On the TAP Free Energy in the Mixed p-Spin Models

Thouless et al. (Phys Mag 35(3):593–601, 1977), derived a representation for the free energy of the Sherrington–Kirkpatrick model, called the TAP free energy, written as the difference of the energy and entropy on the extended configuration space of local magnetizations with an Onsager correction term. In the setting of mixed p-spin models with Ising spins, we prove that the free energy can indeed be written as the supremum of the TAP free energy over the space of local magnetizations whose Edwards–Anderson order parameter (self-overlap) is to the right of the support of the Parisi measure. Furthermore, for generic mixed p-spin models, we prove that the free energy is equal to the TAP free energy evaluated on the local magnetization of any pure state.

[1]  M. Talagrand.  The parisi formula , 2006 .

[2]  M. Mézard,et al.  Spin Glass Theory and Beyond , 1987 .

[3]  Giorgio Parisi,et al.  Infinite Number of Order Parameters for Spin-Glasses , 1979 .

[4]  Giorgio Parisi,et al.  Order parameter for spin-glasses , 1983 .

[5]  F. Guerra.  Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model , 2002, cond-mat/0205123.

[6]  Wei-Kuo Chen,et al.  The Parisi Formula has a Unique Minimizer , 2014, 1402.5132.

[7]  Dmitry Panchenko,et al.  Free energy in the Potts spin glass , 2015, 1512.00370.

[8]  Tim Austin.  Mean field models for spin glasses , 2012 .

[9]  D. Panchenko.  SPIN GLASS MODELS FROM THE POINT OF VIEW OF SPIN DISTRIBUTIONS , 2010, 1005.2720.

[10]  Ronen Eldan,et al.  Decomposition of mean-field Gibbs distributions into product measures , 2017 .

[11]  Dmitry Panchenko,et al.  Free energy in the mixed p-spin models with vector spins , 2015, 1512.04441.

[12]  Aukosh Jagannath,et al.  Approximate Ultrametricity for Random Measures and Applications to Spin Glasses , 2014, 1412.7076.

[13]  M. Talagrand.  Advanced replica-symmetry and low temperature , 2011 .

[14]  Tamar Frankel.  [The theory and the practice...]. , 2001, Tijdschrift voor diergeneeskunde.

[15]  Tim Austin.  Measure concentration and the weak Pinsker property , 2017, 1705.00302.

[16]  A P Young,et al.  Weighted averages and order parameters for the infinite range Ising spin glass , 1983 .

[17]  Ronen Eldan,et al.  Gaussian-width gradient complexity, reverse log-Sobolev inequalities and nonlinear large deviations , 2016, Geometric and Functional Analysis.

[18]  S. Chatterjee.  Spin glasses and Stein’s method , 2007, 0706.3500.

[19]  R. Palmer,et al.  Solution of 'Solvable model of a spin glass' , 1977 .

[20]  P. Anderson.  LECTURES ON AMORPHOUS SYSTEMS , 1984 .

[21]  Gábor Lugosi,et al.  Concentration Inequalities - A Nonasymptotic Theory of Independence , 2013, Concentration Inequalities.

[22]  Dmitry Panchenko,et al.  The Parisi ultrametricity conjecture , 2011, 1112.1003.

[23]  Wei-Kuo Chen,et al.  On the energy landscape of the mixed even p-spin model , 2016, 1609.04368.

[24]  D. Panchenko.  The Sherrington-Kirkpatrick Model , 2013 .

[25]  T. Plefka.  Convergence condition of the TAP equation for the infinite-ranged Ising spin glass model , 1982 .

[26]  Editors , 1986, Brain Research Bulletin.

[27]  Dmitry Panchenko.  Free Energy in the Generalized Sherrington-Kirkpatrick Mean Field Model , 2005 .

[28]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[29]  E. Bolthausen.  An Iterative Construction of Solutions of the TAP Equations for the Sherrington–Kirkpatrick Model , 2012, 1201.2891.

[30]  Wei-Kuo Chen,et al.  Variational representations for the Parisi functional and the two-dimensional Guerra-Talagrand bound , 2015, 1501.06635.

[31]  C. K. Ogden.  Basic by examples , 1933 .

[32]  S. Sen,et al.  On the unbalanced cut problem and the generalized Sherrington–Kirkpatrick model , 2017, 1707.09042.

[33]  Aukosh Jagannath,et al.  A Dynamic Programming Approach to the Parisi Functional , 2015, 1502.04398.

[34]  Wei-Kuo Chen,et al.  Parisi formula for the ground state energy in the mixed p-spin model , 2016, 1606.05335.

[35]  M. Aizenman,et al.  On the structure of quasi-stationary competing particle systems , 2007, 0709.2901.

[36]  S. Kirkpatrick,et al.  Solvable Model of a Spin-Glass , 1975 .

[37]  Giorgio Parisi,et al.  On the formal equivalence of the TAP and thermodynamic methods in the SK model , 2003 .

[38]  A. Bray,et al.  Weighted averages of TAP solutions and Parisi's q(x) , 1984 .

[39]  Antonio Auffinger,et al.  On properties of Parisi measures , 2013, 1303.3573.

[40]  G. Parisi.  A sequence of approximated solutions to the S-K model for spin glasses , 1980 .

[41]  Amir Dembo,et al.  Nonlinear large deviations , 2014, 1401.3495.

[42]  Dmitry Panchenko,et al.  The Parisi formula for mixed $p$-spin models , 2011, 1112.4409.

[43]  Antonio Auffinger,et al.  Thouless-Anderson-Palmer equations for conditional Gibbs measures in the generic p-spin glass model , 2016 .