### Resumé

Originalsprog | Engelsk |
---|---|

Artikelnummer | 114002 |

Tidsskrift | Journal of Physics A: Mathematical and Theoretical |

Vol/bind | 49 |

Udgave nummer | 11 |

Antal sider | 24 |

ISSN | 1751-8113 |

DOI | |

Status | Udgivet - 8 feb. 2016 |

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*Journal of Physics A: Mathematical and Theoretical*,

*49*(11), [114002]. https://doi.org/10.1088/1751-8113/49/11/114002

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*Journal of Physics A: Mathematical and Theoretical*, bind 49, nr. 11, 114002. https://doi.org/10.1088/1751-8113/49/11/114002

**A Theory of Solving TAP Equations for Ising Models with General Invariant Random Matrices.** / Opper, Manfred ; Cakmak, Burak; Winther, Ole.

Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › peer review

TY - JOUR

T1 - A Theory of Solving TAP Equations for Ising Models with General Invariant Random Matrices

AU - Opper, Manfred

AU - Cakmak, Burak

AU - Winther, Ole

PY - 2016/2/8

Y1 - 2016/2/8

N2 - We consider the problem of solving TAP mean eld 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 eective 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 eld only. The TAP magnetizations are stable xed points if an AT stability criterion is fullled. We illustrate our method explicitly for coupling matrices drawn from the random orthogonal ensemble.

AB - We consider the problem of solving TAP mean eld 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 eective 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 eld only. The TAP magnetizations are stable xed points if an AT stability criterion is fullled. We illustrate our method explicitly for coupling matrices drawn from the random orthogonal ensemble.

KW - Dynamical Functional Theory

KW - Iterative Convergent Algorithms

KW - Random Matrices

KW - TAP Equations

KW - Free Probability

U2 - 10.1088/1751-8113/49/11/114002

DO - 10.1088/1751-8113/49/11/114002

M3 - Journal article

VL - 49

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 11

M1 - 114002

ER -