Mean Field Dilute Ferromagnet: High Temperature and Zero Temperature Behavior

We study the mean field dilute model of a ferromagnet. We find and prove an expression for the free energy density at high temperature, and at temperature zero. We find the critical line of the model, separating the phase with zero magnetization from the phase with symmetry breaking. We also compute exactly the entropy at temperature zero, which is strictly positive. The physical behavior at temperature zero is very interesting and related to infinite dimensional percolation, and suggests possible behaviors at generic low temperatures. Lastly, we provide a complete solution for a (partially) annealed model. Our results hold both for the Poisson and the Bernoulli versions of the model.

[1]  S. Ross.  A random graph , 1981 .

[2]  Kanter,et al.  Mean-field theory of spin-glasses with finite coordination number. , 1987, Physical review letters.

[3]  V. Gayrard,et al.  The thermodynamics of the Curie-Weiss model with random couplings , 1993 .

[4]  F. Guerra.  ABOUT THE OVERLAP DISTRIBUTION IN MEAN FIELD SPIN GLASS MODELS , 1996, 1212.2919.

[5]  Svante Janson,et al.  Random graphs , 2000, Wiley-Interscience series in discrete mathematics and optimization.

[6]  Svante Janson,et al.  Random graphs , 2000, ZOR Methods Model. Oper. Res..

[7]  F. Guerra,et al.  The Thermodynamic Limit in Mean Field Spin Glass Models , 2002, cond-mat/0204280.

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

[9]  M. Talagrand,et al.  Spin Glasses: A Challenge for Mathematicians , 2003 .

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

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

[12]  F. Guerra,et al.  The High Temperature Region of the Viana–Bray Diluted Spin Glass Model , 2003, cond-mat/0302401.

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

[14]  F. Guerra.  MATHEMATICAL ASPECTS OF MEAN FIELD SPIN GLASS THEORY , 2004, cond-mat/0410435.

[15]  Random Multi-Overlap Structures and Cavity Fields in Diluted Spin Glasses , 2004, cond-mat/0403506.

[16]  F. Guerra.  Spin Glasses , 2005, cond-mat/0507581.

[17]  S. Salinas,et al.  Replica-symmetric solutions of a dilute Ising ferromagnet in a random field , 2005 .

[18]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[19]  F. Guerra.  Course 5 – An Introduction to Mean Field Spin Glas Theory: Methods and Results , 2006 .

[20]  Stability properties and probability distributions of multi-overlaps in dilute spin glasses , 2006, cond-mat/0612041.

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

[22]  M. Ostilli.  Ising spin glass models versus Ising models: an effective mapping at high temperature: I. General result , 2006, cond-mat/0607498.

[23]  Andrea Montanari,et al.  Reconstruction for Models on Random Graphs , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[24]  A. Gerschcnfeld,et al.  Reconstruction for Models on Random Graphs , 2007, FOCS 2007.

[25]  Sergey N. Dorogovtsev,et al.  Critical phenomena in complex networks , 2007, ArXiv.

[26]  Structural Properties of the Disordered Spherical and Other Mean Field Spin Models , 2006, cond-mat/0607616.

[27]  S. Starr,et al.  Some Observations for Mean-Field Spin Glass Models , 2007, 0707.0031.