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.
Links:
[1] https://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/de/node/3444
[3] https://www.mpim-bonn.mpg.de/de/maninmemorial