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0ВЕСТНИК ОШСКОГО ГОСУДАРСТВЕННОГО УНИВЕРСИТЕТА МАТЕМАТИКА, ФИЗИКА, ТЕХНИКА. 2023, №2
МАТЕМАТИКА
UDC 517.98
https://doi.org/10.52754/16948645 2023 2 187
SOME CONSTRUCTIVE p -ADIC GENERALIZED GIBBS MEASURES FOR THE ISING MODEL ON A CAYLEY TREE
Rahmatullaev Muzaffar Muhammadjanovich, DSc., professor,
mrahmatullaev@rambler.ru
Namangan regional department, Institute of Mathematics named after V.I.Romanovsky,
Namangan, Uzbekistan,
Tukhtabaev Akbarkhuja Mamajonovich, PhD. student, Namangan state university, Namangan, Uzbekistan, akbarxoia.toxtaboyev@mail.ru
Abstract: The paper is devoted to some non-periodic p -adic generalized Gibbs measures for Ising model on a semi-Cayley tree of order к > 1 . We construct uncountable non-periodic p -adic generalized Gibbs measures
for the Ising model on a semi-Cayley tree. We study the boundedness of the measures. Furthermore, we find conditions that guarantee existence of the phase transition.
Keyworsds: p-adic numbers, p-adic Ising model, Cayley tree, Gibbs measure, phase transition.
НЕКОТОРЫЕ КОНСТРУКТИВНО Р-АДИЧЕСКИЕ ОБОБЩЕННЫЕ МЕРЫ ГИББСА ДЛЯ МОДЕЛИ ИЗИНГА НА
ДЕРЕВЕ КЭЛИ
Рахматуллаев Музаффар Мухаммаджанович, д.ф.-м.н., профессор,
mrahmatullaev@rambler. ru Наманганское областное отделение Института математики им. В.И.
Романовского, г. Наманган, Узбекистан, Тухтабаев Акбархужа Мамажонович, PhD.докторант, Наманганский государственный университет,
Наманган, Узбекистан, akbarxoia.toxtaboyev@mail.ru
Аннотация: Статья посвящена изучению некоторых непериодических p -адических обобщенных мер
Гиббса для модели Изинга на полудереве Кэли порядка к > 1. Построено несчетное количество непериодических p -адических обобщенных мер Гиббса для модели Изинга на полудереве Кэли, а также
изучена задача ограниченности этих мер. Кроме того, найдены условия, гарантирующие существование фазового перехода.
Ключевые слова: p-адические числа, p-адическая модель Изинга, дерево Кэли, мера Гиббса, фазовый переход.
Introduction. Let Q be the field of rational numbers. For a fixed prime number p, every
m
rational number x * 0 can be represented in the form x = pr — where,
n
r, m, n e Z, n > 0 and m, n are relatively prime with p . The p -adic norm of x e Q is given by
_Jp-, x * 0, |X|p=|0, x = 0.
This norm is non-Archimedean, i.e., it satisfies the strong triangle inequality \x + y max(lx \P>\ y lP} for all x,jeQ.
The completion of Q with respect to the p -adic norm defines the p -adic field Q p.
Any p -adic number x * 0 can be uniquely represented in the canonical form
x = pr( x) (x0 + XP + X2P2 + •••), where eZ,io^0, x. e {0,1,...,/7-1}, j = 1,2,... .
In this case | x | = p~y{x). For a e Qp and r > 0 we denote
B(a, r) = {xeQp: \ x-a \p</*}. p -adic exponential is defined by
" xn
exp p (x)=Z n,
n=0 n!
f _J_\
which converges for x e B 0, p p-1
v
We set
Ep=\xeQp:\x-\\p<p
This set is the range of the p -adic exponential function (see e.g. [2]). Let (X, B) be a measurable space, where B is an algebra of subsets X. A function p : B —» Qp is said to be a p -adic measure if for any Al,A,,...,Ai eS such that A n A = 0, i * j, the following holds:
HIM =5>(4)-
7=1
A p -adic measure j is called bounded if sup{| j(A) \p. A e B} <<x>. It is said that p -adic measure is probabilistic if j(X) = 1.
