Free energy of bipartite spherical Sherrington–Kirkpatrick model

We consider the free energy of the bipartite spherical Sherrington--Kirkpatrick model. We find the critical temperature and prove the limiting free energy for all non-critical temperature. We also show that the law of the fluctuation of the free energy converges to the Gaussian distribution when the temperature is above the critical temperature, and to the GOE Tracy--Widom distribution when the temperature is below the critical temperature. The result is universal, and the analysis is applicable to a more general setting including the case where the disorders are non-identically distributed.

[1]  D. Thouless,et al.  Spherical Model of a Spin-Glass , 1976 .

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

[3]  D. Ruelle,et al.  Some rigorous results on the Sherrington-Kirkpatrick spin glass model , 1987 .

[4]  J. Fröhlich,et al.  Some comments on the Sherrington-Kirkpatrick model of spin glasses , 1987 .

[5]  A. Crisanti,et al.  The sphericalp-spin interaction spin glass model: the statics , 1992 .

[6]  J. W. Silverstein,et al.  Analysis of the limiting spectral distribution of large dimensional random matrices , 1995 .

[7]  F. Comets,et al.  The Sherrington-Kirkpatrick model of spin glasses and stochastic calculus: The high temperature case , 1995 .

[8]  A. Bovier,et al.  FLUCTUATIONS OF THE FREE ENERGY IN THE REM AND THE P-SPIN SK MODELS , 2000, cond-mat/0007175.

[9]  I. Johnstone.  On the distribution of the largest eigenvalue in principal components analysis , 2001 .

[10]  A. Soshnikov.  A Note on Universality of the Distribution of the Largest Eigenvalues in Certain Sample Covariance Matrices , 2001, math/0104113.

[11]  Noureddine El Karoui.  A rate of convergence result for the largest eigenvalue of complex white Wishart matrices , 2004, math/0409610.

[12]  Z. Bai,et al.  CLT for linear spectral statistics of large dimensional sample covariance matrices with dependent data , 2017, Statistical Papers.

[13]  M. Stephanov,et al.  Random Matrices , 2005, hep-ph/0509286.

[14]  M. Talagrand.  Free energy of the spherical mean field model , 2006 .

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

[16]  J. W. Silverstein,et al.  Spectral Analysis of Large Dimensional Random Matrices , 2009 .

[17]  L. Pastur,et al.  CENTRAL LIMIT THEOREM FOR LINEAR EIGENVALUE STATISTICS OF RANDOM MATRICES WITH INDEPENDENT ENTRIES , 2008, 0809.4698.

[18]  Zhidong Bai,et al.  Functional CLT for sample covariance matrices , 2010, 1011.5729.

[19]  H. Yau,et al.  Rigidity of eigenvalues of generalized Wigner matrices , 2010, 1007.4652.

[20]  P. Forrester.  Log-Gases and Random Matrices , 2010 .

[21]  A. Barra,et al.  Equilibrium statistical mechanics of bipartite spin systems , 2010, 1012.1261.

[22]  N. Pillai,et al.  Universality of covariance matrices , 2011, 1110.2501.

[23]  Oskari Ajanki,et al.  Local semicircle law with imprimitive variance matrix , 2013, 1311.2016.

[24]  Wang Zhou,et al.  Universality for the largest eigenvalue of sample covariance matrices with general population , 2013, 1304.5690.

[25]  H. Yau,et al.  Spectral statistics of Erdős–Rényi graphs I: Local semicircle law , 2011, 1103.1919.

[26]  A. Barra,et al.  Multi-Species Mean Field Spin Glasses. Rigorous Results , 2013, 1307.5154.

[27]  D. Panchenko.  The free energy in a multi-species Sherrington-Kirkpatrick model , 2013, 1310.6679.

[28]  A. Barra,et al.  Mean field bipartite spin models treated with mechanical techniques , 2013, 1310.5901.

[29]  Antonio Auffinger,et al.  Free Energy and Complexity of Spherical Bipartite Models , 2014, 1405.2321.

[30]  J. Lee,et al.  Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population , 2014, 1409.4979.

[31]  H. Yau,et al.  Edge Universality of Beta Ensembles , 2013, 1306.5728.

[32]  Jun Yin,et al.  Anisotropic local laws for random matrices , 2014, 1410.3516.

[33]  F. Guerra.  Spontaneous Replica Symmetry Breaking and Interpolation Methods for Complex Statistical Mechanics Systems , 2015 .

[34]  O. Zeitouni,et al.  The extremal process of critical points of the pure p-spin spherical spin glass model , 2015, 1509.03098.

[35]  Wei-Kuo Chen,et al.  Parisi Formula, Disorder Chaos and Fluctuation for the Ground State Energy in the Spherical Mixed p-Spin Models , 2015, 1512.08492.

[36]  J. Baik,et al.  Fluctuations of the Free Energy of the Spherical Sherrington–Kirkpatrick Model , 2015, Journal of Statistical Physics.

[37]  Wei-Kuo Chen,et al.  Fluctuations of the free energy in the mixed p-spin models with external field , 2015, 1509.07071.

[38]  J. Baik,et al.  Fluctuations of the Free Energy of the Spherical Sherrington–Kirkpatrick Model with Ferromagnetic Interaction , 2016, Annales Henri Poincaré.

[39]  Jianfeng Yao,et al.  GAUSSIAN FLUCTUATIONS FOR LINEAR SPECTRAL STATISTICS OF LARGE RANDOM COVARIANCE MATRICES By , 2016 .

[40]  Philippe Sosoe,et al.  Fluctuations of the overlap at low temperature in the 2-spin spherical SK model , 2019, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques.

[41]  Central limit theorem near the critical temperature for the overlap in the 2-spin spherical SK model , 2018, Journal of Mathematical Physics.