NEW WEAKLY PERIODIC ADIC GENERALIZED GIBBS MEASURE FOR THE ADIC ISING MODEL ON THE CAYLEY TREE OF ORDER TWO Текст научной статьи по специальности «Математика»

ВЕСТНИК ОШСКОГО ГОСУДАРСТВЕННОГО УНИВЕРСИТЕТА МАТЕМАТИКА, ФИЗИКА, ТЕХНИКА. 2023, №2

МАТЕМАТИКА

УДК 517.98 - 519.21

https://doi.org/10.52754/16948645 2023 2 195

NEW WEAKLY PERIODIC P - ADIC GENERALIZED GIBBS MEASURE FOR THE P - ADIC ISING MODEL ON THE CAYLEY TREE

OF ORDER TWO

Raxmatullayev MuzaffarMuxammadjanovich, Dr Sc, professor,

[email protected] Institute of Mathematics named after V.I. Romanovsky of the Academy of Sciences of the Republic of Uzbekistan,

Namangan, Uzbekistan Abdukaxorova Zulxumor Tuxtasinovna, graduate student of PhD,

[email protected] Namangan State University, Namangan, Uzbekistan.

Abstract. In the present paper, we study the P adic Ising model on the Cayley tree of order two. The

H V -

existence of A -weakly periodic (non-periodic) v adic generalized Gibbs measures for this model is proved.

Keywords: Cayley tree, P adic numbers, P adic Ising model, Gibbs measure, weakly periodic Gibbs

measure.

СУЩЕСТВОВАНИЕ СЛАБО ПЕРИОДИЧЕСКИХ ОБОБЩЕННЫХ Р - АДИЧЕСКИХ МЕР ГИББСА ДЛЯ Р - АДИЧЕСКОЙ МОДЕЛИ ИЗИНГА НА ДЕРЕВЕ КЭЛИ ВТОРОГО ПОРЯДКА

Рахматуллаев Музаффар Мухаммаджанович, д.ф.-м.н., профессор,

mrahmatullaev@rambler. ru Институт Математики имени В.И. Романовского Академии Наук Республики Узбекистан, Наманган, Узбекистан Абдукахорова Зулхумор Тухтасиновна, аспирант, zulxumorabdukaxorova@gmail. com Наманганский государственный университет,

Наманган, Узбекистан

Аннотация. В этой статье изучене p - адическая модель Изинга на дереве Кэли второго порядка. Доказано существование HА - слабо периодических (непериодических) p - адических обобщенных мер Гиббса для этой модели.

Ключевые слова: Дерево Кэли, p - адические числа, модель Изинга, мера Гиббса, слабo периодические мерa Гиббса.

Let Q be the field of rational numbers. For a fixed prime p , every rational number x ^ 0

n

can be represented in the form x = pr —, where r,n e Z, m is a positive integer, and n and m

m

are relatively prime with p , r is called the order of x and written r = ordpx. The p -adic norm of x is given by

x i p -r, x * 0

lp [0, x = 0

This norm is non-Archimedean and satisfies the so called strong triangle inequality |x + y\ <max{x| ,y| }

lp N lp r I p 7

for all x, y e Q .

The completion of Q with respect to the p -adic norm defines the p -adic field which is denoted by Q (see [1]).

The completion of the field of rational numbers Q is either the field of real numbers R or one of the fields of p - adic numbers Q (Ostrowski's theorem).

Any p-adic number x * 0 can be uniquely represented in the canonical form x = pr(x)(x0 + x1 p + x2 p2 + ...)

where r(x) e Z and the integers x. satisfy: x0 * 0, xy e {0,1,2,..., p -1}, j e N (see [1]). In

this case Ixl = p ~r( x).

ip r

The Cayley tree rk of order k > 1 is an infinite tree i.e., a graph without cycles, such that exactly k +1 edges originate from each vertex. Denote by V the set of vertices, and by L the set of edges of the Cayley tree rk . Two vertices x and y are called nearest neighbours if there exist an edge l e L connecting them and denote by l = (x, y) (see [2]).

Fix x0 e rk and given vertex x, denote by |x| the number of edges in the shortest path connecting x and x .

For x, y e rk, denote by d(x, y) the number of edges in the shortest path connecting x and y . For x, y e rk, we write x < y if x belongs to the shortest path connecting x0 with y, and we write x < y if x < y and x * y. If x < y and |y| = |x| +1, then we write x ^ y . We set

Wn ={x e V|d (x, x,) = n} Vn = {x e V|d (x, x,) < n} Ln = { = (x, y) e L\x, y e Vn}

S(x) = ^y e V :x ^ y}, Sx(x) = ^y e V :d(x,y) = 1}. The set S(x) is called direct successor of x.

We consider a p - adic Ising model where the spin values take in the set 0 = {-1,1}. We define a configuration a on V by the function a : x e V ^ <y(x) e 0. Similarly, one can be define <rn and an on Vn and WB respectively. Q is the set if all configuration on V and denote Q = 0V

(resp. Qf = 0Vn, Q.W = 0Wn ).

W„

(n)

For given configurations a^ g Qv i and (p n) e we define a configuration in QF as follows

i n\ \ \an-l(x), if x e Vn-1

(a 1 vm )(x)=< .

1 n-1 m A' V (x), if x eWn

A formal p - adic Hamiltonian H: Q ^ Q of the p - adic Ising model is defined by

H(a) = J Za(x)a(y), (1)

{x, y}eL

where 0 < J < p~1/(p-1) for any (x, y) e L.

