑢𝑢
in the model (6) is ]. 1 , 0 [ , , 2 1 2 1 4 15 14 1 ln 1 5 ∈ − − ⋅ + = u t u t J tu β ξ Then, for the kernel ) ,
t K of Hammerstein’s operator we have ]. 1
0 [ , , 2 1 2 1 4 15 14 1 ) , ( 5 ∈ −
− ⋅ + = u t u t u t K
In the second chapter of dissertation it was shown that this model has at least two Gibbs measures and the condition (6) is satisfied. In the second paragraph of chapter 3, we give the Bleher-Ganikhodjaev construction. If an arbitrary edge L l x x l ∈ >= < , 0 is deleted from the Cayley tree k ℑ , it splits into two components–two semi-infinite (half) trees
0 ℑ and k 1 ℑ . Consider the half tree k 0 ℑ , and denote by 0
the set of its vertices. Namely, the root 0
has k nearest neighbors. Denoting ( )
( ) x t f x t h , ln , = we write Eq. (8) as ( ) ( )
( ) ∑ ∫ ∫ ∈ = ) ( 1 0 , 1 0 , ) , 0 ( ) , ( ln ,
S y y u h y u h k du e u K du e u t K x t h
(14)
On the set C[0,1] of continuous functions we define the following nonlinear operator ( ) ( )
( ) 1 0 1 0 ( , ) ln (0, )
f u f u K t u e du Af t K u e du = ∫ ∫ ,
(15)
where 0 ) , ( > u t K .
Condition 12. Assume that 0 ) , ( > u t K is continuous on [ ] 2
, 0 , i.e., 2 ] 1 , 0 [ ) , ( + ∈ ⋅ ⋅ C K , and there is [ )
, 0 ∈ ≡ K α α such that ( )
( ) ( ) ( )
]. 1 , 0 [ ], 1 , 0 [ , , ∈ ∀ ∈ ∀ − ≤ − t C g f t g t f t Ag t Af α
Condition 13. Assume that there are at least two translation invariant solutions: ( ) ( )
[ ] 1 , 0 ,
t h x t h ∈ ≡ and ( ) ( )
] 1 , 0 [ , C t x t h ∈ ≡ η to equation (14). 41
necessary that 1 1 < ≤ α k . We use ) (t h and
) (t η to construct an uncountable set of new solutions to (15). Consider an infinite path ...}
{ 1 0 0 < < = = x x x π (the notation y x < meaning that path from the root to y go through x). Associate to this path a collection ]} 1 , 0 [ , : { 0 , ∈ ∈ =
V x h h x t π π given by = ∈ ∈ = , if , , , if ), ( , , if ), ( , , n x t n n n n x t x x h W x x x t W x x x t h h n η π
(16) n=1,2,…. where n x x (resp. x x n ) means that x is on the left (resp. right) from the path π . Theorem 14. If Conditions 12 and 13 are satisfied, then for any infinite path π
} {
π π
t h h =
Theorem 15. If Condition 12 and 13 are satisfied then for any ] 1 , 0 [ ∈ r there exists a nontranslation invariant Gibbs measure r ν
l r ν ν ≠ if l r ≠
In the third section of Chapter 3, we give the Zachary construction.
0 ) , ( > u t K such that the operator A, (15), is invertible. From (15) we get that , ), ( ) , ( ) ( max min
V x t h x t h t h ∈ ∀ ≤ ≤
(17)
where
) , 0 ( max ) , ( min
ln ) ( ] 1 , 0 [ ] 1 , 0 [ min
u K u t K k t h u u ∈ ∈ = ,
) , 0 ( min
) , ( max ln ) ( ] 1 , 0 [ ] 1 , 0 [ max u K u t K k t h u u ∈ ∈ = .
Under Conditions 13 and 16 we constructed a continuum of distinct functions ξ
t h , , which satisfy the functional equation (11), where ( ) t ξ is such that ( ) ]. 1 , 0 [ ), ( ) ( max
min ∈ ∀ < < t t h t t h ξ
(18)
Theorem 17. If Conditions 13 and 16 are satisfied, then for any ξ
(18) there exists a Gibbs measure ξ µ . Moreover, η ξ µ µ ≠ if η ξ ≠ .
In the forth section of Chapter 3, we considered the SOS (solid-on-solid) model on the Cayley tree of order 2
≥ .
