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Critical Point Shift: The Fractional Moment Method

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

Abstract

This chapter is devoted to showing that, when α ≥ 1 ∕ 2, quenched and annealed critical points are different for every β > 0, with explicit estimates on the difference. Such a result follows from upper bounds on the free energy that are obtained by estimating fractional moments (of order less than one) of the partition function. Estimates for every β > 0, notably for arbitrarily small values of β, are obtained by using a change of measure argument on the law of the disorder and by coarse graining techniques. Proving such estimates becomes harder and harder as α approaches 1 ∕ 2, i.e. the marginal disorder case in the Harris’ sense: for α = 1 ∕ 2 the Harris criterion yields no prediction and whether quenched and annealed critical points differed or not has been a debated issue in the physical literature.

Keywords

  • Partition Function
  • Correlation Length
  • Coarse Graining
  • Random Environment
  • Large Deviation Principle

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Correspondence to Giambattista Giacomin .

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Giacomin, G. (2011). Critical Point Shift: The Fractional Moment Method. 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_6

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