The aim of this paper is to give a review of recent results of Yu. Kondratiev, Yu. Kozitsky, T. Pasurek and myself on the multiplicity of Gibbs states (phase transitions) in infinite spin systems on random configurations, and provide a `pedestrian' route following Georgii–Haggstrom approach to (closely related to phase transitions) percolation problems for a class of random point processes.
Alexei Daletskii, Percolations and phase transitions in a class of random spin systems, Methods Funct. Anal. Topology 21
(2015), no. 3, 225-236.
BibTex
@article {MFAT779,
AUTHOR = {Daletskii, Alexei},
TITLE = {Percolations and phase transitions in a class of random spin systems},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {21},
YEAR = {2015},
NUMBER = {3},
PAGES = {225-236},
ISSN = {1029-3531},
MRNUMBER = {MR3521693},
ZBLNUMBER = {06630269},
URL = {http://mfat.imath.kiev.ua/article/?id=779},
}
References
Leonid Bogachev, Alexei Daletskii, Gibbs cluster measures on configuration spaces, J. Funct. Anal. 264 (2013), no. 2, 508-550. MathSciNetCrossRef
Anton Bovier, Statistical mechanics of disordered systems, Cambridge University Press, Cambridge, 2006. MathSciNetCrossRef
Alexei Daletskii, Yuri Kondratiev, Yuri Kozitsky, Tanja Pasurek, Gibbs states on random configurations, J. Math. Phys. 55 (2014), no. 8, 083513, 17. MathSciNetCrossRef
Alexei Daletskii, Yuri Kondratiev, Yuri Kozitsky, Tanja Pasurek, A phase transition in a quenched amorphous ferromagnet, J. Stat. Phys. 156 (2014), no. 1, 156-176. MathSciNetCrossRef
D. J. Daley, D. Vere-Jones, An introduction to the theory of point processes. Vol. I, Springer-Verlag, New York, 2003. MathSciNet
D. J. Daley, D. Vere-Jones, An introduction to the theory of point processes. Vol. I, Springer-Verlag, New York, 2003. MathSciNet
Hans-Otto Georgii, Gibbs measures and phase transitions, Walter de Gruyter \& Co., Berlin, 1988. MathSciNetCrossRef
H.-O. Georgii, O. Haggstrom, Phase transition in continuum Potts models, Comm. Math. Phys. 181 (1996), no. 2, 507-528. MathSciNet
Hans-Otto Georgii, Olle Haggstrom, Christian Maes, The random geometry of equilibrium phases, in: Phase transitions and critical phenomena, Vol. 18, Academic Press, San Diego, CA, 2001. MathSciNetCrossRef
Olle Haggstrom, Markov random fields and percolation on general graphs, Adv. in Appl. Probab. 32 (2000), no. 1, 39-66. MathSciNetCrossRef
Olav Kallenberg, Random measures, Akademie-Verlag, Berlin; Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1983. MathSciNet
Y. Kondratiev, Y. Kozitsky, T. Pasurek, Gibbs measures of disordered lattice systems with unbounded spins, Markov Process. Related Fields 18 (2012), no. 3, 553-582. MathSciNet
Yu. G. Kondratiev, O. V. Kutoviy, On the metrical properties of the configuration space, Math. Nachr. 279 (2006), no. 7, 774-783. MathSciNetCrossRef
Ronald Meester, Rahul Roy, Continuum percolation, Cambridge University Press, Cambridge, 1996. MathSciNetCrossRef
Mathew D. Penrose, On a continuum percolation model, Adv. in Appl. Probab. 23 (1991), no. 3, 536-556. MathSciNetCrossRef
Chris Preston, Random fields, Springer-Verlag, Berlin-New York, 1976. MathSciNet
Sidney I. Resnick, Extreme values, regular variation, and point processes, Springer-Verlag, New York, 1987. MathSciNetCrossRef
Silvano Romano, Valentin A. Zagrebnov, Orientational ordering transition in a continuous-spin ferrofluid, Phys. A 253 (1998), no. 1-4, 483-497. MathSciNetCrossRef
D. Ruelle, Superstable interactions in classical statistical mechanics, Comm. Math. Phys. 18 (1970), 127-159. MathSciNet
Daniel Raymond Wells, Some Moment Inequalities and a Result on Multivariable Unimodality, Thesis, Indiana University, 1977. MathSciNet