By Itai Benjamini
These lecture notes examine the interaction among randomness and geometry of graphs. the 1st a part of the notes stories numerous easy geometric strategies, earlier than relocating directly to research the manifestation of the underlying geometry within the habit of random methods, in general percolation and random walk.
The examine of the geometry of countless vertex transitive graphs, and of Cayley graphs particularly, in all fairness good built. One target of those notes is to indicate to a few random metric areas modeled through graphs that change into just a little unique, that's, they admit a mixture of houses now not encountered within the vertex transitive international. those comprise percolation clusters on vertex transitive graphs, severe clusters, neighborhood and scaling limits of graphs, lengthy diversity percolation, CCCP graphs acquired via contracting percolation clusters on graphs, and desk bound random graphs, together with the uniform endless 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
If d > 1 and p > 3=4 then 0 is connected to 1 with positive probability. 2. If d D 2 and p < 1=4 then 0 is not connected to 1 with probability 1. Notice that it is enough to show the first part of the theorem for d D 2, since we can think of Z2 as a subgraph of Zd for d 3. The above theorem suggests the existence of a phase transition concerning the existence of an infinite cluster in the percolation process. 2. Let G be a graph with a fixed root . G; / D inff0 Ä p Ä 1 W Pp . $ 1/ > 0g: This is called the critical probability for Bernoulli bond percolation on G rooted at .
2. (Level 3) What is the separation function (in the sense of Sect. 4) of this graph? 1 The Hyperbolic Plane There are several models of hyperbolic geometry. All of them are equivalent, in the sense that there are isometries between them. Which model one wants to work with very much depends on the nature of the problem of interest. The most common models are the Poincaré unit disc model and the half plane model. We now concentrate on the properties of the first. Denote a point in the complex plane C by z D x C iy.
1: 2d d Intuitive explanation of the last theorem is the following: every vertex locally “feels” like in a 2d -regular tree, as the number of crossing paths of a given length is negligible compared to the total number of paths of the same length, for a rather simple proof without the sharp error terms see [ABS04]. 3 Unimodular Random Graphs The metric dloc enables us to speak about convergence of rooted graphs. The root vertices of these graphs play a crucial role in that convergence. In the following, the idea is to randomize the root vertex in order to get a typical view on a given graph.