Journal of Statistical Mechanics: Theory and Experiment
Volume 2007
August 2007
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Andrea Montanari and Antoine Sinton J. Stat. Mech. (2007) P08004 doi:10.1088/1742-5468/2007/08/P08004
A simple one dimensional glassy Kac model
Andrea Montanari1 and Antoine Sinton2
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Figure 1. Factor graph representation of a portion of a KacXOR formula with
k = 3 and R = 3. Empty circles correspond to variables (columns of  ) and filled squares to equations (rows of ). |
Figure 2. Ground state entropy density in the thermodynamic limit  , cf equation (10), for the standard and improved ensembles. Here
k = 3 and γ = 0.4. The horizontal line marks the  limit  (γ) = 0.6. |
Figure 3. Subtracted entropy density  for various values of L and R = 25 constant. We also plot the result of an  extrapolation and the analytical mean field prediction
(0)(γ)–1 + γ (continuous line). |
Figure 4. Subtracted entropy density in the thermodynamic limit:  for various values of R, together with the mean field prediction
(0)(γ)–1 + γ (continuous line). |
| Figure 5. Ground state entropy density R(γ), extrapolated to the thermodynamic limit, versus the inverse interaction range
1/(2R + 1) for various values of γ. Straight lines correspond to the analytic prediction, cf equation (17). |
Figure 6. Subtracted entropy density as a function of γ for several at  fixed. The continuous line corresponds to the analytical prediction  in the limit. |
Figure 7. Correlation  between the boundary of a box of size  and a point in its interior (at distance
n = zR from the centre). In the right frames: blow-up of the region near the boundary. The continuous line (partially hidden by data points) corresponds to the analytic prediction obtained by solving equation (21). |
| Figure 8. Point-to-set correlation length in units of the interaction range
R. The continuous line corresponds to the analytic prediction for and diverges at the glass transition
γs(k = 3) ≈ 0.917 935. |
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