File size: 2,944 Bytes
f61e547 | 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 | import OAI.Probability.FreeIsing.Critical.Percolation
import OAI.Probability.FreeIsing.Trees.CodedPercolation
namespace OAI
noncomputable section
open MeasureTheory
namespace Problem367.CodedPercolation
open EdgeIIDLaw TreeConeCoding
theorem measurePreserving_incoming_cone {b : ℕ}
(r : TreeVertex (b + 1)) (hr : r.1 ≠ []) :
MeasurePreserving
(fun W : TreeVertex (b + 1) → VertexInput => labels r hr (incomingLabel W))
(vertexInputMeasure (b + 1)) (BranchingSurvival.wordMeasure b) := by
have hnon (w : List (Fin b)) : (vertex r hr w).1 ≠ [] := by
intro hnil
have hlen := length_vertex r hr w
have hrlen : 0 < r.1.length := List.length_pos_iff.mpr hr
rw [hnil, List.length_nil] at hlen
omega
let e : List (Fin b) → EdgeLabelCoding.TreeEdge (b + 1) :=
fun w => incomingEdge (vertex r hr w) (hnon w)
have hinj : Function.Injective e := by
intro w z h
apply vertex_injective r hr
exact congrArg (fun a : EdgeLabelCoding.TreeEdge (b + 1) => a.1.2) h
have hm : MeasurePreserving
(fun Y : EdgeLabelCoding.TreeEdge (b + 1) → ℝ => fun w => Y (e w))
(edgeMeasure (b + 1)) (BranchingSurvival.wordMeasure b) :=
⟨by fun_prop, Measure.map_infinitePi_infinitePi_of_inj hinj⟩
exact hm.comp (measurePreserving_vertexEdgeLabels (b + 1))
theorem ae_all_finite_coded_clusters_critical {d : ℕ} (hd : 3 ≤ d) {theta : ℝ}
(h0 : 0 ≤ theta) (h1 : theta ≤ 1)
(hcritical : ((d - 1 : ℕ) : ℝ) * theta ≤ 1) :
∀ᵐ W ∂vertexInputMeasure d, ∀ o : TreeVertex d,
{v | Relation.ReflTransGen (openRelation theta W) o v}.Finite := by
obtain ⟨b, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : d ≠ 0)
have hb : 2 ≤ b := by omega
have hc : (b : ℝ) * theta ≤ 1 := by simpa using hcritical
have hfinite : ∀ᵐ W ∂vertexInputMeasure (b + 1),
∀ r : TreeVertex (b + 1), r.1 ≠ [] →
{v | SubcriticalClusters.DownwardOpen theta (incomingLabel W) r v}.Finite := by
apply ae_all_iff.2
intro r
apply ae_all_iff.2
intro hr
have hext := (measurePreserving_incoming_cone r hr).quasiMeasurePreserving.ae
(BranchingSurvival.ae_exists_extinction_depth hb ⟨h0, h1⟩ hc)
filter_upwards [hext] with W hW
obtain ⟨n, hn⟩ := hW
apply (SubcriticalClusters.finite_downwardOpen_iff theta (incomingLabel W) r).2
refine ⟨n, ?_⟩
intro hsurv
apply hn
apply (survives_iff_rootedPathOpen theta n _).2
exact (survival_iff theta (incomingLabel W) r hr n).1 hsurv
filter_upwards [hfinite, ae_vertexEdgeLabel_symmetric (b + 1)] with W hW hsym
intro o
have hfin := OpenCluster.finite_reachable_of_finite_downwardOpen
(fun v => incomingLabel W v ≤ theta)
(SubcriticalClusters.finite_downwardOpen_of_nonroot theta (incomingLabel W) hW) o
simpa only [openAdj_incoming_eq theta W hsym] using hfin
end Problem367.CodedPercolation
end
end OAI
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