Φ = ℤ . The (formal) Hamiltonian of the SOS model is :
|, ) y ( ) x ( | J = ) ( H L y , x σ σ σ − − ∑ ∈ 〉 〈
(19) 42
where J ∈ ℝ is constant. In the work of K. Kulske, A. LeNey, F. Henning and U.A. Rozikov, the definition gradient Gibbs measures (GGM) is given and construction of periodic such measures is reduced to the solution of the following system of nonlinear equations with infinitely many unknowns: | |
| | 0 | | 0 ( ) = , (0) 1 k i i j j j Z i j j j Z z i z z θ θ ν ν θ − ∈ ∈ + + ∑ ∑ (20) where 𝑧𝑧 𝑖𝑖 = exp(ℎ 𝑖𝑖 ) , 𝑖𝑖 ∈ ℤ. Put k i 0 i z u = u for some 0 > u 0 . Then (20) can be written as 𝑢𝑢 𝑖𝑖 = 𝐶𝐶�∑ 𝜃𝜃 𝐽𝐽 𝑢𝑢 𝑖𝑖−𝑗𝑗
𝑘𝑘
+∞ 𝑗𝑗=1 + 𝑢𝑢
𝑖𝑖 𝑘𝑘 + ∑ 𝜃𝜃 𝐽𝐽 𝑢𝑢 𝑖𝑖+𝑗𝑗 𝑘𝑘
+∞ 𝑗𝑗=1
�, 𝑖𝑖 ∈ ℤ (21)
Proposition 18. A vector = ( , )
u i Z ∈
, with 1 = u 0
only if for ) z (= u k i i
1 1 1 1 = , ,
i i i i u u u u i Z u u τ τ − + − + − ∈ + −
(22)
θ θ τ + −1 = . By this proposition we have
1
0 1 1 1 = . l r u u θ θ
τ − − − + +
+ −
(23) Equations of system (21) for 1 =
− and
1 = i are satisfied independently on values of 1 u
and 1 u and the equation (22) can be separated to the following independent recurrent equations
,
u u ) u u ( = u 1 i i k i 1 1 1 i + − − − − − − − + − + τ τ (24)
, u
u ) u u ( = u 1 i i k i 1 1 1 i − − + − + − + τ τ
(25) where 1 i ≥ , 1 = u 0 and 1 u − , 1 u are some initial numbers. So, if
i u is a solution to (25) then i u
will be a solution for (24). Hence we can consider only equation (25). Let's consider the periodic solutions of (22) i.e., we describe solutions of (22) which have the form
1,
= , = 4 1, , , = 4 1,
if n m u a if n m m Z b if n m − ∈ +
(26) 43
where a and
b some positive numbers. In this case (27) is equivalent to the following system of equations
0. = 2 a a ) b a ( 0 = 2 b b ) b a ( k k − + − + − + − + τ τ τ τ
(27) We described positive solutions of (28) and proved the following Theorem 19. Let 2 k ≥ and b = a . For the SOS-model (20) on the k
tree, with parameter ) ( cosh 2 = β τ
0 >
τ such that the following assertions hold: 1. If c
τ τ
height-periodic boundary laws of the type (26); 2. At c = τ τ
3 heightperiodic boundary law of the type (26); 3. For c > τ τ
Theorem 20. Let 𝑘𝑘 ≥ 2 and b a
. For the SOS-model on the k-regular tree, with parameter ) ( cosh 2 = β τ
1. For any positive fixed b
b 2
τ then there is no any GGM corresponding to nontrivial period-3 height-periodic boundary laws of the type (26); 2. For any positive fixed b
b 2
τ then there is a unique GGM corresponding to nontrivial period-3 height-periodic boundary laws of the type (26).
In the fourth chapter, titled “Ground states for the Potts model with competing interactions” we derived an infinite system of functional equations for the Potts model with competing interactions of radius r = 2 and countable spin values {0,1, … } and nonzero field, on a Cayley tree. In the first section of Chapter 4, we give definitions and known facts about ground states. In the secound section of Chapter 4, we consider functional equation for Potts models with countable set of spin values, i.e. Φ = ℤ.
