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 studies a number of easy geometric strategies, ahead of relocating directly to research the manifestation of the underlying geometry within the habit of random strategies, normally percolation and random walk.

The learn of the geometry of limitless vertex transitive graphs, and of Cayley graphs particularly, within reason good built. One target of those notes is to indicate to a few random metric areas modeled by way of graphs that become just a little unique, that's, they admit a mixture of houses no longer encountered within the vertex transitive international. those contain percolation clusters on vertex transitive graphs, severe clusters, neighborhood and scaling limits of graphs, lengthy variety percolation, CCCP graphs received via contracting percolation clusters on graphs, and desk bound random graphs, together with the uniform limitless 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

Example text

Is unimodular. Proof. We directly verify the MTP. G; / and let f be a transport function. To simplify notation, we write C instead of C. /. :; :; :/ is a transport function and applying the MTP for the original unimodular graph yields the result. 26. We just used a weak property of the Bernoulli percolation in order to say that F is a transport function. This reasoning is valid for any invariant percolation process. 27. G; / be a unimodular random graph. Delete all vertices of degree at least M for some M 0.

Is unimodular. Proof. We directly verify the MTP. G; / and let f be a transport function. To simplify notation, we write C instead of C. /. :; :; :/ is a transport function and applying the MTP for the original unimodular graph yields the result. 26. We just used a weak property of the Bernoulli percolation in order to say that F is a transport function. This reasoning is valid for any invariant percolation process. 27. G; / be a unimodular random graph. Delete all vertices of degree at least M for some M 0.

4 ([BC13]). e. 2 Circle Packing Since random triangulations and UIPT are planer graphs, it is very tempting to try and understand their conformal structures. The theory of Circle Packing is wellsuited for this purpose. A circle packing on the sphere is an arrangement of circles on a given surface (in our case the sphere) such that no overlapping occurs and so that all circles touch another. , the portion of surface covered by them. The contact graph of a circle packing is defined to be the graph with set of vertices which correspond to the set of circles and an edge between two circles if and only if they are tangent.

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