Free energy in the Potts spin glass

We study the Potts spin glass model, which generalizes the Sherrington-Kirkpatrick model to the case when spins take more than two values but their interactions are counted only if the spins are equal. We obtain the analogue of the Parisi variational formula for the free energy, with the order parameter now given by a monotone path in the set of positive-semidefinite matrices. The main idea of the paper is a novel synchronization mechanism for blocks of overlaps. This mechanism can be used to solve a more general version of the Sherrington-Kirkpatrick model with vector spins interacting through their scalar product, which includes the Potts spin glass as a special case. As another example of application, one can show that Talagrand's bound for multiple copies of the mixed $p$-spin model with constrained overlaps is asymptotically sharp. We consider these problems in the subsequent paper, arXiv:1512.04441, and illustrate the main new idea on the technically more transparent case of the Potts spin glass.

[1]  Kanter,et al.  Mean-field theory of the Potts glass. , 1985, Physical review letters.

[2]  Paul Erdös,et al.  A Ramsey-type theorem for bipartite graphs , 2000 .

[3]  R. Tyrrell Rockafellar,et al.  Convex Analysis , 1970, Princeton Landmarks in Mathematics and Physics.

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

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

[6]  D. Sherrington,et al.  Crystal field effects in a general S Ising spin glass , 1977 .

[7]  D. Sherrington,et al.  Novel non-ergodicity in the Potts spin glass , 1983 .

[8]  M. Talagrand.  Mean Field Models for Spin Glasses: Some Obnoxious Problems , 2007 .

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

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

[11]  On the static and dynamical transition in the mean-field Potts glass , 1994, cond-mat/9410093.

[12]  M. Talagrand.  Mean Field Models for Spin Glasses , 2011 .

[13]  A note on the free energy of the coupled system in the Sherrington-Kirkpatrick model , 2004, math/0405359.

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

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

[16]  H. Nishimori,et al.  Gauge-invariant frustrated Potts spin-glass , 1983 .

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

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

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

[20]  On the distribution of the overlaps at given disorder , 2004 .

[21]  Wei-Kuo Chen.  The Aizenman-Sims-Starr scheme and Parisi formula for mixed p-spin spherical models , 2012, 1204.5115.

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

[23]  G. Parisi,et al.  Ultrametricity in an inhomogeneous simplest spin glass model , 1992 .

[24]  Giorgio Parisi,et al.  Glassy Potts model: A disordered Potts model without a ferromagnetic phase , 1999 .

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

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

[27]  G. Parisi,et al.  Free-energy cost for ultrametricity violations in spin glasses , 1993 .

[28]  G. Parisi,et al.  Dynamical critical exponents for the mean-field Potts glass. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[30]  D. Sherrington.  Physics and complexity , 2009, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

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

[32]  D. Sherrington,et al.  The curious case of the Potts spin glass , 1983 .

[33]  F. Guerra,et al.  General properties of overlap probability distributions in disordered spin systems. Towards Parisi ultrametricity , 1998, cond-mat/9807333.

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

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

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