Foundations and the critical point on the square lattice

1 distance ku vk is 1. For brevity, we will also write uv instead of {u, v} for the edge between u and v (note that uv = vu). The boundary @G of G is the set of those vertices u in V for which there is a v 2 Z2 \ V with ku vk = 1. We associate to G a space of spin configurations ⌦G = { 1,+1}V . For 2 ⌦G , v denotes the spin at v. Sometimes we impose positive boundary conditions, meaning that we restrict ourselves to the set of configurations