The (formal) Hamiltonian of the Potts model is: ( )
( ) ( ) ( ) ( )
1 , , , x y x y x y x y H J J σ σ σ σ σ δ δ
> >
= + ∑ ∑
(28) where
𝐽𝐽, 𝐽𝐽 1 ∈ ℝ are constants. Let ℎ: 𝑥𝑥 ⟼ ℎ 𝑥𝑥 = �ℎ
0,𝑥𝑥 , ℎ
1,𝑥𝑥 , … � ∈ ℝ ∞ be a real sequence-valued function of } { \ 0 x V x ∈ . Fix a probability measure 𝜈𝜈 on ℤ + such that 𝜈𝜈(𝑖𝑖) > 0 for all 𝑖𝑖 ∈ ℤ + . Given n=1,2,…, consider the probability distribution n µ on Ω 𝑉𝑉 𝑛𝑛 defined by ( )
( ) ( )
( ) ( )
( ) ∏ ∑ ∈ ∈ − + − = n n V x W x x x n n n n x h H Z σ ν σ β σ µ σ , 1 exp
(29) 44
( ) ( )
,... 2 , 1 , = n n n σ µ , given by (29) for a Cayley tree order two are compatible iff for any } { \ 0
V x ∈
following equation holds: ( ) ,..., 3 , 2 , 1 , , , , * * * , = = i J h h F h z y i x i β
(30) Here ( )
( ) ( )
( ) + − + − = ,... 0 2 ln , 0 1 ln , 0 , 2 , 0 , 1 * ν ν ν ν x x x x x h h h h h ( ) { } ( ) { } . exp
1 exp
1 ln ) , , , ( 0 0 , * , * , 1 0 0 , * , * , 1 * * ∑ ∑ ∞ ≠ + = ∞ ≠ + = + + + + + + + + + + = q p q p z q y p pq ip q p q p z q y p pq ip z y i h h J iq J h h J iq J J h h F βδ δ δ β βδ δ δ β β
In the third section of Chapter 4, we construct periodic ground states for the model. We suppose that M is the set of unit balls with vertices in V . The restriction of a configuration σ on a ball M b ∈ is called a bounded configuration b σ . We define the energy of the configuration b σ on the ball b as
) ( ) ( 2 = ) , ( : , 2 ) ( ) ( , >, ,
1 2
= ) , ( ) ( y x y x d b y x y x b y x y x b b J J J U U σ σ σ σ δ δ σ σ ∑ ∑ ∈ ∈ + ≡ (31)
where 𝐽𝐽 = (𝐽𝐽
1 , 𝐽𝐽
2 ) ∈ ℝ
2 .
𝑘𝑘 ≥ 2 be a natural number and 𝐽𝐽 1 , 𝐽𝐽 2 ∈ ℝ. Then the following statements hold. 1) Let b σ
i c b b = ) ( σ
b c is the center of the ball b ), and , |= 2} = ) ( : { | , |= 1} = ) ( : { | n x x m x x b σ σ r x x l x x |= 4} = ) ( : { | , |= 3} = ) ( : { | σ σ
Then ) ( b U σ
2 2 2 2 2 1 3 3 2 1 2 1 , ) ( ) ( 2 1 = ) , , , , , ( ) ( J C C C C J l l n m J J r l n m U U r l n m i i i i k i b + + + + + + + + ≡ δ δ δ δ σ (32) where 𝑚𝑚, 𝑛𝑛, 𝑙𝑙, 𝑟𝑟 ∈ ℤ + , 𝑚𝑚 + 𝑛𝑛 + 𝑙𝑙 + 𝑟𝑟 = 𝑘𝑘 + 1. 2) For any configuration b σ
𝑈𝑈(𝜎𝜎
𝑏𝑏 ) ∈ {𝑈𝑈
𝑖𝑖,𝑘𝑘 (𝑚𝑚, 𝑛𝑛, 𝑙𝑙, 𝑟𝑟, 𝐽𝐽 1 , 𝐽𝐽
2 ): 𝑚𝑚, 𝑛𝑛, 𝑙𝑙, 𝑟𝑟 ∈ ℤ + , 𝑚𝑚 + 𝑛𝑛 + 𝑙𝑙 + 𝑟𝑟 = 𝑘𝑘 + 1}. Denote ( )
( ) ( ) = { : ( ) = ,
( ) = },
= 1, 2,3, 4; i i p p b k b b b j F F j N c i a p p σ σ σ ≡ ∈ 45
}. |= | , |= | , |= | , = ) ( : { = ) ( 3 ) ( 2 ) ( 1 ) ( , , l F n F m F i c i i i b b b i l n m σ σ Ω
Let 4 S be the group of permutations on {1,2,3,4} . )
= ) ( , , 4 ) ( , , i l n m S i l n m C Ω ∈ π π , where for 4 (4)) (3), (2),
(1), ( = S ∈ π π π π π we put } :
= ) ( ) ( , , ) ( , ,
l n m i l n m Ω ∈ Ω σ πσ π with
)) ( ( = ) )( ( x x σ π πσ .