We define a function h : x ^ hx, Vx e V \ {x0}, hx e Qp and consider p - adic probability generalized Gibbs measure / on Q defined by

/f^n) = ^expp {Hn(an)}nK(xU, n = 1,2,..., (2)

n xeWn

where Z() is the normalizing constant

znh) = Z exp p {Hn (m)}n ha(x),x. (3)

PeQvn xeW„

A p - adic probability generalized Gibbs measure / is said to be consistent if for all n > 1 and <r , eQ , we have

n 1 v n—1

Z/hn)(an-, vp) UVn-1). (4)

PeQwn

In this case, by the p - adic analogue of Kolmogorov theorem there exists a unique measure / on the set Q such that juh ({a v = a n}) = u(n)(an) for all n and an e Qv.(see [3])

Proposition 1.[4] The sequence of p - adic probability distributions {u(")^1, determined by formula (2) is consistent if and only if for any x e V \ {x0}, the following equation holds:

, ___ h +1

hl = n -h++e> (5)

yeS(x) hy +&

where 9 = expp (2J), 9 * 1.

It is known that rk can be represented as a non-commutative group Gk, which is the free product of k +1 cyclic groups of the second order [2].

Let G / G* = {H0,Hl,..., Hr} be a factor group, where G* is a normal subgroup of index

r > 1.

Definition 1. A set h = {hx, x e Gk} of quantities is called G* - periodic if h = hx, for all x e Gk and y e G*.

For x e Gk we denote by x^ the unique point of the set {y e Gk : (x, y)} \ S(x). Definition 2. A set of quantities h = {hx, x e Gk } is called G* - weakly periodic, if hx = htj , for any x e H x± e Hj.

Definition 3. Ap-adic generalized Gibbs measure i is said to be G* - (weakly) periodic if it corresponds to a G* - (weakly) periodic h. We call a Gk - periodic measure a translationinvariant measure. Let

ha =\x e Gk : X^x(ai)-evenf,

ieA

where 0 * A c Nk = {1,2,3,..., k +1}, and ®x(fli) is the number of letters a, in a word x e Gk

. Note that HA - is a normal subgroup of the Gk (see [2]). Note that a weakly periodic Gibbs

measure depends on normal subgroup. According to the selection of the normal subgroup, different weakly periodic Gibbs measures are found (see [3]). The set of weakly periodic Gibbs measures also includes the set of periodic (in particular translation-invariant) Gibbs measures.

We note that in the case |A| = k +1 (where |A| is the number of elements of the set A ),

i.e., A = N, the concept of weak periodicity coincides with ordinary periodicity. Therefore, we

consider A c N such that A * N. In this work, we consider the case |A| = 1. According to (5)

the HA -weakly periodic set of hx has the following form

'hoo, if x e H a , h21 , if x e H A ,

V if x e Gk \ HA ,

^ if x e Gk \ Ha , e Gk \ HA

h =

^ e HA , ^ e Gk \ HA : ^ e HA ,

(6)

By (5) we have

C + 1 h00 /1 , ,2

0 + K10

flh20 + 1 0 + h020 :

h01 =

0K* +1

.0 + h20 y

h10 =

h+1

0 + h2

2 _ Oh2 +1 0+h2

(7)

0h2j +1 0 + h2 '

Consider operator W : R4 ^ R4, defined as follows:

h'2 =

h22

h'2 =

h21

h'2 =

h12

h'2 =

h11

Oh 20 +1

0+ho'

0h020 +1 0 + hV

0K020 +

V°+ h02 y

v0 + h2 y

6h2u +1 Ohl, +1 0 + h2 0 + h2 .

Note that the system of (7) describes fixed points of the operator W , i.e. h =W(h). Lemma 1. The following sets are invariant with respect to the operator W:

/1 = j/* e R 4 : h20 = h21 = h12 = h11 } 12 = {h e R 4 : h20 = h12 = ±h21 }

2

2

Remark 1. [4] It is easy to see that if the function - hx is a solution to equation (5), then the function - h is also a solution. These solutions define the same measure /xh which we consider Ising model on the Cayley tree of order k .

We shall find HA -weakly periodic (non-periodic) p - adic generalized Gibbs measure for the Ising model on the set / .

The system of equation (7) has the following solutions

V2 =±1,

_j-1 + y (0+1)(0- 3)

h223,4 ± 2

_J-1 ¡(0 +1)(0- 3)

h225,6 -± 0

h227 8 =±V-

Lemma 2. The solutions h007 and h00g belong to Q, iff p = 1(mod4). Theorem 1. If p = 1(mod4) then there exists at least one weakly periodic (non-periodic) p - adic generalized Gibbs measure for the p - adic Izing model on the Cayley tree of order two.

Remark 2. In [5] it was proved that for the Ising model on a Cayley tree of order k = 2 with respect to the normal divisor of index 2, there does not exist a weakly periodic (non-translation-invariant) Gibbs measure in real case. In p-adic case in Theorem 1 it was shown that for the Ising model there is at least one new weakly periodic p-adic generalized Gibbs measure.

REFERENCES

1. V. S. Vladimirov, I. V. Volovich and E. V. Zelenov, p -Adic Analysis and Mathematical Physics (World Sci. Publ., Singapore,1994).

2. U. A. Rozikov, Gibbs Measures on Cayley Trees (WorldSci. Publ., Singapore, 2013).

3. Rozikov U. A., Rahmatullaev M. M. Description of weakly periodic Gibbs measures for the Ising model on a Cayley tree. Theor. Math.Phys., 156(2): (2008).

4. Khakimov O. N. On a Generalized p-adic Gibbs Measure for Ising Model on Trees. p-Adic Numbers, Ultrametric Anal. Appl., 6(3), 2014, pp.207-217.

5. Rahmatullaev M. M. "On new weakly periodic Gibbs measures of the Ising model on the Cayley tree of order 6". J. Phys.: Conf Ser., 697 (2016), 012020, pp.7.