The Annals of Probability

Euclidean Gibbs measures on loop lattices: Existence and a priori estimates

Sergio Albeverio, Yuri Kondratiev, Tatiana Pasurek, and Michael Röckner

Full-text: Open access

Abstract

We present a new method to prove existence and uniform a priori estimates for Euclidean Gibbs measures corresponding to quantum anharmonic crystals. It is based first on the alternative characterization of Gibbs measures in terms of their logarithmic derivatives through integration by parts formulas, and second on the choice of appropriate Lyapunov functionals.

Article information

Source
Ann. Probab., Volume 32, Number 1A (2004), 153-190.

Dates
First available in Project Euclid: 4 March 2004

Permanent link to this document
https://projecteuclid.org/euclid.aop/1078415832

Digital Object Identifier
doi:10.1214/aop/1078415832

Mathematical Reviews number (MathSciNet)
MR2040779

Zentralblatt MATH identifier
1121.82005

Subjects
Primary: 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 60G60: Random fields 82B10: Quantum equilibrium statistical mechanics (general)

Keywords
Quantum lattice systems Euclidean Gibbs states smooth measures on vector spaces integration by parts formulae Lyapunov functionals

Citation

Albeverio, Sergio; Kondratiev, Yuri; Pasurek, Tatiana; Röckner, Michael. Euclidean Gibbs measures on loop lattices: Existence and a priori estimates. Ann. Probab. 32 (2004), no. 1A, 153--190. doi:10.1214/aop/1078415832. https://projecteuclid.org/euclid.aop/1078415832