) (
, i l n m C and for any bounded configuration ) ( , ,
l n m b C ∈ σ there exists a periodic configuration ϕ
that ) ( , ,
l n m b C ∈ ′ ϕ for any M b ∈ ′ and b b σ ϕ = . Definition 24. A configuration σ
Hamiltonian H if for any 𝑏𝑏 ∈ 𝑀𝑀 the following holds:
𝑈𝑈(𝜎𝜎
𝑏𝑏 ) = min {𝑈𝑈 𝑖𝑖,𝑘𝑘 (𝑚𝑚, 𝑛𝑛, 𝑙𝑙, 𝐽𝐽 1 , 𝐽𝐽
2 ): 𝑖𝑖 = 1,2,3, 𝑚𝑚, 𝑛𝑛, 𝑙𝑙 ∈ ℤ + ,
𝐽𝐽 1 , 𝐽𝐽 2 ∈ ℝ} Denote ,
2 , 1 2 , 1 2 ( , , ) = {( , ) : ( , , ,
, ) ( , , , , ), i k i k j k A m n l J J U m n l J J U m n l J J ′ ′ ′
≤
𝑓𝑓𝑓𝑓𝑟𝑟 𝑎𝑎𝑙𝑙𝑙𝑙 𝑚𝑚 ′ , 𝑛𝑛 ′ , 𝑙𝑙
′ ∈ ℤ
+ }, 𝑖𝑖 = 1,2,3,4. Let )
GS be the set of all ground states of the relative Hamiltonian. Theorem 25. (i) If 𝐽𝐽 1 = 𝐽𝐽 2 = 0 then Ω = ) (H GS . (ii) If (𝐽𝐽 1 , 𝐽𝐽 2 ) ∈ 𝐴𝐴
𝑖𝑖,𝑘𝑘 (𝑚𝑚, 𝑛𝑛, 𝑙𝑙) then }, :
( { = ) ( 4 ) ( , , S H GS i l n m ∈ π σ π where Ω ∈ ) ( , , i l n m σ such that ) ( , , ) ( , , ) ( i l n m b i l n m Ω ∈ σ for any 𝑏𝑏 ∈ 𝑀𝑀, 0 ≤ 𝑚𝑚 + 𝑛𝑛 + 𝑙𝑙 ≤ 𝑘𝑘 + 1, 𝑖𝑖 = 1, 2, 3, 4. In the fourth section of Chapter 4, we consider Potts model, with competing interactions and countable spin values ( Φ = ℤ) on a Cayley tree of order three. We study periodic ground states for this model. Let *
G be a subgroup of index 1
≥ . Consider the set of right cosets { } * 1 2 / , ,...,
k k r G G H H H = , where * k G is a subgroup. Definition 26. A configuration ( )
x σ
*
σ σ = for all i x H ∈
k G -periodic configuration is a said to be translation-invariant. The period of a periodic configuration is the index of the corresponding subgroup.
( )
σ
*
( )
ij x σ σ = for all i x H ∈
j x H ↓ ∈ . We consider the case k=3 with countable spin values. It is easy to see that ( ) { }
2 12 , ,..., b U U U U σ ∈ for any b σ , where 1 1 2 = 2 6 , U J J +
2 1 2 3 = 3 , 2
J J +
3 1 2 = 2 , U J J +
46
4 1 2 1 = 3 , 2 U J J +
5 2 = 6 , U J 6 1 1 = 2 U J , 7 2 = 3
, U J 8 2 = ,
J 9 1 2 =
J J + ,
10 1 2 1 = , 2 U J J +
11 2 = 2 , U J 12 = 0 U . Using these notations we can give the following definition of ground state of Hamiltonian.
ϕ
( )
{ } 1 2 12 min , ,...,
b U U U U ϕ = for any b M ∈
We set }
) ( : { =
b b i U U C σ σ and
) , ( = ) ( J U J U b i σ
if , = 1, 2, ,12.
σ ∈
For any 1, 2,...,12 i = we put 2 1 2 12 = {
: min{
( ), ( ),
, ( )}}.
i i A J R U U J U J U J ∈ =
We set , , 5 1 0 2 1 A A B A A B ∩ = ∩ = , , 6 9 2 9 2 1 A A B A A B ∩ = ∩ =
, 12 6 3 A A B ∩ = ), ( \ ~ 0 1 1
B A A ∪ = ), ( \ ~ 1 0 2 2 B B A A ∪ = ), ( \ ~ 7 0 5 5 B B A A ∪ = ), ( \ ~ 3 2 6 6 B B A A ∪ = ), ( \ ~ 2 1 9 9 B B A A ∪ = and ). ( \ ~ 7 3 12 12 B B A A ∪ = Let ) (H GS be the
set of all ground states, and let ) (H GS p be the set of all periodic ground states. Theorem 29. a) For any class i C , = 1, 2,...,12 i , and any bounded configuration i b C ∈ σ , there exists a periodic configuration ϕ
such that i b C ∈ ′ ϕ for any M b ∈ ′ and b b σ ϕ = . Theorem 30. A. If ) 0 , 0 ( = J , then Ω = ) (H GS .
