# American Institute of Mathematical Sciences

December  2017, 37(12): 6139-6152. doi: 10.3934/dcds.2017264

## Interactions, specifications, DLR probabilities and the Ruelle operator in the one-dimensional lattice

 1 Departamento de Matemática -UnB, Brasília, DF 70910-900, Brazil 2 Departamento de Matemática -UFRGS, Porto Alegre, RS 91509-900, Brazil

* Corresponding author: Artur O. Lopes

Received  November 2016 Revised  July 2017 Published  August 2017

Fund Project: The authors are partially supported by CNPq and BREUDS

In this paper, we describe several different meanings for the concept of Gibbs measure on the lattice $\mathbb{N}$ in the context of finite alphabets (or state space). We compare and analyze these ''in principle" distinct notions: DLR-Gibbs measures, Thermodynamic Limit and eigenprobabilities for the dual of the Ruelle operator (also called conformal measures).

Among other things we extended the classical notion of a Gibbsian specification on $\mathbb{N}$ in such way that the similarity of many results in Statistical Mechanics and Dynamical System becomes apparent. One of our main result claims that the construction of the conformal Measures in Dynamical Systems for Walters potentials, using the Ruelle operator, can be formulated in terms of Specification. We also describe the Ising model, with $1/r^{2+\varepsilon}$ interaction energy, in the Thermodynamic Formalism setting and prove that its associated potential is in Walters space -we present an explicit expression. We also provide an alternative way for obtaining the uniqueness of the DLR-Gibbs measures.