Export citation

References

  • Albeverio, S. and Høegh-Krohn, R. (1975). Homogeneous random fields and quantum statistical mechanics. J. Funct. Anal. 19 241--272.
  • Albeverio, S., Kondratiev, Yu. G., Kozitzky, Yu. V. and Röckner, M. (2001). Uniqueness of Gibbs states of quantum lattices in small mass regime. Ann. Inst. H. Poincaré Probab. Statist. 37 43--69.
  • Albeverio, S., Kondratiev, Yu. G., Kozitzky, Yu. V. and Röckner, M. (2002). Euclidean Gibbs states of quantum lattice systems. Rev. Math. Phys. 14 1--67.
  • Albeverio, S., Kondratiev, Yu. G., Kozitzky, Yu. V. and Röckner, M. (2004). Small mass implies uniqueness of Gibbs states of a quantum crystal. Comm. Math. Phys. To appear.
  • Albeverio, S., Kondratiev, Yu. G., Pasurek (Tsikalenko), T. and Röckner, M. (2001). Gibbs states on loop lattices: Existence and a priori estimates. C. R. Acad. Sci. Paris Sér. I Math. 33 1--5.
  • Albeverio, S., Kondratiev, Yu. G., Pasurek (Tsikalenko), T. and Röckner, M. (2001). Euclidean Gibbs states of quantum crystals. Mosc. Math. J. 1 1--7.
  • Albeverio, S., Kondratiev, Yu. G., Pasurek (Tsikalenko), T. and Röckner, M. (2002). A priori estimates and existence for Euclidean Gibbs states. Preprint.
  • Albeverio, S., Kondratiev, A. Yu. and Rebenko, L. (1998). Peierls argument and long-range order behaviour of quantum lattice systems with unbounded spins. J. Statist. Phys. 92 1137--1152.
  • Albeverio, S., Kondratiev, Yu. G. and Röckner, M. (1997). Ergodicity of $L^2$-semigroups and extremality of Gibbs states. J. Funct. Anal. 144 394--423.
  • Albeverio, S., Kondratiev, Yu. G. and Röckner, M. (1997). Ergodicity of the stochastic dynamics of quasi-invariant measures and applications to Gibbs states. J. Funct. Anal. 149 415--469.
  • Albeverio, S., Kondratiev, Yu. G., Röckner, M. and Tsikalenko, T. V. (1997). Uniqueness of Gibbs states for quantum lattice systems. Probab. Theory Related Fields 108 193--218.
  • Albeverio, S., Kondratiev, Yu. G., Röckner, M. and Tsikalenko, T. V. (1997). Dobrushin's uniqueness for quantum lattice systems with nonlocal interaction. Comm. Math. Phys. 189 621--630.
  • Albeverio, S., Kondratiev, Yu. G., Röckner, M. and Tsikalenko, T. V. (1999). A priori estimates and existence of Gibbs measures: A simplified proof. C. R. Acad. Sci. Paris Sér. I Math. 328 1049--1054.
  • Albeverio, S., Kondratiev, Yu. G., Röckner, M. and Tsikalenko, T. V. (2000). A priori estimates for symmetrizing measures and their applications to Gibbs states. J. Funct. Anal. 171 366--400.
  • Albeverio, S., Kondratiev, Yu. G., Röckner, M. and Tsikalenko, T. V. (2001). Glauber dynamics for quantum lattice systems. Rev. Math. Phys. 13 51--124.
  • Bell, D. (1985). A quasi-invariance theorem for measures on Banach spaces. Trans. Amer. Math. Soc. 290 851--855.
  • Bellissard, J. and Høegh-Krohn, R. (1982). Compactness and the maximal Gibbs states for random Gibbs fields on a lattice. Comm. Math. Phys. 84 297--327.
  • Barbulyak, V. S. and Kondratiev, Yu. G. (1991). Functional integrals and quantum lattice systems. Rep. Nat. Acad. Sci. Ukraine 8 38--40; 9 31--34; 10 19--21.
  • Betz, V. and Lőrinczi, J. (2000). A Gibbsian description of $P(\phi )_1$-process. Preprint.
  • Bogachev, V. I. and Röckner, M. (2001). Elliptic equations for measures on infinite-dimensional spaces and applications. Probab. Theory Related Fields 120 445--496.
  • Bogachev, V. I., Röckner, M. and Wang, F.-Y. (2001). Elliptic equations for invariant measures on finite and infinite dimensional manifolds. J. Math. Pures Appl. 80 177--221.
  • Bratteli, O. and Robinson, D. W. (1981). Operator Algebras and Quantum Statistical Mechanics$,$ I$,$ II. Springer, New York.
  • Barlow, M. T. and Yor, M. (1982). Semi-martingale inequalities via the Garsia--Rodemich--Rumsey lemma and applications to local times. J. Funct. Anal. 49 198--229.
  • Cerrai, S. (2001). Second Order PDE's in Finite and Infinite Dimensons. A Probabilistic Approach. Lecture Notes in Math. 1762. Springer, New York.
  • Cassandro, M., Olivieri, E., Pellegrinotti, A. and Presutti, E. (1978). Existence and uniqueness of DLR measures for unbounded spin systems. Z. Wahrsch. Verw. Gebiete 41 313--334.
  • Courrège, Ph. and Renouard, P. (1975). Oscillateurs Anharmoniques$,$ Mesures Quasi-Invariantes Sur $\mathcalC(\mathbbR, \mathbbR)$ et Théorie Quantique des Champs en Dimension $\textit1$. Astérisque 22--23. Soc. Math. France, Paris.
  • Deimling, K. (1985). Nonlinear Functional Analysis. Springer, New York.
  • Dobrushin, R. L. (1970). Prescribing a system of random variables by conditional distributions. Theory Probab. Appl. 15 458--486.
