| import Mathlib |
| import OAI.Probability.CriticalPercolation.Model |
| import OAI.Probability.CriticalPercolation.Gluing.FreshTrial |
| import OAI.Probability.CriticalPercolation.Criticality.FiniteEnergy |
| import OAI.Probability.CriticalPercolation.GraphGeometry.GraphBalls |
| import OAI.Probability.CriticalPercolation.GraphGeometry.FiniteInduced |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open Set Function MeasureTheory ProbabilityTheory |
| open scoped Classical BigOperators ENNReal |
|
|
| namespace CriticalPercolation.BernoulliProduct |
| variable {E : Type*} |
|
|
| lemma law_allOpen (q : E → unitInterval) (S : Set E) (hS : S.Finite) : |
| law q (allOpen S)=∏ e∈hS.toFinset, bitMeasure (q e) {True} := by |
| have he : allOpen S=⋂ e∈hS.toFinset, {ω : Set E | (e∈ω)=True} := by |
| ext ω |
| simp [allOpen,Set.subset_def] |
| rw [he,(independent_coordinates q).meas_biInter] |
| · exact Finset.prod_congr rfl (fun e _ => law_coordinate q e True) |
| · intro e _ |
| exact ⟨{True},MeasurableSet.singleton _,rfl⟩ |
|
|
| lemma law_allOpen_lower (q : E → unitInterval) (S : Set E) (hS : S.Finite) |
| (p : unitInterval) (N : ℕ) (hN : S.ncard ≤ N) (hq : ∀ e∈S, (p : ℝ) ≤ q e) : |
| (p : ℝ)^N ≤ (law q).real (allOpen S) := by |
| rw [measureReal_def,law_allOpen q S hS,ENNReal.toReal_prod] |
| calc |
| _ ≤ (p : ℝ)^S.ncard := pow_le_pow_of_le_one p.2.1 p.2.2 hN |
| _ = ∏ e∈hS.toFinset, (p : ℝ) := by simp [Set.ncard_eq_toFinset_card S hS] |
| _ ≤ _ := by |
| apply Finset.prod_le_prod₀ |
| · intro edge hedge; exact p.2.1 |
| · intro e he |
| change (p : ℝ) ≤ (bitMeasure (q e)).real {True} |
| rw [bitMeasure_real_singleton] |
| simpa only [bitWeight,ite_true] using hq e (hS.mem_toFinset.mp he) |
| end BernoulliProduct |
|
|
| namespace BondGraph |
| variable {V E : Type*} (G : BondGraph V E) |
|
|
| lemma vertexBalls_disjoint {v w : V} {n : ℕ} (h : w∉G.vertexBall v (n+n)) : |
| Disjoint (G.vertexBall v n) (G.vertexBall w n) := by |
| apply Set.disjoint_left.mpr |
| intro z hz hz' |
| exact h (G.vertexBall_add hz n (G.vertexBall_comm.mp hz')) |
|
|
| lemma inducedEdges_disjoint {S T : Set V} (h : Disjoint S T) : |
| Disjoint (G.inducedEdges S) (G.inducedEdges T) := by |
| apply Set.disjoint_left.mpr |
| intro e he he' |
| obtain ⟨⟨x,y⟩,hxy⟩ := Sym2.mk_surjective (G.ends e) |
| have hx : x∈G.ends e := by rw [← hxy]; simp |
| exact Set.disjoint_left.mp h (he x hx) (he' x hx) |
|
|
| lemma inward_trials_disjoint (F : V → Set E) (m : ℕ) |
| (hF : ∀ v, F v⊆G.inducedEdges (G.vertexBall v (2*m+1))) |
| {v w : V} (h : w∉G.vertexBall v (4*m+2)) : Disjoint (F v) (F w) := by |
| have h' : w∉G.vertexBall v ((2*m+1)+(2*m+1)) := by |
| simpa only [show (2*m+1)+(2*m+1)=4*m+2 by omega] using h |
| exact (G.inducedEdges_disjoint (G.vertexBalls_disjoint h')).mono (hF v) (hF w) |
|
|
| end BondGraph |
| end CriticalPercolation |
| end |
| end |
| end |
|
|
| end OAI |
|
|
