Jean-Raymond Fontaine

On the uniqueness of the equilibrium state for plane rotators

We study the classical statistical mechanics of the plane rotator, and show that there is a unique translation invariant equilibrium state in zero external field, if there is no spontaneous magnetization. Moreover, this state is then extremal in the equilibrium states. In particular there is a unique phase for the two dimensional rotator, and a unique phase for the three dimensional rotator above the critical temperature. It is also shown that in a sufficiently large external field the Lee-Yang theorem implies uniqueness of the equilibrium state.

Lattice systems with a continuous symmetry

AbstractWe consider perturbations of a massless Gaussian lattice field on ℤd,d≧3, which preserves the continuous symmetry of the Hamiltonian, e.g.,

Lattice systems with a continuous symmetry. I. Perturbation theory for unbounded spins

- H = \sum\limits_{< x,y > } {(\phi _x - \phi _y )^2 + T(\phi _x - \phi _y )^4 ,\phi _x \in \mathbb{R}.}

Surface Tension and Phase Transition for Lattice Systems

It is known that for allT>0 the correlation functions in this model do not decay exponentially. We derive a power law upper bound for all (truncated) correlation functions. Our method is based on a combination of the log concavity inequalities of Brascamp and Lieb, reflection positivity and the Fortuin, Kasteleyn and Ginibre (F.K.G.) inequalities.

Bounds on the decay of correlations for λ(∇φ)4 models

We prove that the expansion in powers of the temperatureT of the correlation functions and the free energy of the plane rotator model on ad-dimensional lattice is asymptotic to all orders inT. The leading term in the expansion is the spin wave approximation and the higher powers are obtained by the usual perturbation series. We also prove the inverse power decay of the pair correlation at low temperatures ford=3.

Absence of symmetry breakdown and uniqueness of the vacuum for multicomponent field theories

AbstractWe investigate a continuous Ising system on a lattice, equivalently an anharmonic crystal, with interactions:

Low-Fugacity Asymptotic Expansion for Classical Lattice Dipole Gases

\sum\limits_{\left\langle {x,y} \right\rangle } {\left( {\phi _x - \phi _y } \right)} ^2 + \lambda \left( {\phi _x - \phi _y } \right)^4 , \phi _x \in \mathbb{R}, x \in \mathbb{Z}^d .