Citation: Leandro Cioletti, Artur O. Lopes. Interactions, specifications, DLR probabilities and the Ruelle operator in the one-dimensional lattice. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6139-6152. doi: 10.3934/dcds.2017264
##### References:
 [1] V. Baladi, Positive Transfer Operators and Decay of Correlations, vol. 16 of Advanced Series in Nonlinear Dynamics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812813633. Google Scholar [2] T. Bousch, La condition de Walters, Ann. Sci. École Norm. Sup.(4), 34 (2001), 287-311. doi: 10.1016/S0012-9593(00)01062-4. Google Scholar [3] A. Bovier, Statistical Mechanics of Disordered Systems, vol. 18 of Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 2006, A mathematical perspective. doi: 10.1017/CBO9780511616808. Google Scholar [4] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, vol. 470 of Lecture Notes in Mathematics, revised edition, Springer-Verlag, Berlin, 2008, With a preface by David Ruelle, Edited by Jean-René Chazottes. Google Scholar [5] L. Cioletti, M. Denker, A. O. Lopes and M. Stadlbauer, Spectral properties of the ruelle operator for product-type potentials on shift spaces, Journal of the London Mathematical Society, 95 (2017), 684-704. doi: 10.1112/jlms.12031. Google Scholar [6] L. Cioletti and A. O. Lopes, Phase transitions in one-dimensional translation invariant systems: A Ruelle operator approach, J. Stat. Phys., 159 (2015), 1424-1455. doi: 10.1007/s10955-015-1202-4. Google Scholar [7] L. Cioletti and A. O. Lopes, Ruelle operator for continuous potentials and DLR-Gibbs measures, preprint, arXiv: 1608.03881.Google Scholar [8] L. Cioletti and A. O. Lopes, Correlation inequalities and monotonicity properties of the ruelle operator, preprint, arXiv: 1703.06126.Google Scholar [9] R. L. Dobrushin, Description of a random field by means of conditional probabilities and conditions for its regularity, Teor. Verojatnost. i Primenen, 13 (1968), 201-229. Google Scholar [10] R. L. Dobrushin, Prescribing a system of random variables by conditional distributions, Theory of Probability & Its Applications, 15 (1970), 458-486. Google Scholar [11] R. S. Ellis, Entropy, Large Deviations, and Statistical Mechanics, Classics in Mathematics, Springer-Verlag, Berlin, 2006, Reprint of the 1985 original. doi: 10.1007/3-540-29060-5. Google Scholar [12] H. Föllmer, Phase transition and Martin boundary, Lecture Notes in Math., 465 (1975), 305-317. Google Scholar [13] S. Friedli and Y. Velenik, Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction., Cambridge University Press, To appear 2017.Google Scholar [14] H. -O. Georgii, Gibbs Measures and Phase Transitions, vol. 9 of de Gruyter Studies in Mathematics, 2nd edition, Walter de Gruyter & Co., Berlin, 2011. doi: 10.1515/9783110250329. Google Scholar [15] C. Gruber, A. Hintermann and D. Merlini, Group Analysis of Classical Lattice Systems, Springer-Verlag, Berlin-New York, 1977, With a foreword by Ph. Choquard, Lecture Notes in Physics, Vol. 60. Google Scholar [16] R. B. Israel, Convexity in the Theory of Lattice Gases, Princeton University Press, Princeton, N. J., 1979, Princeton Series in Physics, With an introduction by Arthur S. Wightman. Google Scholar [17] H. A. Kramers and G. H. Wannier, Statistics of the two-dimensional ferromagnet. I, Phys. Rev.