Phase transition of spin systems

A phase transition is a qualitative change in the macroscopic behaviour of a system as a control parameter — typically temperature — is varied. Familiar examples are the spontaneous magnetisation appearing in the Ising model below a critical temperature, condensation in lattice gases, and percolation thresholds. Mathematically, the limit Gibbs measure changes character: uniqueness fails, and observables such as the correlation length or the susceptibility diverge.

For continuous-spin lattice systems with general single-site potentials, the picture is subtler than for the standard two-state Ising model. Whether a unique infinite-volume Gibbs measure exists depends sensitively on the temperature, on the interaction range, and on the convexity and tail behaviour of the single-site potential.

I am interested in deriving sharp criteria for uniqueness and for the breakdown of uniqueness in this continuous-spin setting, and in the consequences of such criteria for functional inequalities and for the speed of convergence of associated stochastic dynamics.