| import OAI.Probability.FreeIsing.Trees.BranchingExtinction |
| import OAI.Probability.FreeIsing.Trees.ConeSurvival |
| import OAI.Probability.FreeIsing.Trees.PercolationFiniteness |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory Filter |
| open scoped ENNReal Topology |
|
|
| namespace Problem367.SubcriticalClusters |
|
|
| theorem cone_survival_tendsto_zero {b : ℕ} (hb : 2 ≤ b) {p : ℝ} |
| (hp0 : 0 ≤ p) (hp1 : p ≤ 1) (hcritical : (b : ℝ) * p ≤ 1) |
| (r : TreeVertex (b + 1)) (hr : r.1 ≠ []) : |
| Tendsto (fun n => labelMeasure (b + 1) (survivalEvent id p r n)) |
| atTop (𝓝 0) := by |
| let : IsProbabilityMeasure (labelMeasure (b + 1)) := by |
| unfold labelMeasure |
| infer_instance |
| have h := (ENNReal.continuous_ofReal.tendsto 0).comp |
| (BranchingSurvival.probability_tendsto_zero hb ⟨hp0, hp1⟩ hcritical) |
| have heq : (fun n => labelMeasure (b + 1) (survivalEvent id p r n)) = |
| fun n => ENNReal.ofReal (BranchingSurvival.probability b p n) := by |
| funext n |
| rw [← TreeConeCoding.survival_probability_real p r hr n] |
| change labelMeasure (b + 1) (survivalEvent id p r n) = |
| ENNReal.ofReal (labelMeasure (b + 1) (survivalEvent id p r n)).toReal |
| exact (ENNReal.ofReal_toReal (measure_ne_top _ _)).symm |
| rw [heq] |
| simpa only [Function.comp_def, ENNReal.ofReal_zero] using h |
|
|
| theorem ae_all_finite_openClusters_critical {d : ℕ} (hd : 3 ≤ d) {p : ℝ} |
| (hp0 : 0 ≤ p) (hp1 : p ≤ 1) (hcritical : ((d - 1 : ℕ) : ℝ) * p ≤ 1) : |
| ∀ᵐ U ∂labelMeasure d, ∀ o : TreeVertex d, (openCluster p U o).Finite := by |
| obtain ⟨b, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : d ≠ 0) |
| have hb : 2 ≤ b := by omega |
| have hcritical' : (b : ℝ) * p ≤ 1 := by simpa using hcritical |
| apply ae_all_finite_openClusters_of_nonroot_survival_bound |
| (labelMeasure (b + 1)) id p |
| (fun r n => labelMeasure (b + 1) (survivalEvent id p r n)) |
| · intro r hr n |
| exact le_rfl |
| · intro r hr |
| exact cone_survival_tendsto_zero hb hp0 hp1 hcritical' r hr |
|
|
| end Problem367.SubcriticalClusters |
|
|
| end |
|
|
| end OAI |
|
|