Let r+k = (V,L), be a semi-infinite Cayley tree [1] of order k > 1 with the root
x° eV (whose each vertex has exactly k +1 edges, except for the root x°, which has k edges). Here V is the set of vertices and L is the set of edges. The vertices x and y are called nearest neighbors and they are denoted by l = (x, y) if there exists an edge l connecting them. A collection of the pairs (x, x1), (x, x2),..., (xd-1, y) is called a path
from the point x to the point y . The distance d (x, y), x, y eV, on the semi-Cayley tree, is the number of edges of the shortest path from x to y . We set
W = {xeV|d(x,x0) = n}, V ={xeV|d(x,x°)<n}, Ln = {l =< x, y >e L | x,y e Vn}. The set of direct successors of x e W is defined by
5 (x) = { y e Wn+1: d (x, y) = 1}. We recall a coordinate structure in T+k: every vertex x (except for x°) of T+k has
coordinates (i,i2,...,in), here im e {1,2,...,k}, m = 1,n, and for the vertex x° we put (0) . Namely, the symbol (0) constitutes level 0, and the sites (i,i2,...,in) form level n (i.e. d(x°, x) = n) of the lattice. Let us define on r+A binary operation o ; XT^ —> as follows: for any two elements x = (i ,i2,..., in) and y = (j, j2,..., jm ) put
Xoy = (i1,i2,...,in)o(j1J2,...Jm) = (i1J2,...JnJ1J2,...Jm)
By means of the defined operation T+k becomes a noncommutative semigroup with a unit. Let us denote this group by (Gk, °). Using this semigroup structure one defines translations rg:Gk^Gk,geGk by rg(x) = g°x.
Let G c Gk be a sub-semigroup of Gk and h : Gk ^ Y be a Y-valued function defined on Gk. We say that h is G -periodic if h(zg (x)) = h(x) for all g e G and x e Gk. Any Gk -periodic function is called translation-invariant. Now for each m > 2 we put G = {x e Gk: d(x, x0) = 0 (mod m)}. One can check that Gm is a sub-semigroup of x e Gk
We consider p -adic Ising model on the semi-infinite Cayley tree r+A . Let Q be a field of p -adic numbers and 0 = {-1,1} . A configuration 7 on A c V is defined as a function x e A ^ ((x) e O. The set of all configurations on A is denoted by Q^ = OA. For given configurations 7n l e Q and <p(n) eQw we define a configuration in Q as follows
\7„-1( x) if x eV„- ^ L-<p("\x), i/XGWn.
A formal p -adic Hamiltonian H : Q —> Q of the Ising model is defined by
(an_! van)(x)= , b)
H(a) = J £ v(x)a(y) , (1)
< x, y><=L
C i N
where J E B
0, p
i-p
J
for any < x,y >e L .
We are aiming to study some non-periodic p -adic generalized Gibbs measures for the
Ising model on a Cayley tree. Our approach is based on properties Markov random fields on the Cayley tree.
Let h: x e V \ {x0} —» hx e be a function. We define p -adic probability generalized Gibbs distribution jJn) on Q by
Mln)(°n) = iexpp {Hn(an)}nK(x\ n = 1,2,..., (2)
Zn xeW„
where Zn(h) is corresponding normalizing constant:
znh) = Z exp {Hn (an )}n h. (3)
creQv xeW
Vn n
The compatibility conditions for jJn)(&„), n > 1 are given by the equality
Z J\&n-i v^(n)) = jhn-1)(^n-i). (4)
<P( n }^wn
In this case, by the p -adic analogue of Kolmogorov theorem ([3]) there exists a unique measure j on the set Q such that j ({a |V = an}) = jh"-1 (an-1), for all n and
A limiting p -adic distribution generated by (2) is called p -adic generalized Gibbs measure [4].
If there are two different measures J, j and they are bounded, then one says that there is a quasi-phase transition. If there are at least two distinct p -adic generalized Gibbs measures J and V such that J is bounded and V is unbounded, then one says that a phase transition occurs. Moreover, if there is a sequence of sets An such that A eQ with I j( A) \P ^ 0 and | v( A) lp ^^ as n ^ <», then there occurs a strong phase transition [4].
Theorem 1. [4] The sequence of p -adic distributions {jJn)(&n)} determined by
formula (2) is consistent if and only if for any x eV \ {x0}, the following equation holds
, „ h+1
hx2 = n (5)
yeS(x) hy + °
where 6 = exp{2 J}, 6 ^ 1.
