Infinite differentiability for one-dimensional spin system with long range random interaction

Abstract

We consider one-dimensional spin systems with Hamiltonian:

$$H\left( {\sigma _\Lambda } \right) = - \sum\limits_{t,t' \in \Lambda } {\frac{{\varepsilon _{tt'} }}{{\left| {t - t'} \right|^\alpha }}\sigma _t \sigma _{t'} - h\sum\limits_{t \in \Lambda } {\sigma _t } } $$

, where ɛ _tt′ are independent random variables and, using decimation and the cluster expansion, we show that, when α>3/2 andE(ɛ _tt′ )=0, for any magnetic fieldh and inverse temperature β, the correlation functions and the free energy areC ^∞ both inh and β.

Moreover we discuss an example, obtained by a particular choice of the probability distribution of the ɛ _tt′ 's, where the quenched magnetization isC ^∞ but fails to be analytic inh for suitableh and β.