Abstract
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.
Funding Statement
E.B. was partially supported by NSF grant DMS-1902734. Y.S. was partially supported by NSF grant DMS-1954337.
Acknowledgments
We are grateful to Amir Dembo for valuable feedback and suggestions, and to Pax Kivimae for the detection of a computational error in a previous draft. We thank the referee for several corrections resulting from their careful reading.
Citation
Erik Bates. Youngtak Sohn. "Free energy in multi-species mixed p-spin spherical models." Electron. J. Probab. 27 1 - 75, 2022. https://doi.org/10.1214/22-EJP780
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