| import OAI.Probability.FreeIsing.Trees.BranchingSurvivalLaw |
| import OAI.Probability.FreeIsing.Trees.ConePercolation |
|
|
| namespace OAI |
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|
| noncomputable section |
|
|
| open MeasureTheory |
|
|
| namespace Problem367.TreeConeCoding |
|
|
| theorem rootedPathOpen_cons {b : ℕ} (p : ℝ) (U : List (Fin b) → ℝ) |
| (i : Fin b) (w : List (Fin b)) : |
| rootedPathOpen p U (i :: w) ↔ |
| U [i] ≤ p ∧ rootedPathOpen p (BranchingSurvival.branches U i) w := by |
| constructor |
| · intro h |
| constructor |
| · simpa only [List.take_succ_cons, List.take_zero] using h 1 (by omega) (by simp) |
| · intro k hk hkw |
| have hc := h (k + 1) (by omega) (by simp; omega) |
| simpa only [List.take_succ_cons, BranchingSurvival.branches] using hc |
| · rintro ⟨hi, hw⟩ k hk hkw |
| cases k with |
| | zero => omega |
| | succ k => |
| cases k with |
| | zero => simpa only [List.take_succ_cons, List.take_zero] using hi |
| | succ k => |
| have hc := hw (k + 1) (by omega) (by simp at hkw; omega) |
| simpa only [List.take_succ_cons, BranchingSurvival.branches] using hc |
|
|
| theorem survives_iff_rootedPathOpen {b : ℕ} (p : ℝ) (n : ℕ) |
| (U : List (Fin b) → ℝ) : |
| U ∈ BranchingSurvival.survives p n ↔ |
| ∃ w : List (Fin b), w.length = n ∧ rootedPathOpen p U w := by |
| induction n generalizing U with |
| | zero => |
| constructor |
| · intro _ |
| exact ⟨[], rfl, fun k hk hkn => by simp only [List.length_nil] at hkn; omega⟩ |
| · intro _ |
| trivial |
| | succ n ih => |
| constructor |
| · rintro ⟨i, hi, hU⟩ |
| obtain ⟨w, hwlen, hw⟩ := (ih _).mp hU |
| exact ⟨i :: w, by simp [hwlen], (rootedPathOpen_cons p U i w).mpr ⟨hi, hw⟩⟩ |
| · rintro ⟨w, hwlen, hw⟩ |
| cases w with |
| | nil => simp at hwlen |
| | cons i w => |
| obtain ⟨hi, hw'⟩ := (rootedPathOpen_cons p U i w).mp hw |
| exact ⟨i, hi, (ih _).mpr ⟨w, by simpa using hwlen, hw'⟩⟩ |
|
|
| theorem survival_probability_eq_branching {b : ℕ} (p : ℝ) |
| (r : TreeVertex (b + 1)) (hr : r.1 ≠ []) (n : ℕ) : |
| labelMeasure (b + 1) {U | ∃ v : TreeVertex (b + 1), |
| v.1.length = r.1.length + n ∧ SubcriticalClusters.DownwardOpen p U r v} = |
| BranchingSurvival.wordMeasure b (BranchingSurvival.survives p n) := by |
| rw [survival_probability p r hr n] |
| congr 1 |
| ext U |
| exact (survives_iff_rootedPathOpen p n U).symm |
|
|
| theorem survival_probability_real {b : ℕ} (p : ℝ) |
| (r : TreeVertex (b + 1)) (hr : r.1 ≠ []) (n : ℕ) : |
| (labelMeasure (b + 1)).real {U | ∃ v : TreeVertex (b + 1), |
| v.1.length = r.1.length + n ∧ SubcriticalClusters.DownwardOpen p U r v} = |
| BranchingSurvival.probability b p n := by |
| exact congrArg ENNReal.toReal (survival_probability_eq_branching p r hr n) |
|
|
| end Problem367.TreeConeCoding |
|
|
| end |
|
|
| end OAI |
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|
