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Path Properties

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Disorder and Critical Phenomena Through Basic Probability Models

Part of the book series: Lecture Notes in Mathematics ((LNMECOLE,volume 2025))

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Abstract

We present a few selected results that show that localization, respectively delocalization, in the free energy sense does correspond to localized, respectively delocalized, path behavior. We provide also an overview of the literature on path behavior for disorder pinning models, with a particular attention to the main theme of these notes, that is disorder (ir)relevance, and therefore with a particular eye to trying to quantify the differences in path behavior between pure and disordered systems. While certain questions have found satisfactory answers, important (and intriguing) issues remain open.

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References

  1. A. Aharony, A.B. Harris, Absence of self-averaging and universal fluctuations in random systems near critical points. Phys. Rev. Lett. 77, 3700–3703 (1996)

    Article  Google Scholar 

  2. S. Albeverio, X.Y. Zhou, Free energy and some sample path properties of a random walk with random potential. J. Stat. Phys. 83, 573–622 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  3. C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto, G. Scheffer, Sur les inégalités de Sobolev logarithmiques. Panoramas et Synthèses 10, Société Mathématique de France, Paris, 2000

    Google Scholar 

  4. M. Biskup, F. den Hollander, A heteropolymer near a linear interface, Ann. Appl. Probab. 9, 668–687 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. E. Bolthausen, F. den Hollander, Localization transition for a polymer near an interface. Ann. Probab. 25, 1334–1366 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  6. A. Bovier, Statistical Mechanics of Disordered Systems. A Mathematical Perspective. Cambridge Series in Statistical and Probabilistic Mathematics (Cambridge University Press, Cambridge, 2006)

    Google Scholar 

  7. F. Caravenna, G. Giacomin, The weak coupling limit of disordered copolymer models. Ann. Probab. 38, 2322–2378 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  8. J.T. Chayes, L. Chayes, D.S. Fisher, T. Spencer, Correlation length bounds for disordered Ising ferromagnets. Commun. Math. Phys. 120, 501–523 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  9. G. Giacomin, Random Polymer Models (Imperial College Press, London, 2007)

    Book  MATH  Google Scholar 

  10. G. Giacomin, F.L. Toninelli, Estimates on path delocalization for copolymers at selective interfaces. Probab. Theory Relat. Fields 133, 464–482 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. G. Giacomin, F.L. Toninelli, Smoothing effect of quenched disorder on polymer depinning transitions. Commun. Math. Phys. 266, 1–16 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. G. Giacomin, F.L. Toninelli, The localized phase of disordered copolymers with adsorption. ALEA Lat. Am. J. Probab. Math. Stat. 1, 149–180 (2006)

    MATH  MathSciNet  Google Scholar 

  13. G. Giacomin, F.L. Toninelli, On the irrelevant disorder regime of pinning models. Ann. Probab. 37, 1841–1873 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  14. H. Lacoin, The martingale approach to disorder irrelevance for pinning models. Electron. Commun. Probab. 15, 418–427 (2010)

    MathSciNet  Google Scholar 

  15. M. Ledoux, The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs, vol. 89 (American Mathematical Society, Providence, RI, 2001)

    Google Scholar 

  16. C. Monthus, T. Garel, Distribution of pseudo-critical temperatures and lack of self-averaging in disordered Poland-Scheraga models with different loop exponents. Eur. Phys. J. B 48, 393–403 (2005)

    Article  Google Scholar 

  17. J.-C. Mourrat, On the delocalized phase of the random pinning model. arXiv:1010.4671

    Google Scholar 

  18. N. Pétrélis, Copolymer at selective interfaces and pinning potentials: weak coupling limits. Ann. Inst. H. Poincaré (B) Probab. Stat. 45, 175–200 (2009)

    Google Scholar 

  19. Ya. G. Sinai, A random walk with a random potential. Theory Probab. Appl. 38, 382–385 (1993)

    Google Scholar 

  20. M. Talagrand, A new look at independence. Ann. Probab. 24, 1–34 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  21. F.L. Toninelli, Critical properties and finite-size estimates for the depinning transition of directed random polymers. J. Stat. Phys. 126, 1025–1044 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  22. F.L. Toninelli, Localization transition in disordered pinning models. Effect of randomness on the critical properties, in Methods of Contemporary Mathematical Statistical Physics, Lecture Notes in Mathematics, vol. 1970 (2009), pp. 129–176

    Google Scholar 

  23. C. Villani, Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 58 (American Mathematical Society, Providence, RI, 2003)

    Google Scholar 

  24. S. Wiseman, E. Domany, Finite-size scaling and lack of self-averaging in critical disordered systems. Phys. Rev. Lett. 81, 22–25 (1998)

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Département de Mathématiques, Université Paris Diderot, 175 rue du Chevaleret, 75205, Paris, France

    Giambattista Giacomin

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  1. Giambattista Giacomin
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Correspondence to Giambattista Giacomin .

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Giacomin, G. (2011). Path Properties. In: Disorder and Critical Phenomena Through Basic Probability Models. Lecture Notes in Mathematics(), vol 2025. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21156-0_8

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