Physics The Large-Scale Limit of Dyson's

Communications in Mathematical

Commun. Math. Phys. 125, 43-69 (1989)

Physics

© Springer-Verlag 1989

The Large-Scale Limit of Dyson's Hierarchical Vector-Valued Model at Low Temperatures. The marginal case c—yl P. M. Bleher1 and P. Major2 1

Keldysh Institute for Applied Mathematics of the Academy of Sciences of the USSR, Moscow A-47 Miusskaya Square 4, USSR 2 Mathematical Institute of the Hungarian Academy of Sciences, P.O.B. 127, H-1364 Budapest, Hungary

Dedicated to Roland Dobrushin

Abstract. In this paper we construct the equilibrium states of Dyson's vectorvalued hierarchical model with parameter c = ]/ϊ at low temperatures and describe their large-scale limit. The analogous problems for j/2 < c < 2 and 1 < c < j/2 were solved in our papers [1] and [2]. In the present case the largescale limit is similar to the case j/2 < c < 2, i.e. it is a Gaussian self-similar field with long-range dependence in the direction orthogonal to and a field consisting of independent Gaussian random variables in the direction parallel with the magnetization. The main difference between the two cases is that now the normalizing factor in the direction of the magnetization contains, beside the square-root of the volume, a logarithmic term too. 1. Introduction

First we briefly describe the model we are investigating. Dyson's hierarchical model is a one-dimensional classical spin model on the lattice Z = {1,2,...}. Its Hamiltonian function depends on a parameter a, 1 < a < 2, and is defined as *(σ)=-

Σ Σ d(iJΓ"σ(ϊ)σ(J),

U l)

ieZ jeZ n{ίJ)

where d{ίJ) = 2 -\

and

n(ij) = mm{n, there exists some k such that (k —

l)2n0 is a sufficiently small number, and C(t) is an appropriate norming constant which turns po(x) into a density function. For the sake of convenience we shall work in the sequel with the number c = 22~a instead of the parameter a. We investigate the following problem: First we construct an equilibrium state μ = μ(T) at low temperatures with magnetization in the direction of the first coordinate x (1) and then we want to describe its large-scale limit. In more detail, let (1

p

σ = {σ(j) = (σ >(/), ., σ™V)) eR , je Z}

be a random field with the distribution of the equilibrium state μ = μ{T\ and define for all π = 1,2,... the random field

&n0 and to>0 such that ίfOD> n = ί,29...,N

2M

f)s^Dα

\|/2

\]/2j

—. To formulate it we introduce the sequences gM, Am

defined by the recursive relations M = ^ , + ~M

(2-12) for

n0 then a)

ai

N

(4.1)

,

A«o-- . |φ)|^2""/

βn+i=Pn+^2+δ(n),

4

n^N.

(4.2)

and n^N.

(4.3)

for

1/3

/n\ b) For \t\< I — I

and n^N ψn{t) is well-defined, and

\rn/

d3 dt3

for

\t\^(j)

Proof of Lemma ί. Because of Proposition 1 φN(fN) (t) is very close to the Fourier transform of the normal density function φ(χ9 σ) with σ 2 = ——-—-, and the 2(ao-T) analogous result also holds for its derivatives. This implies (4.2) and (4.3) for n = N, since if φN(fN){t) were exactly normal then we would have βN= ——-—- and Λ3

j

2(ao-T)

)

n + 1.

= 0. We prove (4.2) and (4.3) in the general case by induction from n to

P. M. Bleher and P. Major

56

Let us introduce the operator TM by the formula Twι/;(ί) = logTMnexpί/;(ί). It follows from (3.9') that (4.4)

4Ml+βn> and

j/ϊr.

|r

(4.4')

3

16M n ( 1 + l/2Mπ, Since Mn is very large, (4.4') together with our inductive hypothesis imply that n+ lV'3 =

M3

+

i/2Z

"

PW

Because of the identities TMφn(0)= — TnxpN(t)

(4.5)

= 0 it follows from (4.4) and (4.5)

that

where 91 denotes real part. (Observe that 1