Graph Theory

Coarse Geometry and Randomness: École d'Été de Probabilités by Itai Benjamini

By Itai Benjamini

These lecture notes learn the interaction among randomness and geometry of graphs. the 1st a part of the notes reports numerous easy geometric strategies, sooner than relocating directly to study the manifestation of the underlying geometry within the habit of random strategies, more often than not percolation and random walk.

The learn of the geometry of endless vertex transitive graphs, and of Cayley graphs particularly, is reasonably good constructed. One objective of those notes is to indicate to a couple random metric areas modeled through graphs that develop into a bit unique, that's, they admit a mixture of homes no longer encountered within the vertex transitive global. those comprise percolation clusters on vertex transitive graphs, severe clusters, neighborhood and scaling limits of graphs, lengthy diversity percolation, CCCP graphs acquired through contracting percolation clusters on graphs, and desk bound random graphs, together with the uniform countless planar triangulation (UIPT) and the stochastic hyperbolic planar quadrangulation (SHIQ).

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Additional resources for Coarse Geometry and Randomness: École d'Été de Probabilités de Saint-Flour XLI - 2011

Example text

D 1/n 1 p n : If p < proof. 7. 1 . Z / . 1. 8. d -regular tree/ D d 1 1 . 9. , any path from the root to 1 must cross and edge from the set. For example, if we take Z with the root vertex 0, then ff 3; 2g; f5; 6gg is a cut set while ff3; 4g; f10; 11gg is not. 10. G; / is said to be a Minimal Cut Set (MCS) if it is a cut set and any T ¨ S is not a cut set. Next we present the first sufficient condition for pc < 1 which is based on the notion of minimal cut sets. 11. Let G be a connected infinite graph with a root vertex .

13. Z2 / < 1. 14. G/ < 1. G/ < 1. 15. (Level 3) Show that the property pc < 1 is invariant under rough isometries between bounded degree graphs, see Chap. 6 for definitions. Hint: Use domination by product measure (see [LSS97]). G/ < 1, based on an idea of Kesten. Much of the material in this section can be found in [BPP98] and [Per99]. Let 1 ; 2 be two paths of Self-Avoiding Walks (SAW), and denote by j 1 \ 2 j the number of edges in their intersection. 16. j 1 \ 2 j > n/ < Â n . That is, the probability of two independently picked P paths according to having more than n edges in common (intersections), decays exponentially in n.

However, its Euclidean center is closer to origin than its hyperbolic center x, and its Euclidean radius is smaller than its hyperbolic radius r. There are explicit formulas for both these quantities. A comment regarding the Poincaré half plane model. This model for the hyperbolic plane is given by the complex upper half plane fx Ciy W y > 0g together with the metric ds2 WD 4 dx2 C dy2 : y2 In this model, the intersection of the upper half plane with Euclidean circles orthogonal to the real line are infinite geodesics.

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