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b317718 | 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 | import OAI.Probability.FreeIsing.Trees.EdgeIIDLaw
import OAI.Probability.FreeIsing.Trees.PercolationFiniteness
namespace OAI
noncomputable section
open MeasureTheory
namespace Problem367.CodedPercolation
open EdgeIIDLaw
def incomingEdge {d : ℕ} (v : TreeVertex d) (hv : v.1 ≠ []) :
EdgeLabelCoding.TreeEdge d :=
⟨(parentVertex v, v), treeAdj_parentVertex hv, by
have hpos : 0 < v.1.length := List.length_pos_iff.mpr hv
simp only [parentVertex_length]
omega⟩
def incomingLabel {d : ℕ} (W : TreeVertex d → VertexInput) (v : TreeVertex d) : ℝ :=
vertexEdgeLabel W (parentVertex v) v
theorem measurable_incomingLabel {d : ℕ} (v : TreeVertex d) :
Measurable (fun W : TreeVertex d → VertexInput => incomingLabel W v) := by
let : MeasurableSpace (TreeVertex d) := ⊤
let : MeasurableSingletonClass (TreeVertex d) := ⟨fun _ => trivial⟩
exact (EdgeLabelCoding.measurable_edgeLabel (parentVertex v) v).comp
(measurePreserving_splitVertexInputs d).measurable
theorem incoming_all_open {d : ℕ} {theta : ℝ} (h1 : theta ≤ 1)
(S : Finset (TreeVertex d)) (hS : ∀ v ∈ S, v.1 ≠ []) :
vertexInputMeasure d {W | ∀ v ∈ S, incomingLabel W v ≤ theta} =
ENNReal.ofReal theta ^ S.card := by
classical
let f : S → EdgeLabelCoding.TreeEdge d := fun v => incomingEdge v.1 (hS v.1 v.2)
let D : Finset (EdgeLabelCoding.TreeEdge d) := S.attach.image f
have hf : Function.Injective f := by
intro v w h
apply Subtype.ext
exact congrArg (fun e : EdgeLabelCoding.TreeEdge d => e.1.2) h
have hcard : D.card = S.card := by
change (S.attach.image f).card = S.card
rw [Finset.card_image_of_injective _ hf, Finset.card_attach]
have hevent : {W : TreeVertex d → VertexInput | ∀ v ∈ S, incomingLabel W v ≤ theta} =
{W | ∀ e ∈ D, vertexEdgeLabels W e ≤ theta} := by
ext W
constructor
· intro h e he
obtain ⟨v, _, rfl⟩ := Finset.mem_image.mp he
exact h v.1 v.2
· intro h v hv
exact h (f ⟨v, hv⟩) (Finset.mem_image.mpr ⟨⟨v, hv⟩, Finset.mem_attach _ _, rfl⟩)
rw [hevent, vertex_coded_all_open d h1 D, hcard]
def openRelation {d : ℕ} (theta : ℝ) (W : TreeVertex d → VertexInput)
(u v : TreeVertex d) : Prop :=
treeAdj u v ∧ vertexEdgeLabel W u v ≤ theta
theorem measurableSet_openRelation {d : ℕ} (theta : ℝ) (u v : TreeVertex d) :
MeasurableSet {W : TreeVertex d → VertexInput | openRelation theta W u v} := by
let : MeasurableSpace (TreeVertex d) := ⊤
let : MeasurableSingletonClass (TreeVertex d) := ⟨fun _ => trivial⟩
have hm : Measurable (fun W : TreeVertex d → VertexInput => vertexEdgeLabel W u v) :=
(EdgeLabelCoding.measurable_edgeLabel u v).comp
(measurePreserving_splitVertexInputs d).measurable
exact (MeasurableSet.const (treeAdj u v)).inter (measurableSet_le hm measurable_const)
theorem openRelation_relabel {d : ℕ} (theta : ℝ) (g : TreeAut d)
(W : TreeVertex d → VertexInput) (u v : TreeVertex d) :
openRelation theta (relabel g W) (g.toEquiv u) (g.toEquiv v) ↔
openRelation theta W u v := by
simp only [openRelation, g.map_adj, vertexEdgeLabel_relabel]
theorem openAdj_incoming_eq {d : ℕ} (theta : ℝ) (W : TreeVertex d → VertexInput)
(hsym : ∀ u v, treeAdj u v → vertexEdgeLabel W v u = vertexEdgeLabel W u v) :
OpenCluster.openAdj (fun v => incomingLabel W v ≤ theta) = openRelation theta W := by
funext u v
apply propext
by_cases hadj : treeAdj u v
· by_cases hlt : u.1.length < v.1.length
· have hp := treeAdj_parent_of_length_lt hadj hlt
simp only [OpenCluster.openAdj, openRelation, hadj, true_and, ite_eq_left hlt,
incomingLabel, hp]
· have hrev : v.1.length < u.1.length := by
rcases treeAdj_length hadj with h | h <;> omega
have hp := treeAdj_parent_of_length_lt hadj.symm hrev
simp only [OpenCluster.openAdj, openRelation, hadj, true_and, ite_eq_right hlt,
incomingLabel, hp, hsym u v hadj]
· simp only [OpenCluster.openAdj, openRelation, hadj, false_and]
theorem ae_all_finite_coded_clusters {d : ℕ} {theta : ℝ}
(h0 : 0 ≤ theta) (h1 : theta ≤ 1) (hsub : (d - 1 : ℕ) * theta < 1) :
∀ᵐ W ∂vertexInputMeasure d, ∀ o : TreeVertex d,
{v | Relation.ReflTransGen (openRelation theta W) o v}.Finite := by
have hfin := SubcriticalClusters.ae_all_finite_openClusters_of_cylinder_bound
(vertexInputMeasure d) incomingLabel hsub h0
(fun S hS => (incoming_all_open h1 S hS).le)
filter_upwards [hfin, ae_vertexEdgeLabel_symmetric d] with W hW hsym
intro o
have hrel := openAdj_incoming_eq theta W hsym
simpa only [SubcriticalClusters.openCluster, hrel] using hW o
end Problem367.CodedPercolation
end
end OAI
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