| import Mathlib |
| import OAI.Probability.CriticalPercolation.Criticality.RestrictedSeed |
| import OAI.Probability.CriticalPercolation.Gluing.MoveWitness |
| import OAI.Probability.CriticalPercolation.ProductMeasure.MeasurableSetRestrict |
| import OAI.Probability.CriticalPercolation.ProductMeasure.MonotoneIndicatorUpper |
| import OAI.Probability.CriticalPercolation.Gluing.CorridorNormalization |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open Set Function MeasureTheory ProbabilityTheory |
| open scoped Classical ENNReal |
|
|
| namespace SimpleGraph.Walk |
| open scoped _root_.SimpleGraph _root_.SimpleGraph.Walk |
| variable {V : Type*} {H : SimpleGraph V} |
|
|
| lemma bounded_first_crossing (f : V → ℝ) (D T : ℝ) (hD : 0≤D) |
| (hf : ∀ ⦃x y⦄, H.Adj x y → |f x-f y|≤D) |
| {a b : V} (w : H.Walk a b) (ha : f a<T) (hb : T≤f b) : |
| ∃ x, ∃ p : H.Walk a x, p.support⊆w.support ∧ |
| (∀ v∈p.support, f v≤T+D) ∧ T≤f x := by |
| induction w with |
| | nil => exact (not_le_of_gt ha hb).elim |
| | @cons a c b hac w ih => |
| by_cases hc : T≤f c |
| · refine ⟨c,hac.toWalk,?_,?_,hc⟩ |
| · intro v hv |
| simp only [SimpleGraph.Adj.support_toWalk,List.mem_cons,List.not_mem_nil,or_false] at hv |
| rcases hv with rfl|rfl |
| · exact (_root_.SimpleGraph.Walk.cons hac w).start_mem_support |
| · exact List.mem_cons_of_mem a w.start_mem_support |
| · intro v hv |
| simp only [SimpleGraph.Adj.support_toWalk,List.mem_cons,List.not_mem_nil,or_false] at hv |
| rcases hv with rfl|rfl |
| · linarith |
| · have hh := (abs_le.mp (hf hac)).1 |
| linarith |
| · obtain ⟨x,p,hp,hpb,hpx⟩ := ih (lt_of_not_ge hc) hb |
| refine ⟨x,_root_.SimpleGraph.Walk.cons hac p,?_,?_,hpx⟩ |
| · intro v hv |
| rcases List.mem_cons.mp hv with hv|hv |
| · exact List.mem_cons.mpr (Or.inl hv) |
| · exact List.mem_cons.mpr (Or.inr (hp hv)) |
| · intro v hv |
| rcases List.mem_cons.mp hv with hv|hv |
| · rw [hv] |
| linarith |
| · exact hpb v hv |
|
|
| end SimpleGraph.Walk |
| namespace CriticalPercolation.BondGraph |
| variable {V E : Type*} (G : BondGraph V E) |
|
|
| def axisExit (u v : V → ℝ) (A : Set V) (L W D : ℝ) (b : Bool) : Set (Set E) := |
| G.restrictedSeed A {x | L≤Corridor.sign b*u x ∧ Corridor.sign b*u x≤L+D ∧ |
| |u x|≤L+D ∧ |v x|≤W} (G.inducedEdges {x | |u x|≤L+D ∧ |v x|≤W}) |
|
|
| lemma axisExit_upper (u v : V → ℝ) (A : Set V) (L W D : ℝ) (b : Bool) : |
| IsUpperSet (G.axisExit u v A L W D b) := G.restrictedSeed_upper _ _ _ |
| lemma axisExit_finiteWitness (u v : V → ℝ) (A : Set V) (L W D : ℝ) (b : Bool) : |
| BernoulliProduct.HasFiniteOpenWitness (G.axisExit u v A L W D b) := |
| G.restrictedSeed_finiteWitness _ _ _ |
| lemma axisExit_measurable [Countable V] [Countable E] |
| (u v : V → ℝ) (A : Set V) (L W D : ℝ) (b : Bool) : |
| MeasurableSet (G.axisExit u v A L W D b) := G.restrictedSeed_measurable _ _ _ |
|
|
| lemma axisExit_iff (u v : V → ℝ) (A : Set V) (L W D : ℝ) (b : Bool) (ω : Set E) : |
| ω∈G.axisExit u v A L W D b ↔ |
| ∃ a∈A, ∃ x, ∃ w : (G.openGraph ω).Walk a x, |
| (∀ y∈w.support, |u y|≤L+D ∧ |v y|≤W) ∧ |
