Classically, the percolation transition is defined in terms of the emergence of either an infinite or a positive density component, depending on the setting. The percolation model itself can be easily generalized to higher dimensional cell complexes, but the correct analogue of a giant component can be unclear, particularly in the infinite volume case. We will discuss a notion of percolation on a compact manifold motivated by algebraic topology, marked by the appearance of a global loop or its higher dimensional analogue. We prove the existence of a phase transition for this homological percolation in all dimensions for an integer lattice complex on a large torus, both in independent percolation and in a generalization of the random-cluster model. Based on joint work with Benjamin Schweinhart and Matthew Kahle.
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