Abstract
We evaluate the high temperature limit of the free energy of spin glasses on the hypercube with Hamiltonian \(H_N({\underline{\sigma }}) = {\underline{\sigma }}^T J {\underline{\sigma }}\), where the coupling matrix J is drawn from certain symmetric orthogonally invariant ensembles. Our derivation relates the annealed free energy of these models to a spherical integral, and expresses the limit of the free energy in terms of the limiting spectral measure of the coupling matrix J. As an application, we derive the limiting free energy of the random orthogonal model at high temperatures, which confirms non-rigorous calculations of Marinari et al. (J Phys A 27:7647, 1994). Our methods also apply to other well-known models of disordered systems, including the SK and Gaussian Hopfield models.
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Notes
Condition (1.10) can be written as \(\lim _{N\rightarrow \infty }\mathbb {P}\left( \sum _{i=1}^{N} (d_i - \mathbb {E}(d_i))^2 > c \right) \rightarrow 0\) for any constant \(c>0\), since \(W_2^2(\mu _N(D) ,\mu _N(\mathbb {E}(D))) =\frac{1}{N} \sum _{i=1}^{N} (d_i - \mathbb {E}(d_i))^2\). Our results continue to hold if condition (1.10) is replaced by the following slightly general condition: there exists a sequence of deterministic measure \(\nu _N\) supported on N points in \({\mathbb {R}}\) such that \(\lim _{N\rightarrow \infty }\mathbb {P}\left( W_2 (\mu _N(D), \nu _N) > \frac{c}{\sqrt{N}}\right) \rightarrow 0\). However, condition (1.10) suffices in all our applications where we chose \(\nu _N=\frac{1}{N}\sum _{i=1}^N\delta _{\mathbb {E}(d_i)}\).
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Acknowledgments
The authors thank Amir Dembo, Andrea Montanari and Sourav Chatterjee for helpful discussions. S.S. thanks Zhou Fan for help with results about random matrices. S.S. was supported by a William R. and Sara Hart Kimball Stanford Graduate Fellowship. The authors also thank an anonymous referee for carefully reading the manuscript and for providing valuable comments which improved the presentation of the paper.
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Bhattacharya, B.B., Sen, S. High Temperature Asymptotics of Orthogonal Mean-Field Spin Glasses. J Stat Phys 162, 63–80 (2016). https://doi.org/10.1007/s10955-015-1406-7
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DOI: https://doi.org/10.1007/s10955-015-1406-7