Notes on ferromagnetic diluted p-spin model

Abstract

In this paper we develop the interpolating cavity eld technique for the mean eld ferromagnetic p-spin. The model we introduce is a natural extension of the diluted Curie-Weiss model to p > 2 spin interactions. Several properties of the free energy are analyzed and, in particular, we show that it recovers the expressions already known for p = 2 models and for p > 2 fully connected models. Further, as the model lacks criticality, we present extensive numerical simulations to evidence the presence of a rst order phase transition and deepen the behavior at the transition line. Overall, a good agreement is obtained among analytical results, numerics and previous works.

PDE/Statistical Mechanics Duality: Relation Between Guerra’s Interpolated p-Spin Ferromagnets and the Burgers Hierarchy

“…Furthermore, it is one of the simplest models describing the development and propagation of shock waves. A related and well-studied equation is the Sharma-Tasso-Olver (STO) equation [66,74,83], which can be written as…”

“…This equation is known to be integrable (in particular, it presents infinitely many symmetries, a bi-Hamiltonian structure, solitary wave solution and an infinite number of conservation laws [66,74,83,85]). These two equations are the lowest elements of the so-called Burgers hierarchy, which can be presented in the form…”

“…In this paper, we study the relation of Guerra interpolated partition functions of p-spin ferromagnets and prove that the expectation values of the order parameter (the global magnetization) w.r.t. to the associated Boltzmann-Gibbs measure is in 1-1 correspondence with the elements of the Burgers hierarchy [59,66,74,83]. This duality can be explored in both sense: for instance, we can generate specific solutions of the Burgers hierarchy by exploiting the finite size solution of the p-spin systems; on the other hand, we can achieve informations about the thermodynamics of the ferromagnets from the PDE side.…”

Notes on the p-spin glass studied via Hamilton-Jacobi and smooth-cavity techniques

“…its variation w.r.t. the parameter t ruling the strength of the smooth cavity perturbation, that we state without the proof as it is a long but straightforward generalization of the once given for instance in [7,2]:…”

“…In these notes we continue our investigation on the mathematical methods and the physics underlying many body interactions, namely we adapt recent mathematical techniques to the study of equilibrium statistical mechanics of p-spin glasses. In the past we analyzed p-spin systems with the simpler ferromagnetic couplings [11] and p-spin systems with diluted coupling [2], while now we turn to p-spin systems with frustrated couplings, which are termed p-spin glasses [18,15].…”