| 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 |
|
|
