By Itai Benjamini
These lecture notes research the interaction among randomness and geometry of graphs. the 1st a part of the notes reports a number of simple geometric thoughts, sooner than relocating directly to learn the manifestation of the underlying geometry within the habit of random strategies, normally percolation and random walk.
The learn of the geometry of endless vertex transitive graphs, and of Cayley graphs particularly, within reason good constructed. One aim of those notes is to indicate to a few random metric areas modeled by way of graphs that change into slightly unique, that's, they admit a mix of homes now not encountered within the vertex transitive international. those contain percolation clusters on vertex transitive graphs, severe clusters, neighborhood and scaling limits of graphs, lengthy diversity percolation, CCCP graphs acquired by means of 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 info for Coarse Geometry and Randomness: École d'Été de Probabilités de Saint-Flour XLI - 2011
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 . G/ < 1.
X; d / be a metric ı-hyperbolic space with a base point w and let k be a positive integer. T; dT / with a base point t and a map ˆ W X ! T such that 1. e. x; w/; for any point x of X . 2 Hyperbolic Graphs 29 2. x; y/: Proof. We will give the main ideas of the proof here, for the full details see [GH90] Chap. 2 (you will also find a more general variant of this theorem which admits existence of rays in this space). Consider an integer L and a sequence x1 ; : : : ; xL of points in X . xi 2Äi ÄL kı: Now we will define a new pseudometric on X which is 0-hyperbolic.
4 Applications 47 that X0 D . Then we have a probability distribution over the set of (equivalence classes of) graphs with a bi-infinite path on them. The stationarity and reversibility assumption tells us that this probability distribution is invariant under the shift operations which consist of translating the root point of the path by 1 or 1. Hence much of ergodic theory can be applied. Measured Equivalence Relations The MTP has also many connection with so-called measured equivalence relations.