Exact Solution of the Gauge Symmetric p-Spin Glass Model on a Complete Graph

We consider a gauge symmetric version of the p-spin glass model on a complete graph. The gauge symmetry guarantees the absence of replica symmetry breaking and allows to fully use the interpolation scheme of Guerra (Fields Inst. Commun. 30:161, 2001) to rigorously compute the free energy. In the case of pairwise interactions (p=2), where we have a gauge symmetric version of the Sherrington-Kirkpatrick model, we get the free energy and magnetization for all values of external parameters. Our analysis also works for even p≥4 except in a range of parameters surrounding the phase transition line, and for odd p≥3 in a more restricted region. We also obtain concentration estimates for the magnetization and overlap parameter that play a crucial role in the proofs for odd p and justify the absence of replica symmetry breaking. Our initial motivation for considering this model came from problems related to communication over a noisy channel, and is briefly explained.

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

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

[3]  B. Derrida.  Random-energy model: An exactly solvable model of disordered systems , 1981 .

[4]  M. Mézard,et al.  The simplest spin glass , 1984 .

[5]  Nicolas Sourlas,et al.  Spin-glass models as error-correcting codes , 1989, Nature.

[6]  H. Nishimori.  Statistical Physics of Spin Glasses and Information Processing , 2001 .

[7]  西森 秀稔.  Statistical physics of spin glasses and information processing : an introduction , 2001 .

[8]  T. Dorlas,et al.  Large Deviations and the Random Energy Model , 2001 .

[9]  Pisa,et al.  Quadratic replica coupling in the Sherrington-Kirkpatrick mean field spin glass model , 2002, cond-mat/0201091.

[10]  Rome,et al.  The infinite volume limit in generalized mean field disordered models , 2002 .

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

[12]  M. Talagrand.  Spin glasses : a challenge for mathematicians : cavity and mean field models , 2003 .

[13]  Michel Talagrand,et al.  The generalized Parisi formula , 2003 .

[14]  Andrea Montanari,et al.  Tight bounds for LDPC and LDGM codes under MAP decoding , 2004, IEEE Transactions on Information Theory.

[15]  Nicolas Macris,et al.  Sharp Bounds for MAP Decoding of General Irregular LDPC Codes , 2006, 2006 IEEE International Symposium on Information Theory.

[16]  Nicolas Macris,et al.  Exact solution of a p-spin model and its relationship to error correcting codes , 2006, 2006 IEEE International Symposium on Information Theory.

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

[18]  Nicolas Macris,et al.  Exact solution for the conditional entropy of Poissonian LDPC codes over the Binary Erasure Channel , 2007, 2007 IEEE International Symposium on Information Theory.

[19]  Rüdiger L. Urbanke,et al.  Modern Coding Theory , 2008 .

[20]  Nicolas Macris,et al.  Sharp Bounds for Optimal Decoding of Low-Density Parity-Check Codes , 2008, IEEE Transactions on Information Theory.

[21]  Nicolas Macris,et al.  Tight Bounds on the Capacity of Binary Input Random CDMA Systems , 2008, IEEE Transactions on Information Theory.