Random Interlacements and Amenability

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Imani, Sahar | 2018

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 51191 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Haji Mir Sadeghi, Miromid
Abstract:
In this thesis, we consider the model of random interlacement on transient graghs, which was first introduced by Sznitman for the special case of Zd(d > 3) in 2010’s. in Sznitman’s case, it was shown that on Zd: for any intensity u > 0, the interlacement set is almost surely connected. The main result of this thesis says that for transient, transitive graphs, the above property holds if and only if the graph is amenable. In paticular, we show that in nonamenable transitive graphs, for small values of the intensity u the interlacement set has infinitely many infinite clusters. We also provide examples of nonamenable transitive graphs, for which the interlacement set becomes connected for large values of u. Finally, we establish the monotonicity of the transition between the disconnected and the connected phases. providing the uniqueness of the critical value uc where this transition occures
Keywords:
Random Walk ; Graphs ; Amenability ; Poisson Process ; Random Interlacements