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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 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 | 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
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