The Annals of Applied Probability

Poincaré and logarithmic Sobolev constants for metastable Markov chains via capacitary inequalities

André Schlichting and Martin Slowik

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We investigate the metastable behavior of reversible Markov chains on possibly countable infinite state spaces. Based on a new definition of metastable Markov processes, we compute precisely the mean transition time between metastable sets. Under additional size and regularity properties of metastable sets, we establish asymptotic sharp estimates on the Poincaré and logarithmic Sobolev constant. The main ingredient in the proof is a capacitary inequality along the lines of V. Maz’ya that relates regularity properties of harmonic functions and capacities. We exemplify the usefulness of this new definition in the context of the random field Curie–Weiss model, where metastability and the additional regularity assumptions are verifiable.

Article information

Source
Ann. Appl. Probab., Volume 29, Number 6 (2019), 3438-3488.

Dates
Received: May 2017
Revised: January 2019
First available in Project Euclid: 7 January 2020

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1578366319

Digital Object Identifier
doi:10.1214/19-AAP1484

Mathematical Reviews number (MathSciNet)
MR4047985

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 34L15: Eigenvalues, estimation of eigenvalues, upper and lower bounds 49J40: Variational methods including variational inequalities [See also 47J20] 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 82C26: Dynamic and nonequilibrium phase transitions (general)

Keywords
Capacitary inequality harmonic functions logarithmic Sobolev constant mean hitting time metastability Poincaré constant random field Curie–Weiss model

Citation

Schlichting, André; Slowik, Martin. Poincaré and logarithmic Sobolev constants for metastable Markov chains via capacitary inequalities. Ann. Appl. Probab. 29 (2019), no. 6, 3438--3488. doi:10.1214/19-AAP1484. https://projecteuclid.org/euclid.aoap/1578366319


