Critical properties of the random hierarchical model

Abstract

The random hierarchical model is defined by introducing random couplings between spin blocks in Dyson's original hierarchical model for continuous spins. The spin-distribution functions are now site dependent and functions of random parameters that we take to be a quadratic and quartic potential, $r_{l}^{}{}_{}{}^{(m)}$ and $U_{l}^{}{}_{}{}^{(m)}$, respectively, for the mth spin at the $l$ level of the hierarchy. Renormalization-group recursion relations for the cumulant averages are derived in an expansion in ${\ensuremath{\epsilon}}^{\ensuremath{'}}=(2\ensuremath{\sigma}\ensuremath{-}1)\mathrm{ln}(2)$, where $\ensuremath{\sigma}$ is a parameter of the model and a measure of the interaction range. We find that only ${r}_{l}=〈r_{l}^{}{}_{}{}^{(m)}〉$, ${\ensuremath{\Delta}}_{l}=〈{[r_{l}^{}{}_{}{}^{(m)}\ensuremath{-}{r}_{l}]}^{2}〉$, and ${U}_{l}=〈U_{l}^{}{}_{}{}^{(m)}〉$ are relevant physical quantities within the ${\ensuremath{\epsilon}}^{\ensuremath{'}}$ expansion. By solving the equations for ${U}_{l}$ and ${\ensuremath{\Delta}}_{l}$ to $O({\ensuremath{\epsilon}}^{\ensuremath{'}})$ we obtain in general four fixed points of the recursion relations. For $\ensuremath{\sigma}>\frac{1}{2}$, there is no stable fixed point in the physical region $\ensuremath{\Delta}>~0$, $U>~0$; thus the behavior of the system is still open to study. In the particular case of uniform block coupling but random spin distributions, we find a fixed point with ${U}^{*}\ensuremath{\approx}O({|{\ensuremath{\epsilon}}^{\ensuremath{'}}|}^{\frac{1}{2}})$, ${\ensuremath{\Delta}}^{*}\ensuremath{\approx}O({|{\ensuremath{\epsilon}}^{\ensuremath{'}}|}^{\frac{1}{2}})$ for ${\ensuremath{\epsilon}}^{\ensuremath{'}}<0$, but it is unstable.