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Author
Date
2022Type
- Doctoral Thesis
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yes
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Abstract
This thesis is devoted to the study of percolation on graphs of polynomial growth. We mostly focus on Bernoulli bond percolation, introduced by Broadbent and Hammersley in [BH57] as a model for porous media. This model exhibits a phase transition phenomenon, with a subcritical phase where the connection probabilities decay exponentially fast and a supercritical phase where we observe macroscopic structures emerging.
The study of percolation in more general graphs was initiated by Benjamini and Schramm in [BS96]. In particular, the study of the supercritical phase has become one of the main challenges, since many of the results on this phase rely on the difficult Theorem of Grimmett and Marstrand [GM90], which strongly depends on the symmetries of the Euclidean lattice.
In order to extend this result to more complicated geometries, we provide a set of new robust methods in Chapters 2 and 3. This allows us to prove supercritical sharpness on transitive graphs of polynomial growth. As a consequence of this, and a very recent geometrical result by Tessera and Tointon [TT21], in Chapter 4 we are able to show Schramm’s locality conjecture in the case of transitive graphs of polynomial growth.
A key concept to prove the robust version of Grimmett-Marstrand Theorem on the supercritical regime is the local uniqueness. This means that the presence of a unique big cluster is not only true globally, in which case we observe an infinite cluster, but also when restricted to a ball of big radius. To get a quantitative description of this local uniqueness we study how disjoint clusters merge together under the effect of a sprinkling. In Chapter 5 we explore this question for a particular family of graphs, extending the results of Benjamini and Tassion [BT17] to the context of graphs of polynomial growth.
In summary, this work provides new techniques to study supercritical percolation on graphs of polynomial growth. In particular, this adds new perspectives to the understanding of percolation on Z^d, and we hope it will lead to new developments in this theory as well as for other statistical mechanics models. Show more
Permanent link
https://doi.org/10.3929/ethz-b-000585148Publication status
publishedExternal links
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Publisher
ETH ZurichSubject
Percolation theory; Geometric group theoryOrganisational unit
09584 - Tassion, Vincent / Tassion, Vincent
09584 - Tassion, Vincent / Tassion, Vincent
Funding
851565 - Critical and supercritical percolation (EC)
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