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Disorder and Critical Phenomena Through Basic Probability Models

École d’Été de Probabilités de Saint-Flour XL – 2010

  • Book
  • © 2011

Overview

Authors:
  1. Giambattista Giacomin

    1. Département de Mathématiques, Université Paris Diderot, Paris, France

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  • It develops the tools that allow a person with a probability background to access a large amount of theoretical physics literature.
  • Recently developed mathematical techniques are explained in detail for a class of models that demand no background beyond a standard graduate course in probability.
  • It proposes various open questions, along with the explanation of what the answer is expected to be in the physical community.
  • Includes supplementary material: sn.pub/extras

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

Part of the book sub series: École d'Été de Probabilités de Saint-Flour (LNMECOLE)

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Table of contents (8 chapters)

  1. Front Matter

    Pages i-xi
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  2. Introduction

    • Giambattista Giacomin
    Pages 1-4

  3. Homogeneous Pinning Systems: A Class of Exactly Solved Models

    • Giambattista Giacomin
    Pages 5-27

  4. Introduction to Disordered Pinning Models

    • Giambattista Giacomin
    Pages 29-40

  5. Irrelevant Disorder Estimates

    • Giambattista Giacomin
    Pages 41-50

  6. Relevant Disorder Estimates: The Smoothing Phenomenon

    • Giambattista Giacomin
    Pages 51-61

  7. Critical Point Shift: The Fractional Moment Method

    • Giambattista Giacomin
    Pages 63-90

  8. The Coarse Graining Procedure

    • Giambattista Giacomin
    Pages 91-99

  9. Path Properties

    • Giambattista Giacomin
    Pages 101-112

  10. Back Matter

    Pages 113-130
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Keywords

About this book

Understanding the effect of disorder on critical phenomena is a central issue in statistical mechanics. In probabilistic terms: what happens if we perturb a system exhibiting a phase transition by introducing a random environment? The physics community has approached this very broad question by aiming at general criteria that tell whether or not the addition of disorder changes the critical properties of a model: some of the predictions are truly striking and mathematically challenging. We approach this domain of ideas by focusing on a specific class of models, the "pinning models," for which a series of recent mathematical works has essentially put all the main predictions of the physics community on firm footing; in some cases, mathematicians have even gone beyond, settling a number of controversial issues. But the purpose of these notes, beyond treating the pinning models in full detail, is also to convey the gist, or at least the flavor, of the "overall picture," which is, in many respects, unfamiliar territory for mathematicians.

Authors and Affiliations

  • Département de Mathématiques, Université Paris Diderot, Paris, France

    Giambattista Giacomin

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