| import Mathlib | |
| namespace OAI | |
| open Set MeasureTheory ProbabilityTheory | |
| namespace CriticalPercolation | |
| /-- An undirected graph with labelled bonds, including loops and parallel bonds. -/ | |
| structure BondGraph (V E : Type*) where | |
| ends : E → Sym2 V | |
| namespace BondGraph | |
| variable {V E : Type*} (G : BondGraph V E) | |
| /-- The graph of retained bonds. `fromEdgeSet` discards loops. -/ | |
| def openGraph (ω : Set E) : SimpleGraph V := | |
| SimpleGraph.fromEdgeSet (G.ends '' ω) | |
| /-- The underlying connectivity graph with every bond retained. -/ | |
| def fullGraph : SimpleGraph V := G.openGraph Set.univ | |
| /-- Local finiteness counts bonds, not just distinct neighbors. -/ | |
| def LocallyFinite : Prop := ∀ v : V, {e : E | v ∈ G.ends e}.Finite | |
| /-- Automorphisms preserve the vertex and bond structure, including multiplicities. -/ | |
| structure Aut where | |
| vertex : V ≃ V | |
| edge : E ≃ E | |
| preserves : ∀ e, G.ends (edge e) = Sym2.map vertex (G.ends e) | |
| /-- Finitely many vertex orbits under the full automorphism group. -/ | |
| def QuasiTransitive : Prop := | |
| ∃ S : Set V, S.Finite ∧ ∀ v : V, ∃ s ∈ S, ∃ φ : G.Aut, φ.vertex s = v | |
| /-- Finite open paths, including the path of length zero, define the open cluster. -/ | |
| def cluster (ω : Set E) (v : V) : Set V := {w | (G.openGraph ω).Reachable v w} | |
| /-- The event that at least one open cluster is infinite. -/ | |
| def percolates : Set (Set E) := {ω | ∃ v : V, (G.cluster ω v).Infinite} | |
| /-- Independent retention with parameter `p ∈ [0,1]` of each labelled bond. -/ | |
| noncomputable def law (_G : BondGraph V E) (p : unitInterval) : Measure (Set E) := | |
| ProbabilityTheory.setBernoulli Set.univ p | |
| /-- The infimum of parameters with positive probability of an infinite cluster. -/ | |
| noncomputable def criticalProbability : unitInterval := | |
| sInf {p : unitInterval | 0 < G.law p G.percolates} | |
| end BondGraph | |
| end CriticalPercolation | |
| end OAI | |