(2), 60 (1941), 252-262. doi: 10.1103/PhysRev.60.252. Google Scholar [18] O. E. Lanford Ⅲ and D. Ruelle, Observables at infinity and states with short range correlations in statistical mechanics, Comm. Math. Phys., 13 (1969), 194–215, URL http://projecteuclid.org/euclid.cmp/1103841575. doi: 10.1007/BF01645487. Google Scholar [19] F. Ledrappier, Principe variationnel et systémes dynamiques symboliques, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 30 (1974), 185-202. doi: 10.1007/BF00533471. Google Scholar [20] E. W. Montroll, Statistical mechanics of nearest neighbor systems, The Journal of Chemical Physics, 9 (1941), 706-721. doi: 10.1063/1.1750981. Google Scholar [21] L. Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev.(2), 65 (1944), 117-149. doi: 10.1103/PhysRev.65.117. Google Scholar [22] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, (1990), 268pp. Google Scholar [23] C. Preston, Random Fields, Lecture Notes in Mathematics, Vol. 534, Springer-Verlag, Berlin-New York, 1976. Google Scholar [24] D. Ruelle, A variational formulation of equilibrium statistical mechanics and the Gibbs phase rule, Comm. Math. Phys., 5 (1967), 324-329. doi: 10.1007/BF01646446. Google Scholar [25] D. Ruelle, Statistical mechanics of a one-dimensional lattice gas, Comm. Math. Phys., 9 (1968), 267-278. doi: 10.1007/BF01654281. Google Scholar [26] D. Ruelle, Thermodynamic formalism, vol. 5 of Encyclopedia of Mathematics and its Applications, Addison-Wesley Publishing Co., Reading, Mass., 1978, The mathematical structures of classical equilibrium statistical mechanics, With a foreword by Giovanni Gallavotti and Gian-Carlo Rota. Google Scholar [27] D. Ruelle, Dynamical zeta functions and transfer operators, Notices Amer. Math. Soc., 49 (2002), 887-895. Google Scholar [28] D. Ruelle, Thermodynamic Formalism, 2nd edition, Cambridge University Press, Cambridge, 2004, The mathematical structures of equilibrium statistical mechanics. doi: 10.1017/CBO9780511617546. Google Scholar [29] O. Sarig, Lecture notes on thermodynamic formalism for topological markov shifts, Penn State.Google Scholar [30] J. G. Sinaĭ, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64. Google Scholar [31] A. C. D. van Enter, R. Fernández and A. D. Sokal, Regularity properties and pathologies of position-space renormalization-group transformations: scope and limitations of Gibbsian theory, J. Statist. Phys., 72 (1993), 879-1167. doi: 10.1007/BF01048183. Google Scholar [32] P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1975), 937-971. doi: 10.2307/2373682. Google Scholar [33] P. Walters, Invariant measures and equilibrium states for some mappings which expand distances, Trans. Amer. Math. Soc., 236 (1978), 121-153. doi: 10.1090/S0002-9947-1978-0466493-1. Google Scholar [34] P. Walters, Convergence of the Ruelle operator for a function satisfying Bowen's condition, Trans. Amer. Math. Soc., 353 (2001), 327-347. doi: 10.1090/S0002-9947-00-02656-8. Google Scholar [35] P. Walters, Regularity conditions and Bernoulli properties of equilibrium states and $g$ -measures, J. London Math. Soc.(2), 71 (2005), 379-396. doi: 10.1112/S0024610704006076. Google Scholar [36] P. Walters, A natural space of functions for the Ruelle operator theorem, Ergodic Theory Dynam. Systems, 27 (2007), 1323-1348. doi: 10.1017/S0143385707000028. Google Scholar

