Abstract
This chapter develops in detail the most advanced of the two coarse graining techniques employed in the previous chapter. Roughly, it consists in looking at the system in blocks of finite size k, which essentially is the annealed correlation length: if we can get suitable estimates for systems up to that size k, we can bound the fractional moment of the partition function of the (arbitrarily large) system in terms of the partition function of a homogeneous model with pinning parameter that depend on the estimates up to size k.
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References
G. Giacomin, H. Lacoin, F.L. Toninelli, Marginal relevance of disorder for pinning models. Commun. Pure Appl. Math. 63, 233–265 (2010)
G. Giacomin, H. Lacoin, F.L. Toninelli, Disorder relevance at marginality and critical point shift. Ann. Inst. H. Poincaré (B) Probab. Stat. 47, 148–175 (2011)
F.L. Toninelli, Coarse graining, fractional moments and the critical slope of random copolymers. Electron. J. Probab. 14, 531–547 (2009)
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© 2011 Springer-Verlag Berlin Heidelberg
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Giacomin, G. (2011). The Coarse Graining Procedure. 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_7
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DOI: https://doi.org/10.1007/978-3-642-21156-0_7
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