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A theory of solving TAP equations for Ising models with general invariant random matrices
Citation key OpCaWi16
Author Manfred Opper and Burak Cakmak and Ole Winther
Pages 114002
Year 2016
Journal Journal of Physics A: Mathematical and Theoretical
Volume 49
Number 11
Abstract We consider the problem of solving TAP mean field equations by iteration for Ising models with coupling matrices that are drawn at random from general invariant ensembles. We develop an analysis of iterative algorithms using a dynamical functional approach that in the thermodynamic limit yields an effective dynamics of a single variable trajectory. Our main novel contribution is the expression for the implicit memory term of the dynamics for general invariant ensembles. By subtracting these terms, that depend on magnetizations at previous time steps, the implicit memory terms cancel making the iteration dependent on a Gaussian distributed field only. The TAP magnetizations are stable fixed points if a de Almeida–Thouless stability criterion is fulfilled. We illustrate our method explicitly for coupling matrices drawn from the random orthogonal ensemble.
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