show all references

##### References:
 [1] V. Baladi, Positive Transfer Operators and Decay of Correlations, vol. 16 of Advanced Series in Nonlinear Dynamics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812813633. Google Scholar [2] T. Bousch, La condition de Walters, Ann. Sci. École Norm. Sup.(4), 34 (2001), 287-311. doi: 10.1016/S0012-9593(00)01062-4. Google Scholar [3] A. Bovier, Statistical Mechanics of Disordered Systems, vol. 18 of Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 2006, A mathematical perspective. doi: 10.1017/CBO9780511616808. Google Scholar [4] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, vol. 470 of Lecture Notes in Mathematics, revised edition, Springer-Verlag, Berlin, 2008, With a preface by David Ruelle, Edited by Jean-René Chazottes. Google Scholar [5] L. Cioletti, M. Denker, A. O. Lopes and M. Stadlbauer, Spectral properties of the ruelle operator for product-type potentials on shift spaces, Journal of the London Mathematical Society, 95 (2017), 684-704. doi: 10.1112/jlms.12031. Google Scholar [6] L. Cioletti and A. O. Lopes, Phase transitions in one-dimensional translation invariant systems: A Ruelle operator approach, J. Stat. Phys., 159 (2015), 1424-1455. doi: 10.1007/s10955-015-1202-4. Google Scholar [7] L. Cioletti and A. O. Lopes, Ruelle operator for continuous potentials and DLR-Gibbs measures, preprint, arXiv: 1608.03881.Google Scholar [8] L. Cioletti and A. O. Lopes, Correlation inequalities and monotonicity properties of the ruelle operator, preprint, arXiv: 1703.06126.Google Scholar [9] R. L. Dobrushin, Description of a random field by means of conditional probabilities and conditions for its regularity, Teor. Verojatnost. i Primenen, 13 (1968), 201-229. Google Scholar [10] R. L. Dobrushin, Prescribing a system of random variables by conditional distributions, Theory of Probability & Its Applications, 15 (1970), 458-486. Google Scholar [11] R. S. Ellis, Entropy, Large Deviations, and Statistical Mechanics, Classics in Mathematics, Springer-Verlag, Berlin, 2006, Reprint of the 1985 original. doi: 10.1007/3-540-29060-5. Google Scholar [12] H. Föllmer, Phase transition and Martin boundary, Lecture Notes in Math., 465 (1975), 305-317. Google Scholar [13] S. Friedli and Y. Velenik, Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction., Cambridge University Press, To appear 2017.Google Scholar [14] H. -O. Georgii, Gibbs Measures and Phase Transitions, vol. 9 of de Gruyter Studies in Mathematics, 2nd edition, Walter de Gruyter & Co., Berlin, 2011. doi: 10.1515/9783110250329. Google Scholar [15] C. Gruber, A. Hintermann and D. Merlini, Group Analysis of Classical Lattice Systems, Springer-Verlag, Berlin-New York, 1977, With a foreword by Ph. Choquard, Lecture Notes in Physics, Vol. 60. Google Scholar [16] R. B. Israel, Convexity in the Theory of Lattice Gases, Princeton University Press, Princeton, N. J., 1979, Princeton Series in Physics, With an introduction by Arthur S. Wightman. Google Scholar [17] H. A. Kramers and G. H. Wannier, Statistics of the two-dimensional ferromagnet. I, Phys. Rev.(2), 60 (1941), 252-262. doi: 10.1103/PhysRev.60.252. Google Scholar [18] O. E. Lanford Ⅲ and D. Ruelle, Observables at infinity and states with short range correlations in statistical mechanics, Comm. Math. Phys., 13 (1969), 194–215, URL http://projecteuclid.org/euclid.cmp/1103841575. doi: 10.1007/BF01645487. Google Scholar [19] F. Ledrappier, Principe variationnel et systémes dynamiques symboliques, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 30 (1974), 185-202. doi: 10.1007/BF00533471. Google Scholar [20] E. W. Montroll, Statistical mechanics of nearest neighbor systems, The Journal of Chemical Physics, 9 (1941), 706-721. doi: 10.1063/1.1750981. Google Scholar [21] L. Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev.(2), 65 (1944), 117-149. doi: 10.1103/PhysRev.65.117. Google Scholar [22] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, (1990), 268pp. Google Scholar [23] C. Preston, Random Fields, Lecture Notes in Mathematics, Vol. 534, Springer-Verlag, Berlin-New York, 1976. Google Scholar [24] D. Ruelle, A variational formulation of equilibrium statistical mechanics and the Gibbs phase rule, Comm. Math. Phys., 5 (1967), 324-329. doi: 10.1007/BF01646446. Google Scholar [25] D. Ruelle, Statistical mechanics of a one-dimensional lattice gas, Comm. Math. Phys., 9 (1968), 267-278. doi: 10.1007/BF01654281. Google Scholar [26] D. Ruelle, Thermodynamic formalism, vol. 5 of Encyclopedia of Mathematics and its Applications, Addison-Wesley Publishing Co., Reading, Mass., 1978, The mathematical structures of classical equilibrium statistical mechanics, With a foreword by Giovanni Gallavotti and Gian-Carlo Rota. Google Scholar [27] D. Ruelle, Dynamical zeta functions and transfer operators, Notices Amer. Math. Soc., 49 (2002), 887-895. Google Scholar [28] D. Ruelle, Thermodynamic Formalism, 2nd edition, Cambridge University Press, Cambridge, 2004, The mathematical structures of equilibrium statistical mechanics. doi: 10.1017/CBO9780511617546. Google Scholar [29] O. Sarig, Lecture notes on thermodynamic formalism for topological markov shifts, Penn State.Google Scholar [30] J. G. Sinaĭ, Gibbs measures in ergodic theory, Uspehi Mat. Nauk, 27 (1972), 21-64. Google Scholar [31] A. C. D. van Enter, R. Fernández and A. D. Sokal, Regularity properties and pathologies of position-space renormalization-group transformations: scope and limitations of Gibbsian theory, J. Statist. Phys., 72 (1993), 879-1167. doi: 10.1007/BF01048183. Google Scholar [32] P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1975), 937-971. doi: 10.2307/2373682. Google Scholar [33] P. Walters, Invariant measures and equilibrium states for some mappings which expand distances, Trans. Amer. Math. Soc., 236 (1978), 121-153. doi: 10.1090/S0002-9947-1978-0466493-1. Google Scholar [34] P. Walters, Convergence of the Ruelle operator for a function satisfying Bowen's condition, Trans. Amer. Math. Soc., 353 (2001), 327-347. doi: 10.1090/S0002-9947-00-02656-8. Google Scholar [35] P. Walters, Regularity conditions and Bernoulli properties of equilibrium states and $g$ -measures, J. London Math. Soc.(2), 71 (2005), 379-396. doi: 10.1112/S0024610704006076. Google Scholar [36] P. Walters, A natural space of functions for the Ruelle operator theorem, Ergodic Theory Dynam. Systems, 27 (2007), 1323-1348. doi: 10.1017/S0143385707000028. Google Scholar
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