Geometry of critical discrete structures: long-range percolation on the hierarchical lattice and the discrete torus

Arghyadeep Chatterjee, Sourish Maniyar, Sanchayan Sen

Consider (a) balls ${\Lambda }_n$ of growing volumes in the $d$-dimensional hierarchical lattice, and (b) the $d$-dimensional discrete torus ${\mathbb{T}}_n^d$ on $n^d$ vertices. Place edges…