The Parisi formula completed

Abstract

The Parisi formula for the free energy of the Sherrington-Kirkpatrick model is completed to a closed-form generating functional. We first find an integral representation for a solution of the Parisi differential equation and represent the free energy as a functional of order parameters. Then we set stationarity equations for local maxima of the free energy determining the order-parameter function on interval $[0,1]$. Finally we show without resorting to the replica trick that the solution of the stationarity equations leads to a marginally stable thermodynamic state.

References in corpus (6)

• Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model
• The Thermodynamic Limit in Mean Field Spin Glass Models
• Analysis of the infinity-replica symmetry breaking solution of the Sherrington-Kirkpatrick model
• Stability of solutions of the Sherrington-Kirkpatrick model with respect to replications of the phase space
• Hierarchical solutions of the Sherrington-Kirkpatrick model: Exact asymptotic behavior near the critical temperature
• Incompleteness of the Thouless, Anderson, and Palmer mean-field description of the spin-glass phase

Cited by in corpus (5)

• Mean-field solution of the Potts glass near the transition temperature to the ordered phase
• Continuous RSB mean-field solution of the Potts glass
• Continuous replica-symmetry breaking in mean-field spin-glass models: Perturbation expansion without the replica trick
• Replica-symmetry breaking: discrete and continuous schemes in the Sherrington-Kirkpatrick model
• Ergodicity breaking in frustrated disordered systems: Replicas in mean-field spin-glass models