By Itai Benjamini

These lecture notes examine the interaction among randomness and geometry of graphs. the 1st a part of the notes experiences numerous easy geometric techniques, prior to relocating directly to study the manifestation of the underlying geometry within the habit of random methods, usually percolation and random walk.

The research of the geometry of endless vertex transitive graphs, and of Cayley graphs particularly, within reason good built. One objective of those notes is to indicate to a few random metric areas modeled by means of graphs that grow to be a bit unique, that's, they admit a mix of houses no longer encountered within the vertex transitive international. those comprise percolation clusters on vertex transitive graphs, serious clusters, neighborhood and scaling limits of graphs, lengthy diversity percolation, CCCP graphs received by way 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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