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f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 | 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
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