Research

My research mostly deals with understanding rigorously (in the mathematical sense) classical statistical physics models at equilibrium. The models that I often study are the Potts model (and in particular the Ising model) and the Ashkin-Teller model.

Some problems that I (with collaborators) tackled are:

1. The saturation phenomenon: For a general class of models (including the Potts and XY models) and exponentially decaying interaction, we established in [6] sufficient and necessary conditions for the existence of a non-trivial saturation regime, i.e. the existence of a finite temperature T'>T_c  such that the inverse correlation  length in a given direction s is constant on the interval [T',+∞) . This in particular implies that the inverse correlation length is not an analytic function of the temperature in the high temperature regime, and this even in one-dimensional systems. Another interesting feature is that, contrary to what was expected, the Ornstein-Zernike asymptotics do not hold in the whole saturation regime. This was studied perturbatively in [1] and non-perturbatively in [6]. Moreover, we established under suitable hypothesis that the usual Ornstein-Zernike asymptotics hold in the regime (T_c, T') in [5]. Finally, in [8], we established sufficent conditions for the existence of a saturation phenomenon at low temperatures for the Ising model and we established that the Ornstein-Zernike asymptotics hold for the one-dimensional Ising model at T'. This in particular shows that the mass-gap condition (i.e. that the rate of the exponential decay of the direct correlation is stricly larger than the rate of the exponential decay of the total correlation) is not necessary for the Ornstein-Zernike asymptotics to hold.

2. Establishing the complete phase diagram of the ferromagnetic Ashkin-Teller model on the square lattice: In the 1970s, the phase diagram of the symmetric ferromagnetic Ashkin-Teller model on the square lattice was predicted in the physics literature by Kadanoff and Wegner, Wu and Lin, Baxter and others. We proved these predictions rigorously in [7]. In particular, according to the values of the coupling constants, we prove that there exist either one or two critical points. 

3. The study of the correlation functions in the infinite-range systems:  In the high temperature regime, we established a nearly complete picture for the different scenarios of decay that can arise in the ferromagnetic systems. For the Potts model, we proved in [1] and [6] that if the interaction decays subexponentially fast, then the two-point function decays as the same rate as the interaction, and we derived the sharp constant as well. If the interactions decays superexponentially fast, then the two-point function satisfies the Ornstein-Zernike asymptotics (see [5]). If the interaction decays exponentially fast, see the discussion in first point. Thus, the exponential decay of the interaction appears as the critical case as the decay of the interaction varies.