On the probability of staying above a wall for the \((2+1)\) -dimensional SOS

详细信息    查看全文

• 作者：Pietro Caputo ; Fabio Martinelli…
• 关键词：SOS model ; Loop ensembles ; Random surface models ; Entropic repulsion ; Large deviations ; 60K35 ; 60F10 ; 82B41 ; 82C24
• 刊名：Probability Theory and Related Fields
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：163
• 期：3-4
• 页码：803-831
• 全文大小：859 KB
• 参考文献：1.Bolthausen, E., Deuschel, J.-D., Giacomin, G.: Entropic repulsion and the maximum of the two-dimensional harmonic crystal. Ann. Probab. 29(4), 1670-692 (2001)MATH MathSciNet CrossRef
  2.Bolthausen, E., Deuschel, J.-D., Zeitouni, O.: Entropic repulsion of the lattice free field. Commun. Math. Phys. 170(2), 417-43 (1995)MATH MathSciNet CrossRef
  3.Brandenberger, R., Wayne, C.E.: Decay of correlations in surface models. J. Stat. Phys. 27(3), 425-40 (1982)MathSciNet CrossRef
  4.Bricmont, J., El Mellouki, A., Fr?hlich, J.: Random surfaces in statistical mechanics: roughening, rounding, wetting, \(\ldots \) . J. Stat. Phys. 42(5-), 743-98 (1986)CrossRef
  5.Caputo, P., Lubetzky, E., Martinelli, F., Sly, A., Toninelli, F.L.: Dynamics of 2+1 dimensional sos surfaces above a wall: slow mixing induced by entropic repulsion. Ann. Probab. 42, 1516-589 (2014)MATH MathSciNet CrossRef
  6.Caputo, P., Lubetzky, E., Martinelli, F., Sly, A., Toninelli, F.L.: The shape of the (2+1)D SOS surface above a wall. C. R. Math. Acad. Sci. Paris 350(13-4), 703-06 (2012)MATH MathSciNet CrossRef
  7.Caputo, P., Lubetzky, E., Martinelli, F., Sly, A., Toninelli, F.L.: Scaling limit and cube-root fluctuations in sos surfaces above a wall. J. Eur. Math. Soc. arXiv:-302.-941 (2013, preprint)
  8.Deuschel, J.-D.: Entropic repulsion of the lattice free field. II. The \(0\) -boundary case. Commun. Math. Phys. 181(3), 647-65 (1996)MATH MathSciNet CrossRef
  9.Deuschel, J.-D., Giacomin, G.: Entropic repulsion for massless fields. Stoch. Process. Appl. 89(2), 333-54 (2000)MATH MathSciNet CrossRef
  10.Deuschel, J.-D., Giacomin, G., Ioffe, D.: Large deviations and concentration properties for \(\nabla \phi \) interface models. Probab. Theory Relat. Fields 117(1), 49-11 (2000)MATH MathSciNet CrossRef
  11.Dobrushin, R., Kotecky, R., Shlosman, S.: Wulff construction, Translations of Mathematical Monographs, vol. 104. American Mathematical Society, Providence (1992) (A global shape from local interaction, Translated from the Russian by the authors)
  12.Fortuin, C.M., Kasteleyn, P.W., Ginibre, J.: Correlation inequalities on some partially ordered sets. Commun. Math. Phys. 22, 89-03 (1971)MATH MathSciNet CrossRef
  13.Kotecky, R., Preiss, D.: Cluster expansion for abstract polymer models. Commun. Math. Phys. 103(3), 491-98 (1986)MATH CrossRef
  14.Lebowitz, J.L., Maes, C.: The effect of an external field on an interface, entropic repulsion. J. Stat. Phys. 46(1-), 39-9 (1987)MathSciNet CrossRef
  15.Lubetzky, E., Martinelli F., Sly, A.: Harmonic pinnacles in the Discrete Gaussian model. arXiv:-405.-241 (2014, preprint)
  16.Velenik, Y.: Localization and delocalization of random interfaces. Probab. Surv. 3, 112-69 (2006)MATH MathSciNet CrossRef
  
• 作者单位：Pietro Caputo (1)
  Fabio Martinelli (1)
  Fabio Lucio Toninelli (2)
  
  1. Dipartimento di Matematica e Fisica, Università Roma Tre, Largo S. Murialdo 1, 00146, Rome, Italy
  2. Université de Lyon, CNRS and Institut Camille Jordan, Université Lyon 1, 43 bd du 11 novembre 1918, 69622, Villeurbanne, France
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Probability Theory and Stochastic Processes
  Mathematical and Computational Physics
  Quantitative Finance
  Mathematical Biology
  Statistics for Business, Economics, Mathematical Finance and Insurance
  Operation Research and Decision Theory
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-2064

文摘

We obtain sharp asymptotics for the probability that the \((2+1)\)-dimensional discrete SOS interface at low temperature is positive in a large region. For a square region \(\Lambda \), both under the infinite volume measure and under the measure with zero boundary conditions around \(\Lambda \), this probability turns out to behave like \(\exp (-\tau _\beta (0) L \log L )\), with \(\tau _\beta (0)\) the surface tension at zero tilt, also called step free energy, and L the box side. This behavior is qualitatively different from the one found for continuous height massless gradient interface models (Bolthausen et al., Commun Math Phys 170(2):417-43, 1995; Deuschel et al., Stochastic Process Appl 89(2):333-54, 2000). Keywords SOS model Loop ensembles Random surface models Entropic repulsion Large deviations