Adelic Percolation

Models of long range percolations on lattices and on hierarchical lattices appear at first to represent very different random geometries. However, both can be reduced to building blocks of a similar nature through an adelic perspective suggested by Manin's "reflections on arithmetical physics". Indeed these two types of percolation models can be related through the use of three intermediate geometries: a 1-parameter deformation based on the power mean function, relating lattice percolation to a percolation model governed by the toric volume form; the adelic product formula for a function field, relating the hierarchical lattice model to an adelic percolation model; and the adelic product formula for a number field that relates the toric percolation model on the lattice given by its ring of integers in the Minkowski embedding to another adelic percolation model.