Adelic Models of Percolation

TL;DR

This paper introduces adelic models to unify various long-range percolation models on lattices and hierarchical structures using number theoretic and geometric frameworks.

Contribution

It develops a novel approach connecting lattice, hierarchical, and adelic percolation models through the use of the adelic product formula and geometric deformations.

Findings

Unified framework for percolation models via adelic structures

Relation between lattice and hierarchical percolation models

Application of number theory to percolation theory

Abstract

Models of long range percolations on lattices and on hierarchical lattices are 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 number fields that relates the toric percolation model on the lattice given by the ring of integers in the Minkowski embedding to another adelic percolation model.