This paper proves that the free energy is an analytic (very smooth) function in the disordered regime for three well-known models of statistical mechanics. The models are the classical XY model in any dimension, the two-dimensional lattice Coulomb gas, and the square well model in any dimension. For the 2D Coulomb gas the authors also prove a strong form of Debye screening, and they show that this screened phase is the complement of the Berezinskii–Kosterlitz–Thouless (BKT) phase.

By “free energy is analytic” the authors mean there are no singularities or phase-transition type kinks in that function of temperature inside the disordered range. Concretely, they obtain analyticity in the subcritical interval [0, βc) for the XY model and for the 2D lattice Coulomb gas. The square well model is shown to be disordered at every positive temperature, and so its free energy is analytic for all positive temperatures.

The technical route is probabilistic. The authors show that the Gibbs measures for these models can be written as “factors of i.i.d.”. That phrase means one can generate a configuration of the model from independent random labels by applying the same local rule at every site. Moreover, the regions of the random labels that actually affect a given site — the information clusters — have volumes that decay exponentially in size. To get these statements they build and analyse dynamics (Glauber-type updates), prove space–time mixing, and use constructions related to “coupling from the past” and cluster-expansion ideas.

Why this matters: analyticity of the free energy is a strong mathematical form of “no thermodynamic singularity” in the disordered phase. For the Coulomb gas, Debye screening means charges are effectively neutralised at long distance so interactions between local observables decay exponentially. Showing that the Debye phase coincides with the complement of the BKT phase links a probabilistic screening picture with the more physical description of the BKT transition.