1 ~
J ∈
{ }
∈ =
H GS i p : ) ( ) ( ϕ . 2. If 2 ~ A J ∈
{ }
l m l H GS lm p ≠ Φ ∈ = , , : ) ( ) ( 2 ϕ
3. If 5 ~ A J ∈
{ }
l m l H GS lm p ≠ Φ ∈ = , , : ) ( ) ( 5 ϕ
4. If 6 ~ A J ∈
{ }
m p l p n m l H GS lmnp p ≠ ≠ ≠ Φ ∈ = , , , , : ) ( ) ( 6 ϕ . 5. If 9 ~ A J ∈
{ }
l m l H GS lm p ≠ Φ ∈ = , , : ) ( ) ( 9 ϕ
6. If 12 ~ A J ∈
{ }
∈ =
H GS l p : ) ( 12 ϕ . C. 1. If ) 0 , 0 ( \ B J ∈
{ }
l m l i H GS lm i p ≠ Φ ∈ = , , , : , ) ( ) ( 2 ) ( ϕ ϕ . 2. If ) 0 , 0 ( \ 0
J ∈
{ }
l m l i H GS lm i p ≠ Φ ∈ = , , , : , ) ( ) ( 5 ) ( ϕ ϕ . 3. If ) 0 , 0 ( \ 1
J ∈
{ }
l m l i H GS lm lm p ≠ Φ ∈ = , , , : , ) ( ) ( 9 ) ( 2 ϕ ϕ
4. If ) 0 , 0 ( \ 2
J ∈
{ }
n m l p n m l H GS lm lmnp p ≠ ≠ ≠ Φ ∈ = , , , , : , ) ( ) ( 9 ) ( 6 ϕ ϕ
5. If ) 0 , 0 ( \ 3
J ∈
{ }
n m l p n m l H GS l lmnp p ≠ ≠ ≠ Φ ∈ = , , , , : , ) ( ) ( 12 ) ( 6 ϕ ϕ
47
8
∈ , then periodic configuration l lmnp lmn lmn lm 12 ) ( 8 ) ( 7 ) ( 7 ) ( 5 , , , , ϕ ϕ ψ ξ ϕ
ground states, and weakly periodic configuration ) ( 7 lmn ξ
states, where p n m l p n m l ≠ ≠ ≠ Φ ∈ , , , , .
48
The dissertation work is devoted to the study of Gibbs measures for lattice systems with an infinite set of spin values. The main results of the research are as follows: 1. The critical value of temperature that provide the phase transitions for the XY model on the Cayley tree of arbitrary order is determined; 2. For a model with a continuum set of spin values on the Cayley tree of arbitrary order a critical temperature is found, such that for temperatures lower from this critical value there are exactly three translation invariant Gibbs measures. In case of Cayley trees of order two and three the explicit solutions corresponding to the translation invariant Gibbs measures are found. 3. Several models with a continuum set of spin values and with nearest- neighbor interactions are constructed, which have at least two periodic Gibbs measures; 4. Using ART (Akin, Rozikov, Temir), Blexer-Ganikhodjaev, Zachary constructions continuum sets of new Gibbs measures are constructed. 5. For the SOS (Solid on Solid) model with a countable set of spin values gradient Gibbs measures are found; 6. For the Potts model with competing interactions and a countable set of spin values on Cayley tree a system of functional equations is dirived solutions of which corresponds to Gibbs measures. Weakly periodic ground states (configurations) of this model are constructed.
49
УЧЕНЫХ СТЕПЕНЕЙ ПРИ НАЦИОНАЛЬНОМ УНИВЕРСИТЕТЕ УЗБЕКИСТАНА
БОТИРОВ ГОЛИБЖОН ИСРОИЛОВИЧ МЕРЫ ГИББСА ДЛЯ РЕШЕТЧАТЫХ СИСТЕМ С БЕСКОНЕЧНЫМ МНОЖЕСТВОМ ЗНАЧЕНИЙ СПИНА 01.01.01 – Математический анализ (физико-математические науки) АВТОРЕФЕРАТ ДИССЕРТАЦИИ ДОКТОРА (DSc) ФИЗИКО-МАТЕМАТИЧЕСКИХ НАУК ТАШКЕНТ–2020 |
