Structure of Finite-RSB Asymptotic Gibbs Measures in the Diluted Spin Glass Models

We suggest a possible approach to proving the Mézard–Parisi formula for the free energy in the diluted spin glass models, such as diluted K-spin or random K-sat model at any positive temperature. In the main contribution of the paper, we show that a certain small modification of the Hamiltonian in any of these models forces all finite-RSB asymptotic Gibbs measures in the sense of the overlaps to satisfy the Mézard–Parisi ansatz for the distribution of spins. Unfortunately, what is still missing is a description of the general full-RSB asymptotic Gibbs measures. If one could show that the general case can be approximated by finite-RSB case in the right sense then one could a posteriori remove the small modification of the Hamiltonian to recover the Mézard–Parisi formula for the original model.

[1]  Dmitry Panchenko,et al.  Structure of $$1$$1-RSB Asymptotic Gibbs Measures in the Diluted $$p$$p-Spin Models , 2013, 1308.1944.

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

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

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

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

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

[7]  M. Mézard,et al.  The Bethe lattice spin glass revisited , 2000, cond-mat/0009418.

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

[9]  M. Aizenman,et al.  On the Stability of the Quenched State in Mean-Field Spin-Glass Models , 1997, cond-mat/9712129.

[10]  Michele Leone,et al.  Replica Bounds for Optimization Problems and Diluted Spin Systems , 2002 .

[11]  Tim Austin,et al.  Exchangeable random measures , 2013, 1302.2116.

[12]  Dmitry Panchenko,et al.  A hierarchical version of the de Finetti and Aldous-Hoover representations , 2013, 1301.1259.

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

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

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

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

[17]  M. Talagrand,et al.  Bounds for diluted mean-fields spin glass models , 2004, math/0405357.

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

[19]  Dmitry Panchenko,et al.  Hierarchical exchangeability of pure states in mean field spin glass models , 2013, 1307.2207.

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

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

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

[23]  Olav Kallenberg,et al.  On the representation theorem for exchangeable arrays , 1989 .

[24]  D. Aldous.  Representations for partially exchangeable arrays of random variables , 1981 .

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

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