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Following the Ground States of Full‐RSB Spherical Spin Glasses
Author(s) -
Subag Eliran
Publication year - 2021
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21922
Subject(s) - spin glass , mathematics , eigenvalues and eigenvectors , hessian matrix , upper and lower bounds , ground state , hamiltonian (control theory) , combinatorics , replica , energy (signal processing) , mathematical analysis , physics , quantum mechanics , mathematical optimization , statistics , art , visual arts
We focus on spherical spin glasses whose Parisi distribution has support of the form [0, q ] . For such models we construct paths from the origin to the sphere that consistently remain close to the ground‐state energy on the sphere of corresponding radius. The construction uses a greedy strategy, which always follows a direction corresponding to the most negative eigenvalues of the Hessian of the Hamiltonian. For finite mixtures ξ ( x ) it provides an algorithm of time complexity O ( N deg( ξ ) ) to find w.h.p. points with the ground‐state energy, up to a small error. For the pure spherical models, the same algorithm reaches the energy − E ∞ , the conjectural terminal energy for gradient descent. Using the TAP formula for the free energy, for full‐RSB models with support [0, q ] , we are able to prove the correct lower bound on the free energy (namely, prove the lower bound from Parisi's formula), assuming the correctness of the Parisi formula only in the replica symmetric case. © 2020 Wiley Periodicals LLC