Generalized Curie-Weiss model and quadratic pressure in ergodic theory

Abstract

We extend results on quadratic pressure and convergence of Gibbs mesures from [12] to the Curie-Weiss-Potts model. We define the notion of equilibrium state for the quadratic pressure and show that under some conditions on the maxima for some auxiliary function, the Gibbs measure converges to a convex combination of eigen-measures for the Transfer Operator. This extension works for dynamical systems defined by infinite-to-one maps. As an example, we compute the equilibrium for the mean-field XY model as the number of particles goes to +∞.

Nonlinear thermodynamical formalism

“…(see, e.g., [Phe01, Proposition 1.2]) The accumulation points can indeed fail to be equilibrium measures, e.g., in the Curie-Weiss model when there are two asymmetric equilibrium measures and one chooses symmetric Gibbs ensembles, see [LW19].…”

“…In fact, this failure of uniqueness can occur even for a topologically transitive subshift of finite type with a Hölder-continuous potential (see e.g. [LW19] and Section 4 below). However uniqueness holds for generic non-linearities for any 𝑑 ⩾ 1 (Proposition 3.22).…”

“…In the 1970s, Sinai, Ruelle, Bowen, and others (see, e.g., [Bow08,Rue04,Sin72]) developed a thermodynamical approach to dynamical systems inspired by the statistical mechanics of lattice systems. In a recent work [LW19], the third named author and Watbled applied this program to the Curie-Weiss mean-field theory: they introduced a new thermodynamical formalism over the full shift where the energy functional is quadratic. They obtained precise results using the specific structure of this setting.…”

On the Dimension of the Space of Harmonic Functions on Transitive Shift Spaces

“…The potential defined in (8) is inspired in the mean-field or Curie-Weiss model for ferromagnetism. It is one of the simplest models in Equilibrium Statistical Mechanics exhibiting the phase transition phenomenon, see [11,25,27,46,47,57] for more details from the Statistical Mechanics point of view.…”

Higher-dimensional nonlinear thermodynamic formalism

“…Building on work on the Curie-Weiss mean-field theory in [LW17], the nonlinear topological pressure was introduced in [BL20] as a generalization of (1) as follows (more precisely, we give an equivalent formulation using separated sets instead of covers). Given a continuous function F : R → R, the nonlinear topological pressure of a continuous function ϕ : X → R is given by…”