Broadly speaking, percolation theory is the branch of probability theory concerned with the emergence of long range order in a random environment. As is often the case with several areas of probability theory, independence among the components of the underlying random environment serves as a very natural and fruitful set-up. In this “classical” setting, the area has seen tremendous progress over the past several decades. Recently, there has been a splurge of interest in studying percolation phenomena in environments which are strongly dependent and exhibit significant structural rigidity compared to the independent models. Such rigidities may arise from various sources like the algebraic decay of correlation or the analytic properties of an underlying random field, or the topological properties of some random objects like the random walk or loop-soup etc.

In the first part of the talk, I will give an introduction to these developments. In the second part of the talk, I will discuss some recent results (several from some ongoing works) pertaining to the percolation of level-sets of Gaussian free fields on lattices and finite graphs, vacant sets of random walks on torus and lattice, and random interlacements. Some of these results illustrate phenomenologies that are either absent or extremely hard to prove using current techniques in the independent set-up.