Introduced in 1957 by Broadbent and Hammersley as a simple probabilistic model for movement of fluid through porous media, percolation theory has emerged as one of the most active and richest areas of research in modern probability theory. As one the most well-known models demonstrating an order-disorder phase transition, the techniques developed in the context of percolation theory have been widely successful in rigorous understanding of models in classical statistical physics, e.g. Ising and Potts. Furthermore in two dimensions it enjoys a deep connection with conformal field theory. In this talk I will attempt to provide a very brief tour of some horizons of classical percolation theory including some very recent results. I will then move onto percolation models in dependent media which pose new challenges to overcome. In this context I will present some recent results about a model related to the geometry of level sets of a canonical random Gaussian function (Gaussian free field) on lattice graphs. I will also briefly discuss some directions for future research. Some of the recent results are based on joint works with Hugo Duminil-Copin, Aran Raoufi, Pierre-Francois Rodriguez, Franco Severo and Ariel Yadin. No knowledge about percolation theory is assumed on the part of the audience.

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Last updated: 17 May 2024