Free energy in multi-species mixed \(p\)-spin spherical models. (English) Zbl 1492.60264

Bates, Erik; Sohn, Youngtak

Electron. J. Probab. 27, Paper No. 52, 75 p. (2022).

Summary: We prove a Parisi formula for the limiting free energy of multi-species spherical spin glasses with mixed \(p\)-spin interactions. The upper bound involves a Guerra-style interpolation and requires a convexity assumption on the model’s covariance function. Meanwhile, the lower bound adapts the cavity method of Chen so that it can be combined with the synchronization technique of Panchenko; this part requires no convexity assumption. In order to guarantee that the resulting Parisi formula has a minimizer, we formalize the pairing of synchronization maps with overlap measures so that the constraint set is a compact metric space. This space is not related to the model’s spherical structure and can be carried over to other multi-species settings.

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
60G15	Gaussian processes
82B44	Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
82D30	Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)

Keywords:

Aizenman-Sims-Starr scheme; cavity method; free energy; Guerra interpolation; multi-species spin glass; Parisi formula; spherical spin glass; synchronization

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