Parisi formula, disorder chaos and fluctuation for the ground state energy in the spherical mixed p-spin models

Abstract

We show that the limiting ground state energy of the spherical mixed p-spin model can be identified as the infimum of certain variational problem. This complements the well-known Parisi formula for the limiting free energy in the spherical model. As an application, we obtain explicit formulas for the limiting ground state energy in the replica symmetry, one level of replica symmetry breaking and full replica symmetry breaking phases at zero temperature. In addition, our approach leads to new results on disorder chaos in spherical mixed even p-spin models. In particular, we prove that when there is no external field, the location of the ground state energy is chaotic under small perturbations of the disorder. We also establish that in the spherical mixed even p-spin model, the ground state energy superconcentrates in the absence of external field, while it obeys a central limit theorem if the external field is present.

40 pages, restructured presentation

References in corpus (3)

• Fluctuations of the free energy of the spherical Sherrington-Kirkpatrick model
• Disorder chaos in spin glasses
• Disorder chaos in the spherical mean-field model

Cited by in corpus (21)

• Complex energy landscapes in spiked-tensor and simple glassy models: ruggedness, arrangements of local minima and phase transitions
• Low Temperature Asymptotics in Spherical Mean Field Spin Glasses
• Ferromagnetic to paramagnetic transition in spherical spin glass
• Triviality of the geometry of mixed $p$-spin spherical Hamiltonians with external field
• Free energy in multi-species mixed $p$-spin spherical models
• On the energy landscape of the mixed even $p$-spin model
• A connection between MAX $\kappa$-CUT and the inhomogeneous Potts spin glass in the large degree limit
• Bounding flows for spherical spin glass dynamics
• Spherical spin glass model with external field
• Crisanti-Sommers formula and simultaneous symmetry breaking in multi-species spherical spin glasses
• Concentration of the complexity of spherical pure $p$-spin models at arbitrary energies
• Optimizing Mean Field Spin Glasses with External Field
• Bounding the Complexity of Replica Symmetry Breaking for Spherical Spin Glasses
• A spin glass model for reconstructing nonlinearly encrypted signals corrupted by noise
• Existence of two-step replica symmetry breaking for the spherical mixed p-spin glass at zero temperature
• Sharp complexity asymptotics and topological trivialization for the (p, k) spiked tensor model
• Replica-symmetry breaking transitions in the large deviations of the ground-state of a spherical spin-glass
• Overlap of a spherical spin glass model with microscopic external field
• Controlling the Glass Transition through Active Fluctuating Interactions
• Free Energy Subadditivity for Symmetric Random Hamiltonians
• On Marginal Stability in Low Temperature Spherical Spin Glasses