publication . Article . Preprint . 2018

Entropic repulsion and lack of the $g$-measure property for Dyson models

Bissacot, Rodrigo; Endo, Eric O.; van Enter, Aernout C. D.; Le Ny, Arnaud;
Open Access English
  • Published: 01 Nov 2018
Abstract
We consider Dyson models, Ising models with slow polynomial decay, at low temperature and show that its Gibbs measures deep in the phase transition region are not $g$-measures. The main ingredient in the proof is the occurrence of an entropic repulsion effect, which follows from the mesoscopic stability of a (single-point) interface for these long-range models in the phase transition region.
Subjects
free text keywords: DIMENSIONAL ISING-MODEL, LATTICE MODELS, RUELLE OPERATOR, Mathematical Physics, Mathematics - Dynamical Systems, GLOBAL MARKOV PROPERTY, POTTS MODELS, To be checked by Faculty, Mathematics - Probability, GENERAL-THEORY, Condensed Matter - Statistical Mechanics, PHASE-TRANSITION, LONG-RANGE INTERACTIONS, COMPLETE CONNECTIONS, GIBBS MEASURES
68 references, page 1 of 5

[1] M. Aizenman, J. Chayes, L. Chayes, C. Newman. Discontinuity of the Magnetization in the One-Dimensional 1/ | x − y |2 Percolation, Ising and Potts Models. J. Stat. Phys. 50, no 1/2:1-40, 1988.

[2] H. Berbee. Chains with infinite connections: Uniqueness and Markov representation. Prob. Th. Rel. Fields. 76: 243-253, 1987. [OpenAIRE]

[3] N. Berger, C. Hoffman, V. Sidoravicius. Nonuniqueness for Specifications in l2+ǫ. arXiv:math/0312344, 2003. To appear in Erg. Th. Dyn. Syst., DOI: https://doi.org/10.1017/etds.2016.101, 2017.

[4] S. Berghout, R. Ferna´ndez, E. Verbitskiy. On the Relation between Gibbs and gmeasures. Preprint, 2017.

[5] R. Bowen. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. 2nd Edition (J.-R. Chazottes ed.), Springer Lecture Notes in Mathematics 470, 2008.

[6] M. Bramson, S. Kalikow. Non-Uniqueness in g-functions. Israel J. Math. 84 153-160, 1993.

[7] J. Bricmont, J. Lebowitz, C.-E. Pfister. On the Equivalence of Boundary Conditions. J. Stat. Phys. 21, No 5: 573-582, 1979.

[8] G. Brown, A.H. Dooley. Odometer Actions on G-measures. Erg. Th. Dyn. Syst. 11:279-307, 1991.

[9] G. Brown, A.H. Dooley. On G-measures and Product Measures. Erg. Th. Dyn. Syst. 18:95-107, 1998.

[10] M. Cassandro, P.A. Ferrari, I. Merola, E. Presutti. Geometry of Contours and Peierls Estimates in d = 1 Ising Models with Long Range Interactions. J. Math. Phys. 46(5), 0533305, 2005.

[11] M. Cassandro, I. Merola, P. Picco. Phase Separation for the Long Range Onedimensional Ising Model. J. Stat. Phys. 167, no 2: 351-382, 2017.

[12] M. Cassandro, I. Merola, P. Picco, U. Rozikov. One-Dimensional Ising Models with Long Range Interactions: Cluster Expansion, Phase-Separating Point. Comm. Math. Phys. 327:951-991, 2014 .

[13] M. Cassandro, E. Orlandi, P. Picco. Phase Transition in the 1D Random Field Ising Model with Long Range Interaction. Comm. Math. Phys. 288:731-744, 2009.

[14] L. Cioletti, A.O. Lopes. Interactions, Specifications, DLR Probabilities and the Ruelle Operator in the One-Dimensional Lattice. Preprint arXiv:1404.3232, 2014. [OpenAIRE]

[15] L. Cioletti, A.O. Lopes. Phase Transitions in One-Dimensional Translation Invariant Systems: a Ruelle Operator Approach. J. Stat. Phys. 159 (6):1424-1455, 2015.

