Renormalization of Dyson's hierarchical vector valued \(\Phi ^ 4\) model at low temperatures
The paper proves a very nice central limit theorem for a pure supercritical equilibrium state of Dyson's hierarchical \({\mathbb{R}}^ m\)- valued model with parameter \(c=2^{2-a}\in (\sqrt{2},2)\) and with \(\Phi^ 4\) single spin distribution. The theorem requires the usual \((volume)^{-1/2}=(2^ n)^{-1/2}\) scaling in the direction of the spontaneous magnetization, but a \(((4/c)^ n)^{-1/2}\) scaling in the orthogonal directions. More precisely, for \(h,\beta,u>0\), \(e_ 1=(1,0,...,0)\in {\mathbb{R}}^ m\), we are given on \(\sigma =(\sigma_ k=(\sigma^ i_ k\), \(i=1,...,m)\), \(k=1,...,2^ n)\in {\mathbb{R}}^{m2^ n}\) the Gibbs measure \[ \mu^ h_ n(d\sigma)=\exp \{\frac{\beta}{2}\sum_{1\leq k\neq \ell \leq 2^ n}2^{- an(k,\ell)}<\sigma_ k,\sigma_{\ell}>+\sum_{1\leq k\leq 2^ n}[\beta h<e_ 1,\sigma_ k>-u| \sigma_ k|^ 4/4-| \sigma_ k|^ 2/2]\}d\sigma /Z^ h_ n, \] where n(k,\(\ell)\) is the smallest n' with \((p-1)2^{n'}<k,\ell \leq p2^{n'}\) for some p, \(d\sigma\) is the Lebesgue measure on \({\mathbb{R}}^{m2^ n}\) and \(Z^ h_ n\) is the normalizing constant. With \(\mu^+(d\sigma)=\lim_{h\to 0}\lim_{n\to \infty}\mu^ h_ n(d\sigma),\) one knows that for \(\beta >\beta_{cr}\), \(\int \sigma^ i_ k\mu^+(d\sigma)=\delta_{i1}m\) with \(m=m_{\beta}>0\) and the Kronecker symbol \(\delta_{i1}\). Now, for \(k=1,...,2^ r\), let \[ (T^ n\sigma)^ i_ k=\alpha_ i^{-n/2}\sum_{(k-1)2^ n<\ell \leq k2^ n}(\sigma^ i_{\ell}-\delta_{i1}m)\quad with\quad \alpha_ 1=2\quad but\quad \alpha_ i=4/c\in (2,\sqrt{8}) \] for \(i=2,...,m\). Under \(\mu^+\), the random variables \(((T\sigma)_ 1,...,(T\sigma)_{2^ r})\) converge to a Gaussian variable (\({\bar \sigma}{}^ i_ k\), \(i=1,...,m\); \(k=1,...,2^ r)\) with \({\bar \sigma}^ 1_ 1,...,{\bar \sigma}^ 1_{2^{\ell}},({\bar \sigma}^ 2_ j, 1\leq j\leq 2^ r),...,({\bar \sigma}^ m_ j, 1\leq j\leq 2^ r)\) independent. This result is heavily based on the first author's earlier results on large deviations for the hierarchical model [see ''A theorem on large deviations in the neighbourhood of \(\Phi^ 4\)-hierarchical model''. Teor. Veroyatn. Primen. 30, No.3, 499-510 (1985)].
scientific article; zbMATH DE number 3942683