An Introduction to Random Interlacements

1. Front Matter

   Pages i-x

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2. Random Walk, Green Function, and Equilibrium Measure

   • Alexander Drewitz, Balázs Ráth, Artëm Sapozhnikov

   Pages 1-9

3. Random Interlacements: First Definition and Basic Properties

   • Alexander Drewitz, Balázs Ráth, Artëm Sapozhnikov

   Pages 11-18

4. Random Walk on the Torus and Random Interlacements

   • Alexander Drewitz, Balázs Ráth, Artëm Sapozhnikov

   Pages 19-29

5. Poisson Point Processes

   • Alexander Drewitz, Balázs Ráth, Artëm Sapozhnikov

   Pages 31-35

6. Random Interlacement Point Process

   • Alexander Drewitz, Balázs Ráth, Artëm Sapozhnikov

   Pages 37-50

7. Percolation of the Vacant Set

   • Alexander Drewitz, Balázs Ráth, Artëm Sapozhnikov

   Pages 51-60

8. Source of Correlations and Decorrelation via Coupling

   • Alexander Drewitz, Balázs Ráth, Artëm Sapozhnikov

   Pages 61-73

9. Decoupling Inequalities

   • Alexander Drewitz, Balázs Ráth, Artëm Sapozhnikov

   Pages 75-86

10. Phase Transition of \({\mathcal{V}}^{u}\)

    • Alexander Drewitz, Balázs Ráth, Artëm Sapozhnikov

    Pages 87-95

11. Coupling of Point Measures of Excursions

    • Alexander Drewitz, Balázs Ráth, Artëm Sapozhnikov

    Pages 97-113

12. Back Matter

    Pages 115-120

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This book gives a self-contained introduction to the theory of random interlacements. The intended reader of the book is a graduate student with a background in probability theory who wants to learn about the fundamental results and methods of this rapidly emerging field of research. The model was introduced by Sznitman in 2007 in order to describe the local picture left by the trace of a random walk on a large discrete torus when it runs up to times proportional to the volume of the torus. Random interlacements is a new percolation model on the d-dimensional lattice. The main results covered by the book include the full proof of the local convergence of random walk trace on the torus to random interlacements and the full proof of the percolation phase transition of the vacant set of random interlacements in all dimensions. The reader will become familiar with the techniques relevant to working with the underlying Poisson Process and the method of multi-scale renormalization, which helps in overcoming the challenges posed by the long-range correlations present in the model. The aim is to engage the reader in the world of random interlacements by means of detailed explanations, exercises and heuristics. Each chapter ends with short survey of related results with up-to date pointers to the literature.

• Percolation
• Potential theory
• Random interlacements
• Random walk
• Random walk on torus
• Renormalization

• Department of Mathematics, Columbia University, New York City, USA

  Alexander Drewitz

• Department of Mathematics, University of British Columbia, Vancouver, Canada

  Balázs Ráth

• Max-Planck Institute of Mathematics in the Sciences, Leipzig, Germany

  Artëm Sapozhnikov

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