  • Driesler, W., Landau, L. and Perez, J. F. (1979). Estimates of critical length and critical temperatures for classical and quantum lattice systems. J. Statist. Phys. 20 123--162.
  • Da Prato, G. and Zabczyk, J. (1996). Ergodicity for Infinite-Dimensional Systems. Cambridge Univ. Press.
  • Daletskii, Yu. L. and Sokhadze, G. A. (1988). Absolute continuity of smooth measures. Funct. Anal. Appl. 22 149--150.
  • Föllmer, H. (1988). Random fields and diffusion processes. Lecture Notes in Math. 1362 101--204. Springer, New York.
  • Fritz, J. (1982). Stationary measures of stochastic gradient dynamics, infinite lattice models. Z. Wahrsch. Verw. Gebiete 59 479--490.
  • Fröhlich, J. (1977). Schwinger functions and their generating functionals II. Adv. Math. 33 119--180.
  • Funaki, T. (1991). The reversible measures of multi-dimensional Ginzburg--Landau type continuum model. Osaka J. Math. 28 462--494.
  • Faris, W. G. and Minlos, R. A. (1999). A quantum crystal with multidimensional anharmonic oscillators. J. Statist. Phys. 94 365--387.
  • Garet, O. (2002). Harmonic oscillators on an Hilbert space: A Gibbsian approach. Potential Anal. 17 65--88.
  • Georgii, H.-O. (1988). Gibbs Measures and Phase Transitions. 9. Walter de Gruyter, New York.
  • Glimm, J. and Jaffe, A. (1981). Quantum Physics. A Functional Integral Point of View. Springer, New York.
  • Globa, S. A. and Kondratiev, Yu. G. (1990). The construction of Gibbs states of quantum lattice systems. Selecta Math. Sov. 9 297--307.
  • Helffer, B. (1998). Splitting in large dimensions and infrared estimates. II. Moment inequalities. J. Math. Phys. 39 760--776.
  • Hariya, Y. (2001). A new approach to constructing Gibbs measures on $C(\mathbbR;\mathbbR^d)$---an application for hard-wall Gibbs measures on $C(\mathbbR,\mathbbR)$. Preprint.
  • Holley, R. and Stroock, D. (1976). $L_2$-theory for the stochastic Ising model. Z. Wahrsch. Verw. Gebiete 35 87--101.
  • Iwata, K. (1985). Reversible measures of a $P(\varphi )_1$-time evolution. Proc. Taniguchi Symp. PMMP 195--209.
  • Kirillov, A. I. (1995). On the reconstruction of measures from their logarithmic derivatives. Izvestiya RAN$:$ Ser. Mat. 59 121--138.
  • Kusuoka, S. (1982). Dirichlet forms and diffusion processes on Banach space. J. Fac. Science Univ. Tokyo 29 79--95.
  • Klein, A. and Landau, L. (1981). Stochastic processes associated with KMS states. J. Funct. Anal. 42 368--428.
  • Lörinczi, J. and Minlos, R. A. (2001). Gibbs measures for Brownian paths under the effect of an external and a small pair potential. J. Statist. Phys. 105 607--649.
  • Lörinczi, J., Minlos, R. A. and Spohn, H. (2002). Infrared regular representation of the three dimensional masslesss Nelson model. Lett. Math. Phys. 59 189--198.
  • Lebowitz, J. L. and Presutti, E. (1976). Statistical mechanics of systems of unbounded spins. Comm. Math. Phys. 50 195--218.
  • Malyshev, V. A. and Minlos, R. A. (1995). Linear Infinite-Particle Operators. AMS Translations 143.
  • Moulin-Ollagnier, J. (1985). Ergodic Theory and Statistical Mechanics. Lecture Notes in Math. 1115. Springer, New York.
  • Minlos, R. A., Roelly, S. and Zessin, H. (2000). Gibbs states on space-time. Potential Anal. 13 367--408.
  • Minlos, R. A., Verbeure, A. and Zagrebnov, V. (2000). A quantum crystal model in the light mass limit: Gibbs state. Rev. Math. Phys. 12 981--1032.
  • Osada, H. and Spohn, H. (1999). Gibbs mesures relative to Brownian motion. Ann. Probab. 27 1183--1207.
  • Preston, C. (1976). Random Fields. Lecture Notes in Math. 534. Springer, Berlin.
  • Park, Y. M. and Yoo, H. J. (1994). A characterization of Gibbs states of lattice boson systems. J. Statist. Phys. 75 215--239.
  • Park, Y. M. and Yoo, H. J. (1995). Uniqueness and clustering properties of Gibbs states for classical and quantum unbounded spin systems. J. Statist. Phys. 80 223--271.
  • Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Springer, New York.
  • Royer, G. (1975). Unicité de certaines mesures quasi-invariantes sur $C(\mathbbR)$. Ann. Scient. Éc. Norm. Sup. 4$^e$ 8 319--338.
  • Royer, G. (1977). Étude des champs Euclidiens sur un resau $\mathbbZ^\gamma $. J. Math. Pures Appl. 56 455--478.
  • Royer, G. and Yor, M. (1976). Représentation intégrale de certains mesures quasi-invariantes sur $\mathcalC(\mathbbR)$; mesures extrémales et propriété de Markov. Ann. Inst. Fourier 26 7--24.
  • Ruelle, D. (1969). Statistical Mechanics. Rigorous Results. Benjamin, New York.
  • Simon, B. (1974). The $P(\varphi )_2$ Euclidean Field Theory. Princeton Univ. Press.
  • Simon, B. (1979). Functional Integration and Quantum Physics. Academic Press, New York.