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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 | 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
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