I will discuss the scaling limit of the simple random walk on infinite random graphs embedded into R^d. The prime examples are the infinite cluster of supercritical bond percolation on Z^d and the Voronoi graph arising from a homogeneous Poisson point process. Under the usual diffusive scaling, the path distribution converges to that of a non-degenerate Brownian motion. The proofs are based on the consideration of a harmonic deformation of the graph, which is an embedding that makes the random walk a martingale. The size of the deformation is controlled using harmonicity arguments (planar graphs) and/or heat kernel estimates. Time permitting I will mention connections to homogenization theory and sketch a calculation which indicates that the deformation has Gaussian free field as its scaling limit. Based on joint works with N. Berger and T. Prescott.
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