| import Mathlib |
| import OAI.Probability.CriticalPercolation.ProductMeasure.FiniteOpenWitness |
| import OAI.Probability.CriticalPercolation.Gluing.CorridorNormalization |
| import OAI.Probability.CriticalPercolation.Criticality.RestrictedSeed |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open Set Function MeasureTheory ProbabilityTheory |
| open scoped Classical ENNReal |
|
|
| namespace CriticalPercolation.BondGraph |
| variable {V E : Type*} (G : BondGraph V E) |
|
|
| def boxMove (t z : V → ℝ) (m : ℕ) (precision radius : ℝ) (y : V) (s : ℝ) : Set (Set E) := |
| {ω | ∃ a∈G.vertexBall y m, ∃ b, ∃ w : (G.openGraph ω).Walk a b, |
| (∀ x∈w.support, |t x-t y|≤radius ∧ |z x-z y|≤radius) ∧ |
| |t b-(t y+1)|≤precision ∧ |z b-(z y+s)|≤precision} |
|
|
| lemma coordinateMove_eq_boxMove (t z : V → ℝ) (m : ℕ) (e : ℝ) (y : V) (s : ℝ) : |
| G.coordinateMove t z m e y s=G.boxMove t z m (e/4) 100 y s := rfl |
|
|
| lemma walk_induced_support {S : Set V} {ω : Set E} {a b : V} |
| (w : (G.openGraph (ω∩G.inducedEdges S)).Walk a b) (hb : b∈S) : |
| ∀ x∈w.support, x∈S := by |
| induction w with |
| | nil => |
| intro x hx |
| simp only [SimpleGraph.Walk.support_nil,List.mem_singleton] at hx |
| subst x |
| exact hb |
| | @cons a c b hac w ih => |
| intro x hx |
| rcases (List.mem_cons.mp hx) with hxa|hx |
| · rw [hxa] |
| exact ((G.open_induced_adj S ω a c).mp hac).2.1 |
| · exact ih hb x hx |
|
|
| lemma boxMove_eq_restrictedSeed (t z : V → ℝ) (m : ℕ) (q B : ℝ) (y : V) (s : ℝ) : |
| G.boxMove t z m q B y s=G.restrictedSeed (G.vertexBall y m) |
| {b | (|t b-(t y+1)|≤q ∧ |z b-(z y+s)|≤q) ∧ |t b-t y|≤B ∧ |z b-z y|≤B} |
| (G.inducedEdges {x | |t x-t y|≤B ∧ |z x-z y|≤B}) := by |
| ext ω |
| constructor |
| · rintro ⟨a,ha,b,w,hw,ht,hz⟩ |
| exact Set.mem_iUnion₂.mpr ⟨a,ha,b,G.walk_reachable_induced _ ω w hw,⟨ht,hz⟩,hw b w.end_mem_support⟩ |
| · intro h |
| obtain ⟨a,ha,b,⟨w⟩,hb⟩ := Set.mem_iUnion₂.mp h |
| let w' := w.mapLe (G.openGraph_mono Set.inter_subset_left) |
| refine ⟨a,ha,b,w',?_,hb.1.1,hb.1.2⟩ |
| intro x hx |
| rw [SimpleGraph.Walk.support_mapLe_eq_support] at hx |
| exact G.walk_induced_support w hb.2 x hx |
|
|
| lemma restrictedSeed_upper (A T : Set V) (S : Set E) : IsUpperSet (G.restrictedSeed A T S) := by |
| intro ω τ h hω |
| exact G.seedHitEvent_upper A T (Set.inter_subset_inter_left S h) hω |
|
|
| lemma restrictedSeed_finiteWitness (A T : Set V) (S : Set E) : |
| BernoulliProduct.HasFiniteOpenWitness (G.restrictedSeed A T S) := by |
| intro ω hω |
| obtain ⟨a,ha,b,hr,hb⟩ := Set.mem_iUnion₂.mp hω |
| obtain ⟨K,hK,hKsub,hKr⟩ := G.mem_cluster_finite_witness hr |
| refine ⟨K,hK,fun _ he => (hKsub he).1,?_⟩ |
| change K∩S∈G.seedHitEvent A T |
| rw [Set.inter_eq_left.mpr (fun _ he => (hKsub he).2)] |
| exact Set.mem_iUnion₂.mpr ⟨a,ha,b,hKr,hb⟩ |
|
|
| lemma boxMove_upper (t z : V → ℝ) (m : ℕ) (q B : ℝ) (y : V) (s : ℝ) : |
| IsUpperSet (G.boxMove t z m q B y s) := by |
| rw [G.boxMove_eq_restrictedSeed] |
| exact G.restrictedSeed_upper _ _ _ |
|
|
| lemma boxMove_finiteWitness (t z : V → ℝ) (m : ℕ) (q B : ℝ) (y : V) (s : ℝ) : |
| BernoulliProduct.HasFiniteOpenWitness (G.boxMove t z m q B y s) := by |
| rw [G.boxMove_eq_restrictedSeed] |
| exact G.restrictedSeed_finiteWitness _ _ _ |
|
|
| lemma boxMove_measurable [Countable V] [Countable E] |
| (t z : V → ℝ) (m : ℕ) (q B : ℝ) (y : V) (s : ℝ) : |
| MeasurableSet (G.boxMove t z m q B y s) := by |
| rw [G.boxMove_eq_restrictedSeed] |
| exact G.restrictedSeed_measurable _ _ _ |
|
|
| lemma lower_two_boxMoves [Countable E] (t z : V → ℝ) (m : ℕ) (q B : ℝ) |
| (o : V) (pc : unitInterval) (hpc : 0<pc) (eta : ℝ) |
| (hm : ∀ b : Bool, 1-eta<(BernoulliProduct.law (fun _ : E => pc)).real |
| (G.boxMove t z m q B o (Corridor.sign b))) : |
| ∃ p : unitInterval, (pc : ℝ)/2≤p ∧ p<pc ∧ ∀ b : Bool, |
| 1-eta<(BernoulliProduct.law (fun _ : E => p)).real |
| (G.boxMove t z m q B o (Corridor.sign b)) := |
| BernoulliProduct.exists_lower_parameter _ (fun _ => G.boxMove_upper _ _ _ _ _ _ _) |
| (fun _ => G.boxMove_finiteWitness _ _ _ _ _ _ _) pc hpc (fun _ => 1-eta) hm |
|
|
| end CriticalPercolation.BondGraph |
| end |
| end |
| end |
|
|
| end OAI |
|
|
