Summary
It has been known for some time that one-dimensional continuous Gibbs systems specified by long range hard core pair potentials exhibit no phase transition, i.e. are uniquely determined by the potential, provided the tail of the potential has a finite first moment. The present paper deals with long range potentials which have no hard core but diverge appropriately at zero, and proves that they admit unique “regular” Gibbs systems at arbitrary temperature and activity level. Regularity here means that the expected numbers of particles in intervals of equal length are uniformly bounded.
References
DeMasi, A.: One-dimensional DLR invariant measures are regular. Comm. Math. Phys. 67, 43–50 (1979)
Dobrushin, R.L.: Analyticity of correlation functions in one-dimensional classical systems with slowly decreasing potentials. Comm. Math. Phys. 32, 269–289 (1973)
Dobrushin, R.L.: Conditions for the absence of phase transitions in one-dimensional classical systems (in Russian). Mat. Sb. (N.S.) 93 (135), 29–49 (1974)
Gallavotti, G., Miracle-Sole, S.: Absence of phase transitions in hard-core one-dimensional systems with long-range interactions. J. Math. Phys. 11, 147–154 (1969)
Papangelou, F.: On the absence of phase transition in one-dimensional random fields. (I) Sufficient conditions. Z. Wahrscheinlichkeitstheor. Verw. Geb. 67, 239–254 (1984)
Papangelou, F.: On the absence of phase transition in one-dimensional random fields. (II) Superstable spin systems. Z. Wahrscheinlichkeitstheor. Verw. Geb. 67, 255–263 (1984)
Preston, C.: Random fields. Lect. Notes Math. 534. Berlin Heidelberg New York: Springer 1976
Ruelle, D.: Superstable interactions in classical statistical mechanics. Comm. Math. Phys. 18, 127–159 (1970)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Papangelou, F. On the absence of phase transition in continuous one-dimensional Gibbs systems with no hard core. Probab. Th. Rel. Fields 74, 485–496 (1987). https://doi.org/10.1007/BF00363511
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00363511
Keywords
- Phase Transition
- Stochastic Process
- Probability Theory
- Long Range
- Statistical Theory