×
Caputo, Pietro; Labbé, Cyril; Lacoin, Hubert

Spectral gap and cutoff phenomenon for the Gibbs sampler of \(\nabla \varphi\) interfaces with convex potential. (English. French summary) Zbl 1502.37032

Ann. Inst. Henri Poincaré, Probab. Stat. 58, No. 2, 794-826 (2022).
Summary: We consider the Gibbs sampler, or heat bath dynamics associated to log-concave measures on \({\mathbb{R}^N}\) describing \(\nabla \varphi\) interfaces with convex potentials. Under minimal assumptions on the potential, we find that the spectral gap of the process is always given by \({\text{gap}_N}=1-\cos (\pi /N)\), and that for all \(\epsilon \in (0,1)\), its \(\varepsilon \)-mixing time satisfies \({T_N}(\epsilon )\sim \frac{\log N}{2{\text{gap}_N}}\) as \(N\to \infty \), thus establishing the cutoff phenomenon. The results reveal a universal behavior in that they do not depend on the choice of the potential.

Cited in 6 Documents

MSC:

37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
37A25 Ergodicity, mixing, rates of mixing
37A50 Dynamical systems and their relations with probability theory and stochastic processes
60J25 Continuous-time Markov processes on general state spaces
82C22 Interacting particle systems in time-dependent statistical mechanics

Keywords:

cutoff; mixing time; spectral gap
Cite Review PDF
Full Text: DOI arXiv

References:

[1] F. Barthe and E. Milman. Transference principles for log-Sobolev and spectral-gap with applications to conservative spin systems. Comm. Math. Phys. 323 (2013) 575-625. · Zbl 1297.82011 · doi:10.1007/s00220-013-1782-2
[2] F. Barthe and P. Wolff. Remarks on non-interacting conservative spin systems: The case of gamma distributions. Stochastic Process. Appl. 119 (2009) 2711-2723. · Zbl 1169.60325 · doi:10.1016/j.spa.2009.02.004
[3] T. Bodineau, D. Ioffe and Y. Velenik. Rigorous probabilistic analysis of equilibrium crystal shapes. J. Math. Phys. 41 (2000) 1033-1098. · Zbl 0969.00035 · doi:10.1063/1.533180
[4] P. Caputo. Uniform Poincaré inequalities for unbounded conservative spin systems: The non-interacting case. Stochastic Process. Appl. 106 (2003) 223-244. · Zbl 1075.60581 · doi:10.1016/S0304-4149(03)00044-9
[5] P. Caputo, C. Labbé and H. Lacoin. Mixing time of the adjacent walk on the simplex. Ann. Probab. 48 (2020) 2449-2493. · Zbl 1456.60185 · doi:10.1214/20-AOP1428
[6] P. Caputo, F. Martinelli and F. L. Toninelli. Mixing times of monotone surfaces and sos interfaces: A mean curvature approach. Comm. Math. Phys. 311 (2012) 157-189. · Zbl 1276.82034 · doi:10.1007/s00220-012-1425-z
[7] A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications. Stochastic Modelling and Applied Probability 38. Springer-Verlag, Berlin, 2010. Corrected reprint of the second (1998) edition. · Zbl 1177.60035 · doi:10.1007/978-3-642-03311-7
[8] R. Durrett. Probability—Theory and Examples. Cambridge Series in Statistical and Probabilistic Mathematics 49. Cambridge University Press, Cambridge, 2019. Fifth edition of [MR1068527]. · Zbl 1440.60001 · doi:10.1017/9781108591034
[9] T. Funaki. Stochastic interface models. In Lectures on Probability Theory and Statistics 103-274. Lecture Notes in Math. 1869. Springer, Berlin, 2005. · Zbl 1119.60081 · doi:10.1007/11429579_2
[10] G. Giacomin. Aspects of Statistical Mechanics of Random Surfaces. Lecture Notes for Course Given at IHP, 2002.
[11] C. Labbé and H. Lacoin. Cutoff phenomenon for the asymmetric simple exclusion process and the biased card shuffling. Ann. Probab. 47 (2019) 1541-1586. · Zbl 1466.60152 · doi:10.1214/18-AOP1290
[12] H. Lacoin. Mixing time and cutoff for the adjacent transposition shuffle and the simple exclusion. Ann. Probab. 44 (2016) 1426-1487. · Zbl 1408.60061 · doi:10.1214/15-AOP1004
[13] D. A. Levin and Y. Peres. Markov Chains and Mixing Times. American Mathematical Society, Providence, RI, 2017. Second edition of [MR2466937], With contributions by Elizabeth L. Wilmer, With a chapter on “Coupling from the past” by James G. Propp and David B. Wilson. · Zbl 1390.60001 · doi:10.1090/mbk/107
[14] T. M. Liggett. Interacting Particle Systems. Classics in Mathematics. Springer-Verlag, Berlin, 2005. Reprint of the 1985 original. · doi:10.1007/b138374
[15] F. Martinelli and A. Sinclair. Mixing time for the solid-on-solid model. Ann. Appl. Probab. 22 (2012) 1136-1166. · Zbl 1283.60122 · doi:10.1214/11-AAP791
[16] G. Menz and F. Otto. Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential. Ann. Probab. 41 (2013) 2182-2224. · Zbl 1282.60096 · doi:10.1214/11-AOP715
[17] Y. Peres and P. Winkler. Can extra updates delay mixing? Comm. Math. Phys. 323 (2013) 1007-1016. · Zbl 1277.82036 · doi:10.1007/s00220-013-1776-0
[18] V. V. Petrov. Sums of Independent Random Variables. Ergebnisse der Mathematik und Ihrer Grenzgebiete 82. Springer-Verlag, New York-Heidelberg, 1975. Translated from the Russian by A. A. Brown. · Zbl 0322.60042
[19] G. Posta. Spectral gap for an unrestricted kawasaki type dynamics. ESAIM Probab. Stat. 1 (1997) 145-181. · Zbl 0869.60091 · doi:10.1051/ps:1997106
[20] C. J. Preston. A generalization of the \[\text{FKG}\] inequalities. Comm. Math. Phys. 36 (1974) 233-241.
[21] J. Ville. Étude critique de la notion de collectif. NUMDAM, 1939. · Zbl 0021.14601
[22] D. B. Wilson. Mixing times of Lozenge tiling and card shuffling Markov chains. Ann. Appl. Probab. 14 (2004) 274-325 · Zbl 1040.60063 · doi:10.1214/aoap/1075828054
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.