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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 | 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
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