FREE ENERGY IN THE POTTS SPIN GLASS 1

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 will consider these problems in the subsequent paper and illustrate the main new idea on the technically more transparent case of the Potts spin glass.

[1]  Subhabrata Sen,et al.  Optimization on sparse random hypergraphs and spin glasses , 2016, Random Struct. Algorithms.

[2]  A connection between MAX $κ$-CUT and the inhomogeneous Potts spin glass in the large degree limit , 2017 .

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

[4]  Andrea Montanari,et al.  Extremal Cuts of Sparse Random Graphs , 2015, ArXiv.

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

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

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

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

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

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

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

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

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

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

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

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

[17]  D. Panchenko.  Free Energy in the Generalized Sherrington-Kirkpatrick Mean Field Model , 2004, math/0405362.

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

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

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

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

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

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

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

[25]  G. Parisi,et al.  On the static and dynamical transition in the mean-field Potts glass , 1994, cond-mat/9410093.

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

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

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

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

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

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

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

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

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

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

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

[37]  Ihrer Grenzgebiete,et al.  Ergebnisse der Mathematik und ihrer Grenzgebiete , 1975, Sums of Independent Random Variables.

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