**Authors**:
G. Benfatto, G. Gallavotti, F. Nicolo`

**Title**:
*
The Dipole Phase in the Two-Dimensional Hierarchical Coulomb Gas: Analyticity
and Correlations Decay
*

Communication in Mathematical Physics, **106**, 277-288, 1986

**Abstract:** We illustrate the mechanism producing the
dipole phase in a two dimensional Coulomb system by a detailed analysis of a
hierarchical model. We prove the analyticity of the pressure and of the
correlations for $\alpha^2=e^2\beta>8\pi$ (i.e. right above the usually
conjectured value for the Kosterlitz- Thouless phase transition). We find also
a power law decay for the correlations with exponent $\alpja^2/2\pi$ as the
hierarchical distance goes to infinity.

**Key words: ** Quantum fields, Perturbation theory,
Renormalization theory, Dipole gas, Sine-Gordon model,
$2$-dimensional systems