Main results. On the semi-Cayley tree of order two, we denote by hf) (i = 0,1, 2) and h(p) (j = 1, 2) the translation-invariant and G2 -periodic solutions of the equation (5), respectively. It is known [4] that if p = 1 (mod 4) then
hs-=1, c=6z1^Em+iL, ^=1 -6U(6-1)2 -462 (6)
2 26
Let k > 3, k0 = 2. For x e V, by S^ (x) we denote an arbitrary set of k0 vertices of the set S(x), and remaining k - k0 vertices is denoted by (x). Let k - k0 = a + b + c, where a and b are even, c is even or odd. We define the set of quantities h = {hx, x e V} (where h e {1,hf),fcf),h(p),h(p)}) as follows:
if at vertex x we have hx = hf) (i = 1,2) (hx = h(p) or hx = 1), then the function h ,
which gives p -adic values to each vertex y e S(x) is defined by the following rule (7) (resp. (8) or (9)).
on a /2 + 2 vertices of S(x),
hy =
hf) (t)
on a / 2
h( p) on b /2
hp 1
on b /2 on c
vertices of S(x), vertices of S(x), vertices of S(x), vertices of S(x).
(7)
h =
hf) (t)
h =
vertices of S(x), vertices of S(x), vertices of S(x), on b /2 + 2 vertices of S(x), on c vertices of S(x).
ht) on a /2 h(% on a /2 Kp) on b /2
on a / 2
^ on a /2
hp) on b/2
h(p) 1
(8)
h3( p on b /2
vertices of S(x), vertices of S(x), vertices of S(x), vertices of S(x), vertices of S(x).
(9)
on c + 2
Lemma 1. Let p = 1 (mod 4). Then any set of quantities according to the rules (7), (8) and (9) on the Cayley tree F k+ satisfy the functional equation (5).
Remark 1. 1) If a = b = c = 0 in (7) and (9) then p -adic generalized Gibbs measures corresponding to set of quantities h are translation-invariant, the figure for case (8), we get p -adic generalized G2(2) -periodic Gibbs measures (see [4]);
2) If a = b = 0, c ^ 0 in (9) and (7), (8) then p -adic generalized Gibbs measures corresponding to set of quantities h are translation-invariant (see [4]) and ART Gibbs measures, respectively (see [6]);
3) If b = c = 0, a ^ 0 in (7) then p -adic generalized Gibbs measures corresponding to set of quantities h are (k0) -translation-invariant (see [7]);
4) If a = c = 0, b ^ 0 in (8) then p -adic generalized Gibbs measures corresponding to set of quantities h are (k ) -periodic (see [7]);
5) In other cases, we get new measures except to previous known ones.
In real case Bleher-Ganikhodjaev construction was studied in [8]. We are going to investigate this construction in p-adic case. Consider an infinite path n = x0 = x0 < xx <... on
the semi-Cayley tree Fk+ (the notation x < y meaning that paths from the root to y go through
x). We assign the set of p -adic numbers hn = {h, x eV} satisfying the equation (5) to the path
n . For n = 1, 2,..., x e W , the set hn is unambiguously defined by the conditions
hi =
—, if x < x„ , x e W„,
K n * (10)
K, if xn <X, X<EWn, where x -< Xn (resp. Xn -< x) means that x is on the left (resp. right) from the path n and h is translation-invariant solution of the equation (5).
Lemma 2. For any infinite path n, there exists a unique set of numbers hn = {hn, x e V} satisfying (5) and (10).
Remark 2. The measure which is defined in Lemma 2, depends on the path n. The cardinality of the measures is uncountable (see [8]).
Theorem 2. Let p = 1 (mod 4). Then for the measures correspond to the set of
quantities according to the rules (7), (8) and (9) the followings hold
1) If a2 + b2 ^ 0, then the measures are unbounded;
2) If a = b = 0, then the measures are bounded.
Theorem 3. Let p > 3, hi be the set of quantities defined by (10). Then the measures
correspond to the set of quantities h are bounded if and only if h = 1.
Theorem 4. Let p>3, Q)p be a field of p -adic numbers in which there exist
translation-invariant solutions of the functional equation (5). Then there exists a phase transition in the field Q .
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