Abstract
We prove a duality principle that connects the thermodynamic limits of the free energies of the Hamiltonians and their squared interactions. Under the main assumption that the limiting free energy is concave in the squared temperature parameter, we show that this relation is valid in a large class of disordered systems. In particular, when applied to mean field spin glasses, this duality provides an interpretation of the Parisi formula as an inverted variational principle, establishing a prediction of Guerra.
Highlights
A fundamental goal in spin glass theory is to study the thermodynamic limit of random Hamiltonian systems that simultaneously exhibit ferromagnetic and anti-ferromagnetic properties
We study the thermodynamic limit of the free energy associated to the temperature β ∈ R, F (β) log ZN (β), where
It is motivated by an observation of Francesco Guerra [13], who suggested that the thermodynamic limit of the free energy in the mixed p-spin model is concave in the squared temperature
Summary
A fundamental goal in spin glass theory is to study the thermodynamic limit of random Hamiltonian systems that simultaneously exhibit ferromagnetic and anti-ferromagnetic properties. The aim of this paper is to provide a rigorous framework that connects the Parisi formula and the classical approaches It is motivated by an observation of Francesco Guerra [13], who suggested that the thermodynamic limit of the free energy in the mixed p-spin model is concave in the squared temperature. From such concavity, he conjectured a Legendre duality between the Parisi formula and the Legendre transform Γ of the scaled Parisi functional P√β, where the temperature and the functional order parameter are conjugate variables. Our result extends to the spherical mixed p-spin model as well, where it explains the nature of the Crisanti-Sommers representation of the limiting free energy
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