Non Uniqueness of p-adic Gibbs Distribution for the Ising Model on the Lattice Zd Текст научной статьи по специальности «Математика»

УДК 517.9

Non Uniqueness of p-adic Gibbs Distribution for the Ising Model on the Lattice Zd

Zohid T. Tugyonov*

Institute of Mathematics Durmon Yuli, 29, Tashkent, 100125 Uzbekistan

Received 05.08.2015, received in revised form 24.09.2015, accepted 28.12.2056

In this paper, we show non uniqueness of p-adic Gibbs distribution for the Ising model on the Zd. Moreover, we prove that a p-adic Gibbs distribution is bounded if and only if p = 2.

Keywords: Gibbs distribution, Ising model, lattice. DOI: 10.17516/1997-1397-2016-9-1-123-127.

Introduction

Real Gibbs measures arise in many problems of probability theory and statistical mechanics. This measure, related to the Boltzmann distribution, generalizes the notion of canonical ensemble. In addition, Gibbs measure is unique measure that maximizes the entropy of the expected energy. But non-archimedean (p-adic) analogue of Gibbs measures have been little studied. It is known that in the case of real numbers concepts of Gibbs measure and Markov random field are identical. But in the p-adic case, the class of p-adic Markov random fields is wider than the class of p-adic Gibbs measures [1]. One of the main problems of physics is to study the set of all p-adic Gibbs measures (see e.g. [1,2]).

Let us present some main definitions from the theory of p-adic numbers (see [3-5]). Let p be

n

a prime number. Every rational number x = 0 can be represented in the form x = pr —, where

m

r,n G Z, m is a positive number, (n, m) = 1, where m and n are not divisible by p. A p-adic

n

norm of rational number x = pr — is defined as follows

m

, = ( p-r, if x = 0, \x\p = \ 0, if x = 0.

The completion of the set of rational numbers Q under p-adic norm leads to the field of p-adic numbers Qp for every prime p. This p-adic norm satisfies the strong triangle inequality:

x + y\p < max{|x|p, \y\p}. (1)

This property shows that p-adic norm is a non-Archimedean norm. It immediately follows from the strong triangle inequality that

1) if \x\p = \y\p, then \x + y\p = max{|x|p, \y\p};

2) if \x\p = \y\p, then \x + y\p < \x\p; For a G Qp and r > 0 we introduce

B(a, r) = {x G Qp : \x — a\p < r}.

* zohidboy@mail.ru © Siberian Federal University. All rights reserved

The p-adic logarithm is defined as

CO / \ n

logp(x) = logp(1 + (x — 1)) = ]T( —1)^.

n

n=1

The power series converges for x £ B(1,1). The p-adic exponent is defined as

xn

expp(x) = 53 n!

n!

n=0

and the power series converges for x £ B(0,p-1/(p-1)). Let x £ B(0,p-1/(p1). Then

I expp(x)|p = 1, | expp(x) - 1|p = \x\p, | logp(1 + x)|p = \x\p,

logp(expp(x)) = x, expp(logp(1 + x)) = 1 + x.

Let (X, B) be a measurable space, where B is an algebra of subsets of X. The function H : B ^ Qp is a p-adic distribution if for any A1,... ,An £ B such that Ai n Aj = 0 (i = j) we have the equality

(n \ n

U Aj ) = £rtAj). j=1 ) j=1 A p-adic distribution is a measure if

sup{|^(A)|p : A £ B} < <x.

A p-adic measure is a probability measure if n(X) = 1.

Let us consider the set of points x = (x1;..., xd), xi £ Z, i = 1,..., d, i.e., the d is dimensional

d

integer lattice Zd with metric p(x'; x'') = x — x'( |. Two points x and y of the lattice Zd are

i=1

called nearest-neighbors if p(x; y) = 1. It is symbolized by < x,y >.

Let Qp be the set of p -adic numbers and $ = { — 1, +1} is the set of spin values. By standard way we can define the real (see. [6]) or p-adic (see [1]) Gibbs distribution for the Ising model. For some natural number n the set

Mn = |x = (x1,..., xd), xi ^ 0, i = 1,..., d : p1(x, xo) ^ n|

is called the fundamental quadrate. A point x £ Zd \ Mn is called nearest-neighbor of the set Mn if there exists y £ Mn such that < x,y >, i.e., p(x; y) = 1. The set of all nearest-neighbors of Mn is called the quadrate contour of Mn and it is designated as dMn. Configuration a defined on the set Zd is a function x £ Zd ^ a(x) £ Restriction of a on any subset V c Zd is designated as a(V), i.e., a(V) = {a(x), x £ V}. The set of all configurations on Mn is denoted by Qn and the set of all configurations on Zd is denoted by Q. Hamiltonian of the p-adic Ising model is defined as follows:

H(a) = — J V a(x)a(y), where JL < 1. (2)

p

Conditional Hamiltonian Hn(a | a'), a, a' £ Q of the p-adic Ising model on Mn has the form Hn(a | a') = — J a(x)a(y) — J ^ a(x)a'(y).

<x,y> <x,y>

x,yeMn xeMn.yeSMn

In this paper, p-adic Gibbs distributions for the p-adic Ising model on Zd are studied.

1. The p-adic Gibbs distributions

The p-adic conditional Gibbs distribution for configurations a on the quadrate Mn and a' on Zd \ Mn with the Hamiltonian Hn(a | a') is the following p-adic number

Hn{a | a')

expp (-Hn(a I a')) E expp (-Hn(a I a'))'

(3)

Let us consider the following configurations:

a+ = {a(x) = 1}, a" = {a(x) = -1}.