| L≤Corridor.sign b*u x ∧ Corridor.sign b*u x≤L+D := by |
| constructor |
| · intro h |
| obtain ⟨a,ha,x,⟨w⟩,hx⟩ := Set.mem_iUnion₂.mp h |
| refine ⟨a,ha,x,w.mapLe (G.openGraph_mono Set.inter_subset_left),?_,hx.1,hx.2.1⟩ |
| intro y hy |
| rw [SimpleGraph.Walk.support_mapLe_eq_support] at hy |
| exact G.walk_induced_support w hx.2.2 y hy |
| · rintro ⟨a,ha,x,w,hw,hx⟩ |
| exact Set.mem_iUnion₂.mpr ⟨a,ha,x,G.walk_reachable_induced _ ω w hw, |
| hx.1,hx.2,hw x w.end_mem_support⟩ |
|
|
| lemma axisExit_of_crossing (u v : V → ℝ) (A : Set V) (L W D : ℝ) |
| (hD : 0≤D) (hu : ∀ ⦃x y⦄, G.fullGraph.Adj x y → |u x-u y|≤D) |
| {ω : Set E} {a x : V} (ha : a∈A) (hal : |u a|<L) |
| (w : (G.openGraph ω).Walk a x) (hx : L≤|u x|) |
| (hv : ∀ y∈w.support, |v y|≤W) : |
| ω∈G.axisExit u v A L W D true ∪ G.axisExit u v A L W D false := by |
| have hu' : ∀ ⦃x y⦄, (G.openGraph ω).Adj x y → |abs (u x)-abs (u y)|≤D := by |
| intro x y hxy |
| exact (abs_abs_sub_abs_le _ _).trans (hu (G.openGraph_mono (Set.subset_univ _) hxy)) |
| obtain ⟨y,p,hp,hpb,hpy⟩ := _root_.OAI.SimpleGraph.Walk.bounded_first_crossing (fun x => |u x|) D L hD hu' w hal hx |
| have hpath : ∀ z∈p.support, |u z|≤L+D ∧ |v z|≤W := |
| fun z hz => ⟨hpb z hz,hv z (hp hz)⟩ |
| by_cases hy : 0≤u y |
| · apply Or.inl |
| apply (G.axisExit_iff u v A L W D true ω).mpr |
| refine ⟨a,ha,y,p,hpath,?_,?_⟩ |
| · simpa [Corridor.sign,abs_of_nonneg hy] using hpy |
| · simpa [Corridor.sign,abs_of_nonneg hy] using hpb y p.end_mem_support |
| · apply Or.inr |
| apply (G.axisExit_iff u v A L W D false ω).mpr |
| have hy' : u y≤0 := (lt_of_not_ge hy).le |
| refine ⟨a,ha,y,p,hpath,?_,?_⟩ |
| · simpa [Corridor.sign,abs_of_nonpos hy'] using hpy |
| · simpa [Corridor.sign,abs_of_nonpos hy'] using hpb y p.end_mem_support |
|
|
| lemma favored_of_union [Countable E] (q : E → unitInterval) |
| (A B : Set (Set E)) (hA : IsUpperSet A) (hB : IsUpperSet B) |
| (hwA : BernoulliProduct.HasFiniteOpenWitness A) |
| (hwB : BernoulliProduct.HasFiniteOpenWitness B) |
| (a : ℝ) (ha : 0≤a) (hu : 1-a^2<(BernoulliProduct.law q).real (A∪B)) : |
| 1-a<(BernoulliProduct.law q).real A ∨ 1-a<(BernoulliProduct.law q).real B := by |
| have hH := BernoulliProduct.harris_finiteWitness q A B hA hB hwA hwB |
| have hBmeas := BernoulliProduct.measurableSet_finiteWitness B hB hwB |
| have hunion := measureReal_union_add_inter (s := A) hBmeas (μ := BernoulliProduct.law q) |
| (measure_ne_top _ _) (measure_ne_top _ _) |
| have hAl : (BernoulliProduct.law q).real A≤1 := measureReal_le_one |
| have hBl : (BernoulliProduct.law q).real B≤1 := measureReal_le_one |
| by_contra hn |
| push Not at hn |
| have hm : a^2≤(1-(BernoulliProduct.law q).real A)*(1-(BernoulliProduct.law q).real B) := by |
| nlinarith [mul_le_mul (by linarith : a≤1-(BernoulliProduct.law q).real A) |
| (by linarith : a≤1-(BernoulliProduct.law q).real B) ha (by linarith : 0≤1-(BernoulliProduct.law q).real A)] |
| nlinarith |
|
|
| end CriticalPercolation.BondGraph |
| end |
| end |
| end |
|
|
| end OAI |
|
|
