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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Criticality.AutDisplacement
import OAI.Probability.CriticalPercolation.GraphGeometry.UniformGeometry
import OAI.Probability.CriticalPercolation.ProductMeasure.BernoulliErgodicity
namespace OAI
section
noncomputable section
open Set MeasureTheory
open scoped Classical
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
theorem invariant_event_zero_one [Infinite V] (hc : G.fullGraph.Connected)
(hl : G.LocallyFinite) (hq : G.QuasiTransitive) (p : unitInterval)
(A : Set (Set E)) (hA : MeasurableSet A)
(hinv : ∀ φ : G.Aut, (fun ω : Set E => φ.edge '' ω) ⁻¹' A = A) :
G.law p A = 0 ∨ G.law p A = 1 := by
have hh := BernoulliProduct.invariant_zero_one p (fun φ : G.Aut => (Aut.symm G φ).edge)
(fun S => by
obtain ⟨φ,hφ⟩ := G.aut_displaces_finite_bonds hc hl hq (S : Set E) S.finite_toSet
exact ⟨φ,hφ⟩) A hA (fun φ => hinv (Aut.symm G φ))
simpa only [BernoulliProduct.law_const,law] using hh
end CriticalPercolation.BondGraph
end
end
end OAI
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