Abstract
We develop a method to study extremality of the disordered state ℙβ for the Ising model on a general countable tree T. It is shown that the tail σ- field is ℙβ -trivial as soon as β is less than the spin glass critical inverse temperature \( \beta _c^{SG} \), which is determined from the relation tanh\( (\beta _c^{SG} ) = {1 \mathord{\left/ {\vphantom {1 {\sqrt {br(T)} }}} \right. \kern-\nulldelimiterspace} {\sqrt {br(T)}}} \). The method is based on the FK representation of ferromagnetic systems and recursive estimates on conditional expectations of the spin at the root. Similar estimates in the context of the bit reconstruction problem on general trees were originally obtained in [EKPS] using different methods.
Résumé
On développe une méthode pour étudier l’extrémalité de l’état désordonné ℙβ pour le modèle d’Ising sur un arbre dénombrable général T. On montre que la tribu de queue est triviale dès que β est inférieur à\( \beta _c^{SG} \), (inverse de la température critique du verre de spin) déterminé par la relation:
La méthode est fondée sur la représentation FK des systèmes ferromagnétiques et sur des estimées récursives des espérances conditionnelles du spin à la racine. Des estimées similaires dans le contexte du problème de la reconstruction du spin sur des arbres généraux ont été obtenus d’abord dans [EKPS] par W. Evans, C.Kenyon, Y.Peres, L.Schulman en utilisant des méthodes différentes.
The work was partially supported by the NSF grant DMS 9504513 and by the Commission of the European Union under the contract CHRX-CT93-0411
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© 1996 Birkhäuser Verlag Basel
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Ioffe, D. (1996). Extremality of the Disordered State for the Ising Model on General Trees. In: Chauvin, B., Cohen, S., Rouault, A. (eds) Trees. Progress in Probability, vol 40. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9037-3_1
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DOI: https://doi.org/10.1007/978-3-0348-9037-3_1
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