By Itai Benjamini
These lecture notes research the interaction among randomness and geometry of graphs. the 1st a part of the notes experiences numerous easy geometric suggestions, sooner than relocating directly to learn the manifestation of the underlying geometry within the habit of random strategies, 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 constructed. One objective of those notes is to indicate to a couple random metric areas modeled by means of graphs that change into a little unique, that's, they admit a mixture of homes no longer encountered within the vertex transitive global. those contain percolation clusters on vertex transitive graphs, severe clusters, neighborhood and scaling limits of graphs, lengthy variety percolation, CCCP graphs acquired via 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).