Abstract
This paper investigates spin distributions for a generic spherical p-spin model; we give a representation of spin distributions in terms of a stochastic process. In order to do this, we find a series of invariance principles analogous to the cavity equations used for the hypercube. The construction of these equations is a multistep process; we first change coordinates to write the sphere as a product space for some cutoff parameters R and C. We then construct an auxiliary Hamiltonian that writes the effect of the final spin as an independent Gaussian process, e.g., with and decoupled from . The error rate in this approximation decays as . The final step is to justify the double limit to derive exact results; this double limit can be interpreted as a novel renormalization procedure.
Acknowledgments
The author is grateful to Aukosh Jagannath for introducing him to this problem and for useful discussions.
Citation
Arka Adhikari. "Spin distributions for generic spherical spin glasses." Electron. J. Probab. 27 1 - 43, 2022. https://doi.org/10.1214/22-EJP755
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