68 references, page 1 of 5
Abstract
We consider Dyson models, Ising models with slow polynomial decay, at low temperature and show that its Gibbs measures deep in the phase transition region are not $g$-measures. The main ingredient in the proof is the occurrence of an entropic repulsion effect, which follows from the mesoscopic stability of a (single-point) interface for these long-range models in the phase transition region.
Subjects
free text keywords: DIMENSIONAL ISING-MODEL, LATTICE MODELS, RUELLE OPERATOR, Mathematical Physics, Mathematics - Dynamical Systems, GLOBAL MARKOV PROPERTY, POTTS MODELS, To be checked by Faculty, Mathematics - Probability, GENERAL-THEORY, Condensed Matter - Statistical Mechanics, PHASE-TRANSITION, LONG-RANGE INTERACTIONS, COMPLETE CONNECTIONS, GIBBS MEASURES
68 references, page 1 of 5

[1] M. Aizenman, J. Chayes, L. Chayes, C. Newman. Discontinuity of the Magnetization in the One-Dimensional 1/ | x − y |2 Percolation, Ising and Potts Models. J. Stat. Phys. 50, no 1/2:1-40, 1988.

[2] H. Berbee. Chains with infinite connections: Uniqueness and Markov representation. Prob. Th. Rel. Fields. 76: 243-253, 1987. [OpenAIRE]

[3] N. Berger, C. Hoffman, V. Sidoravicius. Nonuniqueness for Specifications in l2+ǫ. arXiv:math/0312344, 2003. To appear in Erg. Th. Dyn. Syst., DOI: https://doi.org/10.1017/etds.2016.101, 2017.

[4] S. Berghout, R. Ferna´ndez, E. Verbitskiy. On the Relation between Gibbs and gmeasures. Preprint, 2017.

[5] R. Bowen. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. 2nd Edition (J.-R. Chazottes ed.), Springer Lecture Notes in Mathematics 470, 2008.

[6] M. Bramson, S. Kalikow. Non-Uniqueness in g-functions. Israel J. Math. 84 153-160, 1993.

[7] J. Bricmont, J. Lebowitz, C.-E. Pfister. On the Equivalence of Boundary Conditions. J. Stat. Phys. 21, No 5: 573-582, 1979.

[8] G. Brown, A.H. Dooley. Odometer Actions on G-measures. Erg. Th. Dyn. Syst. 11:279-307, 1991.

[9] G. Brown, A.H. Dooley. On G-measures and Product Measures. Erg. Th. Dyn. Syst. 18:95-107, 1998.

[10] M. Cassandro, P.A. Ferrari, I. Merola, E. Presutti. Geometry of Contours and Peierls Estimates in d = 1 Ising Models with Long Range Interactions. J. Math. Phys. 46(5), 0533305, 2005.

[11] M. Cassandro, I. Merola, P. Picco. Phase Separation for the Long Range Onedimensional Ising Model. J. Stat. Phys. 167, no 2: 351-382, 2017.

[12] M. Cassandro, I. Merola, P. Picco, U. Rozikov. One-Dimensional Ising Models with Long Range Interactions: Cluster Expansion, Phase-Separating Point. Comm. Math. Phys. 327:951-991, 2014 .

[13] M. Cassandro, E. Orlandi, P. Picco. Phase Transition in the 1D Random Field Ising Model with Long Range Interaction. Comm. Math. Phys. 288:731-744, 2009.

[14] L. Cioletti, A.O. Lopes. Interactions, Specifications, DLR Probabilities and the Ruelle Operator in the One-Dimensional Lattice. Preprint arXiv:1404.3232, 2014. [OpenAIRE]

[15] L. Cioletti, A.O. Lopes. Phase Transitions in One-Dimensional Translation Invariant Systems: a Ruelle Operator Approach. J. Stat. Phys. 159 (6):1424-1455, 2015.

68 references, page 1 of 5
Powered by OpenAIRE Open Research Graph
Any information missing or wrong?Report an Issue
publication . Article . Preprint . 2018

Entropic repulsion and lack of the $g$-measure property for Dyson models

Bissacot, Rodrigo; Endo, Eric O.; van Enter, Aernout C. D.; Le Ny, Arnaud;