The Parisi formula for mixed $p$-spin models

The Parisi formula for the free energy in the Sherrington-Kirkpatrick and mixed $p$-spin models for even $p\geq2$ was proved in the seminal work of Michel Talagrand [Ann. of Math. (2) 163 (2006) 221-263]. In this paper we prove the Parisi formula for general mixed $p$-spin models which also include $p$-spin interactions for odd $p$. Most of the ideas used in the paper are well known and can now be combined following a recent proof of the Parisi ultrametricity conjecture in [Ann. of Math. (2) 177 (2013) 383-393].

[1]  D. Ruelle.  A mathematical reformulation of Derrida's REM and GREM , 1987 .

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

[3]  V. Sudakov,et al.  Gram-de finetti matrices , 1984 .

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

[5]  F. Guerra,et al.  Ju l 1 99 8 General properties of overlap probability distributions in disordered spin systems , 1998 .

[6]  Michel Talagrand,et al.  Construction of pure states in mean field models for spin glasses , 2010 .

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

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

[9]  F. Rosati,et al.  Some exact results on the ultrametric overlap distribution in mean field spin glass models (I) , 2000, cond-mat/0002342.

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

[11]  M. Aizenman,et al.  Extended variational principle for the Sherrington-Kirkpatrick spin-glass model , 2003 .

[12]  E. Bolthausen,et al.  On Ruelle's Probability Cascades and an Abstract Cavity Method , 1998 .

[13]  S. Chatterjee,et al.  Random overlap structures: properties and applications to spin glasses , 2010, 1011.1823.

[14]  On the Dovbysh-Sudakov representation result , 2009, 0905.1524.

[15]  M. Mézard,et al.  Nature of the Spin-Glass Phase , 1984 .

[16]  D. Panchenko.  PR ] 4 O ct 2 00 8 A connection between Ghirlanda-Guerra identities and ultrametricity , 2008 .

[17]  D. Panchenko.  Ghirlanda-Guerra identities and ultrametricity: An elementary proof in the discrete case , 2011, 1106.3984.

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

[19]  Spin Glasses: A Challenge for Mathematicians.: A Challenge for Mathematicians. , 2005 .

[20]  D. Panchenko.  The Ghirlanda-Guerra identities for mixed p-spin model , 2010, 1002.2190.

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

[22]  D. Panchenko.  A note on Talagrand's positivity principle , 2007, 0708.2453.

[23]  M. Talagrand.  On Guerra's broken replica-symmetry bound , 2003 .

[24]  M. Mézard,et al.  Replica symmetry breaking and the nature of the spin glass phase , 1984 .

[25]  M. Aizenman,et al.  An Extended Variational Principle for the SK Spin-Glass Model , 2003, cond-mat/0306386.

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

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