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Flatness Is a Criterion for Selection of Maximizing Measures

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Abstract

For the one-dimensional classical spin system, each spin being able to get Np+1 values, and for a non-positive potential, locally proportional to the distance to one of N disjoint configurations set {(j−1)p+1,…,jp}ℤ, we prove that the equilibrium state converges as the temperature goes to 0.

The main result is that the limit is a convex combination of the two ergodic measures with maximal entropy among maximizing measures and whose supports are the two shifts where the potential is the flattest.

In particular, this is a hint to solve the open problem of selection, and this indicates that flatness is probably a/the criterion for selection as it was conjectured by A.O. Lopes.

As a by product we get convergence of the eigenfunction at the log-scale to a unique calibrated subaction.

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Notes

  1. β is the inverse of the temperature, thus β→+∞ means Temp→0, which means we freeze the system.

  2. Note that these Bernoulli shifts are empty interior compact sets.

  3. We emphasize that κ can be assume to be smaller than 4 if β is chosen sufficiently big.

  4. These are different from the truncated sums defined above.

  5. Note that \(\gamma=\alpha_{1}\frac{\theta}{1-\theta }+\alpha\) and F 1 behaves like \(e^{-I_{1}}\frac{1}{\beta^{r}g(\beta)}e^{(\gamma -\alpha_{1}\frac{\theta}{1-\theta} )\beta }\).

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  1. Laboratoire de Mathématiques de Bretagne Atlantique-UMR 6205, Université de Brest, 6, rue Victor Le Gorgeu, 29285, Brest, France

    Renaud Leplaideur

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  1. Renaud Leplaideur
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Correspondence to Renaud Leplaideur.

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Part of this research was supported by DynEurBraz IRSES FP7 230844 & ANR Project WeakKam.

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Leplaideur, R. Flatness Is a Criterion for Selection of Maximizing Measures. J Stat Phys 147, 728–757 (2012). https://doi.org/10.1007/s10955-012-0497-7

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