For configurations a+ and a~ we define two p-adic Gibbs conditional distributions on the quadrate Mn:

tiFn(a)= n{a(Mn) I ab(Zd \ Mn))

exPp (-Hn(a))

e = -, + ,

where

Hn(a) = -J a(x)a(y) - eJ ^ a(x),

<x,y> <x,y>

x,yEMn xEMn,yEdMn

zn = J2 exPp (-Hn(a)) ■

(4)

Lemma 1. If p = 2, \ J\p ^ - then \H(a)\p ^ -.

pp

Proof. Taking into account the strong triangle inequality and using \a(x)\p = 1, we obtain

1

IH (a)

J aix)a(y)

<x,y>

< IJIp ■

p

Lemma 2. Ifp = 2 then z+ = zn and \z+\p = \zn\p = 1.

Proof. Let p = 2. From H+(a) = H-(—a) we obtain z+ = z—. Let us consider the following chain of equalities:

lz+lp =

]T expp (-H+(a)) £ (expp (-H+(a)) - 1)

]T (expp (-H+(a)) - 1) + IQn\

; = PnI =

p Pj p

1.

Let us introduce the following notation

Hn (a) = - J J2 <x,y> a(x)a(y).

Theorem 1. Let p = 2. Suppose that there exist limits

lim v±(a) =

n ^^

Then p-adic measures and are different. In addition, for any r G N a configuration a can be found such that \y+(a) — n~(a)\p = \ J\p-p~r.

ь

n

p

p

p

max

p

Proof. Let an(a) = \|+(a) - i— (a)\p ■ Using Lemmas 1 and 2, we have

an(<r)

exPp (-H+(a)) exp„ (-H- (a))

z+ z n

z+ zn

\z+ p

exPp (Hn (a))

expp

J a(x) I - exPp

<x,y> \ xeMn,yeBMn

-j y^ a(x)

<x,y> \ xeMn,yeBMn

exPp

2J J2 a(x) I - 1

. <x,y>

\ xeMn,yedMn

2J E a(x)

<x,y> xeMn,yedMn

2J (2dnd- 2s—(a(n)))

where a(n) = {a(x), x £ dMn} and s (a(n)) is the number of elements of the set

{x £ dMn : a(x) = —1}.

Let n0 =

( pr \ W

. Note that for all n > n0 the difference 2dnd 1 —pr is a natural number.

Then there exists a configuration a such that a\dMn = a(n) and s (a(n)) = 2n — pr■ For such n the equality an(a(n)) = \4J\p ■ p—r = \ J\p ■ p—r holds. Therefore we have lim an(a) = \J\p ■ p—r.

n

□

Proposition 1. A cardinality of the set {a : \|+(a) — l—(a)\p = \J\p ■ p—r} is continuum.

2. Boundedness of p-adic Gibbs measures.

In this section we obtain the condition for p-adic Gibbs distribution to be a measure (bounded) for the Ising model.

Theorem 2. A p-adic Gibbs distribution for the Ising model is bounded if and only if p = 2. Proof. Consider the following expression for \^(an)\p:

\l(a„

exp„(H (an))

E exPp(H(^n))

Vn £

1

1

E (expp (-H(a)) - 1) + \fin\

E (expp (-H(a)) - l) + 2n

if p = 2.

To prove unboundedness of i for p = 2 one needs only to show that i is not bounded on some path in Zd. Let n = {■■■,xn1,x0,x1, ■■■} be a path in Zd. Marginal distributions on n is of the form

n — 1

In (wn ) = Pw(x-n) n P

w(xm)w(xm+l),

(5)

p

1

p

p

p

p

p

1

d

p

p

m=—n

where wn is a configuration on {x-n,... ,x0,..., xn} and Pi are coordinates of invariant vector of the matrix

p = (p ) = _1_ ( expp(J) expp(-J)

V exPp(J ) + exPp(-J )\expp(-J) expp(J)

We have

P'2 I expp(J) + expp(-J)|2 |(expp(J) - 1) + (expp(-J) - 1) + 2|2 ^ 2 (6) It follows from (5) and (6) that ц is unbounded for p = 2. □

References

[1] U.A.Rozikov, O.N.Khakimov, p-adic Gibbs measures and Markov random fields on countable graphs, TMPh, 175(2013), no. 1, 518-525.

[2] U.A.Rozikov, Gibbs measures on Cayley trees, World Sci., Publ. Singapore, 2013.

[3] V.S.Vladimirov, I.V.Volovich, E.V.Zelenov, p-adic Analysis and Mathematical Physics, World Sci., Singapore, 1994.

[4] N.Koblitz, p-adic numbers, p-adic analysis, and zeta-functions, New York, 1984.

[5] A.Yu.Khrennikov, Non-archimedean analysis and its applications, Moscow, Fizmatlit, 2003 (in Russian).

[6] H.-O.Georgii, Gibbs Measures and Phase Transitions, W. de Gruyter, Berlin, 1988.

Неединственность p-адического распределения Гиббса для модели Изинга на решетке Zd

Зохид Т. Тугенов

В данной работе для модели Изинга на Zd мы покажем неединственность p-адического распределения Гиббса. Кроме того, докажем, что p-адические гиббсовские распределение являются ограниченными тогда и только тогда, когда p = 2.

Ключевые слова: распределение Гиббса, модель Изинга, решетка.