Export citation

References

  • [1] Amaro de Matos, J. M. G., Patrick, A. E. and Zagrebnov, V. A. (1992). Random infinite-volume Gibbs states for the Curie–Weiss random field Ising model. J. Stat. Phys. 66 139–164.
  • [2] Ané, C., Blachère, S., Chafaï, D., Fougères, P., Gentil, I., Malrieu, F., Roberto, C. and Scheffer, G. (2000). Sur les Inégalités de Sobolev Logarithmiques. Panoramas et Synthèses [Panoramas and Syntheses] 10. Société Mathématique de France, Paris.
  • [3] Bakry, D., Gentil, I. and Ledoux, M. (2014). Analysis and Geometry of Markov Diffusion Operators. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 348. Springer, Cham.
  • [4] Barthe, F., Cattiaux, P. and Roberto, C. (2006). Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry. Rev. Mat. Iberoam. 22 993–1067.
  • [5] Barthe, F. and Roberto, C. (2003). Sobolev inequalities for probability measures on the real line. Studia Math. 159 481–497.
  • [6] Beltrán, J. and Landim, C. (2015). A martingale approach to metastability. Probab. Theory Related Fields 161 267–307.
  • [7] Berglund, N. and Dutercq, S. (2016). The Eyring–Kramers law for Markovian jump processes with symmetries. J. Theoret. Probab. 29 1240–1279.
  • [8] Bianchi, A., Bovier, A. and Ioffe, D. (2009). Sharp asymptotics for metastability in the random field Curie–Weiss model. Electron. J. Probab. 14 1541–1603.
  • [9] Bianchi, A., Bovier, A. and Ioffe, D. (2012). Pointwise estimates and exponential laws in metastable systems via coupling methods. Ann. Probab. 40 339–371.
  • [10] Bianchi, A. and Gaudillière, A. (2016). Metastable states, quasi-stationary distributions and soft measures. Stochastic Process. Appl. 126 1622–1680.
  • [11] Bobkov, S. G. and Götze, F. (1999). Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 1–28.
  • [12] Bobkov, S. G. and Tetali, P. (2006). Modified logarithmic Sobolev inequalities in discrete settings. J. Theoret. Probab. 19 289–336.
  • [13] Bovier, A. (2004). Metastability and ageing in stochastic dynamics. In Dynamics and Randomness II. Nonlinear Phenom. Complex Systems 10 17–79. Kluwer Academic, Dordrecht.
  • [14] Bovier, A. (2006). Metastability: A potential theoretic approach. In International Congress of Mathematicians. Vol. III 499–518. Eur. Math. Soc., Zürich.
  • [15] Bovier, A. and den Hollander, F. (2015). Metastability: A Potential-Theoretic Approach. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 351. Springer, Cham.
  • [16] Bovier, A., Eckhoff, M., Gayrard, V. and Klein, M. (2001). Metastability in stochastic dynamics of disordered mean-field models. Probab. Theory Related Fields 119 99–161.
  • [17] Bovier, A., Eckhoff, M., Gayrard, V. and Klein, M. (2002). Metastability and low lying spectra in reversible Markov chains. Comm. Math. Phys. 228 219–255.
  • [18] Bovier, A., Eckhoff, M., Gayrard, V. and Klein, M. (2004). Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times. J. Eur. Math. Soc. (JEMS) 6 399–424.
  • [19] Bovier, A., Gayrard, V. and Klein, M. (2005). Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues. J. Eur. Math. Soc. (JEMS) 7 69–99.
  • [20] Bovier, A. and Manzo, F. (2002). Metastability in Glauber dynamics in the low-temperature limit: Beyond exponential asymptotics. J. Stat. Phys. 107 757–779.
  • [21] Burke, C. J. and Rosenblatt, M. (1958). A Markovian function of a Markov chain. Ann. Math. Stat. 29 1112–1122.
  • [22] Cassandro, M., Galves, A., Olivieri, E. and Vares, M. E. (1984). Metastable behavior of stochastic dynamics: A pathwise approach. J. Stat. Phys. 35 603–634.
  • [23] Cheeger, J. (1970). A lower bound for the smallest eigenvalue of the Laplacian. In Problems in Analysis (Papers Dedicated to Salomon Bochner, 1969) 195–199. Princeton Univ. Press, Princeton, NJ.
  • [24] Chen, M.-F. (2005). Capacitary criteria for Poincaré-type inequalities. Potential Anal. 23 303–322.
  • [25] Chen, M.-F. (2005). Eigenvalues, Inequalities, and Ergodic Theory. Probability and Its Applications (New York). Springer, London.
  • [26] Diaconis, P. and Saloff-Coste, L. (1996). Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 695–750.
  • [27] Diaconis, P. and Shahshahani, M. (1987). Time to reach stationarity in the Bernoulli–Laplace diffusion model. SIAM J. Math. Anal. 18 208–218.
  • [28] Freidlin, M. I. and Wentzell, A. D. (1998). Random Perturbations of Dynamical Systems, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 260. Springer, New York.
  • [29] Helffer, B., Klein, M. and Nier, F. (2004). Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach. Mat. Contemp. 26 41–85.
  • [30] Higuchi, Y. and Yoshida, N. (1995). Analytic conditions and phase transition for ising models. Lecture notes in Japanese.
  • [31] Holley, R. and Stroock, D. (1987). Logarithmic Sobolev inequalities and stochastic Ising models. J. Stat. Phys. 46 1159–1194.
  • [32] Lawler, G. F. and Sokal, A. D. (1988). Bounds on the $L^{2}$ spectrum for Markov chains and Markov processes: A generalization of Cheeger’s inequality. Trans. Amer. Math. Soc. 309 557–580.
  • [33] Lee, T.-Y. and Yau, H.-T. (1998). Logarithmic Sobolev inequality for some models of random walks. Ann. Probab. 26 1855–1873.
  • [34] Levin, D. A., Luczak, M. J. and Peres, Y. (2010). Glauber dynamics for the mean-field Ising model: Cut-off, critical power law, and metastability. Probab. Theory Related Fields 146 223–265.
  • [35] Levin, D. A., Peres, Y. and Wilmer, E. L. (2009). Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI.
  • [36] Madras, N. and Zheng, Z. (2003). On the swapping algorithm. Random Structures Algorithms 22 66–97.
  • [37] Mathieu, P. and Picco, P. (1998). Metastability and convergence to equilibrium for the random field Curie–Weiss model. J. Stat. Phys. 91 679–732.
  • [38] Maz’ja, V. G. (1972). Certain integral inequalities for functions of several variables. In Problems of Mathematical Analysis, No. 3: Integral and Differential Operators, Differential Equations (Russian) 33–68. Izdat. Leningrad. Univ., Leningrad.
  • [39] Maz’ya, V. (2011). Sobolev Spaces with Applications to Elliptic Partial Differential Equations. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 342. Springer, Heidelberg.
  • [40] Menz, G. and Schlichting, A. (2014). Poincaré and logarithmic Sobolev inequalities by decomposition of the energy landscape. Ann. Probab. 42 1809–1884.
  • [41] Miclo, L. (1999). An example of application of discrete Hardy’s inequalities. Markov Process. Related Fields 5 319–330.
  • [42] Muckenhoupt, B. (1972). Hardy’s inequality with weights. Studia Math. 44 31–38. Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, I.
  • [43] Olivieri, E. and Vares, M. E. (2005). Large Deviations and Metastability. Encyclopedia of Mathematics and Its Applications 100. Cambridge Univ. Press, Cambridge.
  • [44] Rao, M. M. and Ren, Z. D. (2002). Applications of Orlicz Spaces. Monographs and Textbooks in Pure and Applied Mathematics 250. Dekker, New York.
  • [45] Salinas, S. R. and Wreszinski, W. F. (1985). On the mean-field Ising model in a random external field. J. Stat. Phys. 41 299–313.
  • [46] Schlichting, A. (2012). The Eyring–Kramers formula for Poincaré and logarithmic Sobolev inequalities. Ph.D. thesis, Universität Leipzig.
  • [47] Slowik, M. (2012). Contributions to the potential theoretic approach to metastability with applications to the random field Curie-Weiss-Potts model. Ph.D. thesis, Technische Univ. Berlin.
  • [48] Sugiura, M. (1995). Metastable behaviors of diffusion processes with small parameter. J. Math. Soc. Japan 47 755–788.