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

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 uncomplicated geometric options, sooner than relocating directly to learn the manifestation of the underlying geometry within the habit of random approaches, often percolation and random walk.

The learn of the geometry of limitless vertex transitive graphs, and of Cayley graphs specifically, in all fairness good constructed. One target of those notes is to indicate to a few random metric areas modeled by way of graphs that become slightly unique, that's, they admit a mix of houses now not encountered within the vertex transitive global. those contain percolation clusters on vertex transitive graphs, severe clusters, neighborhood and scaling limits of graphs, lengthy diversity percolation, CCCP graphs bought through 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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**Sample text**

N / i converges in distribution when n ! 1. The last part of the theorem follows from the remarks made on the volume growth inside T1 . 32. T; / and for r 0 denote r the first time the walk reach distance r from the root. Show that Ä1 r 1=˛ Ä EŒ r Ä Ä2 r 1=˛ for some 0 < Ä1 < Ä2 < 1. Chapter 6 Random Planar Geometry What is a typical random surface? This question has arisen in the theory of twodimensional quantum gravity where discrete triangulations have been considered as a discretization of a random continuum Riemann surface.

2. 3. 4. x 0 // x 0 j Ä c for every x 0 2 Y . The first two conditions mean that f and g are nearly Lipschitz if we are looking from far away. The two other conditions mean that f and g are nearly inverse of each other. It is easy to check that the composition of two rough-isometries is also a rough-isometry. Thus, rough isometries provide an equivalence relation on the class of metric spaces. 11. (Level 2) 1. 9. Find rough isometry constants between X and T . 2. Show that a regular (infinite) tree and a hyperbolic plane are not rough isometric.

1/. We continue this process until we obtain a subdivision consisting of ˛b with lengths between 12 and 1. len. // C 1 steps. Recall that since the space is hyperbolic, all the triangles with the sides ˛b , ˛b0 , ˛b1 are ı-thin. Since d. R// > 4ı it follows that d. 0/ on ˛ such that d. 0// < ı. 0/. We Continue in the same manner. len. n/ lies on some geodesic of length not greater than 1. n C 1// < 1. n C 1// > R C r by definition. p/ > 2 r 1 ı 1 t u and completes the proof. A metric tree is one of the most basic examples of a hyperbolic space.