Hierarchical Spherical Model from a Geometric Point of View
Abstract
A continuous version of the hierarchical spherical model at dimension d=4 is investigated. Two limit distributions of the block spin variable Xγ, normalized with exponents γ=d+2 and γ=d at and above the critical temperature, are established. These results are proven by solving certain evolution equations corresponding to the renormalization group (RG) transformation of the O(N) hierarchical spin model of block size Ld in the limit L↓1 and N→∞. Starting far away from the stationary Gaussian fixed point the trajectories of these dynamical system pass through two different regimes with distinguishable crossover behavior. An interpretation of this trajectories is given by the geometric theory of functions which describe precisely the motion of the Lee–Yang zeroes. The large-N limit of RG transformation with Ld fixed equal to 2, at the criticality, has recently been investigated in both weak and strong (coupling) regimes by Watanabe (J. Stat. Phys. 115:1669–1713, 2004) . Although our analysis deals only with N=∞ case, it complements various aspects of that work.
- Publication:
-
Journal of Statistical Physics
- Pub Date:
- September 2008
- DOI:
- 10.1007/s10955-008-9568-1
- arXiv:
- arXiv:0802.3317
- Bibcode:
- 2008JSP...132..811M
- Keywords:
-
- Hierarchical model;
- Local potential approximation;
- Lee–Yang zeroes;
- Limit distributions;
- Geometric theory;
- Conformal mapping;
- Mathematical Physics;
- 82B28;
- 30C20;
- 30C15;
- 35F20
- E-Print:
- 27 pages, 6 figures, submitted to Journ. Stat. Phys