By Itai Benjamini
These lecture notes examine the interaction among randomness and geometry of graphs. the 1st a part of the notes experiences a number of easy geometric recommendations, prior to relocating directly to study the manifestation of the underlying geometry within the habit of random methods, regularly percolation and random walk.
The learn of the geometry of limitless vertex transitive graphs, and of Cayley graphs particularly, in all fairness good constructed. One target of those notes is to indicate to a couple random metric areas modeled through graphs that become 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 variety percolation, CCCP graphs got by means of 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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Extra info for Coarse Geometry and Randomness: École d'Été de Probabilités de Saint-Flour XLI - 2011
Let us start by defining hyperbolic spaces and state some of their basic properties. The most general definition uses the notion of the Gromov product. 3. X; d / be a metric space and x; y; z 2 X three points in it. xjy/z is the following: Up to some additive constant it describes the distance from z to any x-y geodesic. xjy/z is in fact precisely this distance. We now give the definition of ı-hyperbolic space. 4. yjz/w g ı: This definition can be rewritten in another form. There exist three possibilities to divide these four points into pairs.
The n n-grid tori converge towards Z2 as n ! 1. 4. Gn / ! 1 as n ! 1. Then Gn converges towards a d -regular tree with respect to dloc . 5. r/. 6. r/, then there is c < 1 such that it is cr sofic? 0; r/ as an r-ball? d / can be? Note that for trees this is the girth problem. 10). Let fGn gn 1 be an expander family. Let G be a graph such that Gn ! G with respect to dloc as n ! 1. Then: 1. G/ then there exists ˛ > 0 such that Â Pp there is an open cluster in Gn of size ˛jGn j Ã ! 1 2. G/ then for every ˛ > 0 Â Pp there is an open cluster in Gn of size ˛jGn j Ã !
1) is often omitted from the definition of the hyperbolic metric. We remark that it is also common to identify points of H2 with points in the open unit disc in the Euclidean plane rather than in the complex plane. t/ W 0 Ä t Ä 1g has length Z 1 L. 1 dx dy : x 2 y 2 /2 If z1 ; z2 2 H2 , then the geodesic between them (that is, the shortest curve that starts at z1 and ends at z2 ) is either a segment of an Euclidean circle that intersects the boundary of D orthogonally, or a segment of a straight line that passes through the origin.