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b317718 | 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 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 | import OAI.Probability.FreeIsing.Critical.Branching
import OAI.Probability.FreeIsing.Trees.BranchingSurvivalLaw
namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped Topology
namespace Problem367.BranchingSurvival
theorem probability_eq_survival (b : ℕ) {p : ℝ} (hp0 : 0 ≤ p) (hp1 : p ≤ 1)
(n : ℕ) : probability b p n = CriticalBranching.survival b p n := by
induction n with
| zero => simp [CriticalBranching.survival]
| succ n ih =>
rw [probability_succ b hp0 hp1, ih]
rfl
theorem probability_le_reciprocal {b : ℕ} (hb : 1 ≤ b) {p : ℝ}
(hp : p ∈ Set.Icc (0 : ℝ) 1) (hcritical : (b : ℝ) * p ≤ 1) (n : ℕ) :
probability b p n ≤ 1 / (1 + (((b : ℝ) - 1) * p ^ 2) * n) := by
rw [probability_eq_survival b hp.1 hp.2]
exact CriticalBranching.survival_le_reciprocal hb hp hcritical n
theorem probability_tendsto_zero {b : ℕ} (hb : 2 ≤ b) {p : ℝ}
(hp : p ∈ Set.Icc (0 : ℝ) 1) (hcritical : (b : ℝ) * p ≤ 1) :
Tendsto (probability b p) atTop (𝓝 0) := by
have heq : probability b p = CriticalBranching.survival b p :=
funext (probability_eq_survival b hp.1 hp.2)
rw [heq]
exact CriticalBranching.survival_tendsto_zero hb hp hcritical
lemma survives_succ_subset (b : ℕ) (p : ℝ) (n : ℕ) :
@survives b p (n + 1) ⊆ survives p n := by
induction n with
| zero => exact Set.subset_univ _
| succ n ih =>
rintro U ⟨i, hi, hchild⟩
exact ⟨i, hi, ih hchild⟩
lemma antitone_survives (b : ℕ) (p : ℝ) : Antitone (@survives b p) :=
antitone_nat_of_succ_le (survives_succ_subset b p)
theorem measure_survives_forever {b : ℕ} (hb : 2 ≤ b) {p : ℝ}
(hp : p ∈ Set.Icc (0 : ℝ) 1) (hcritical : (b : ℝ) * p ≤ 1) :
wordMeasure b (⋂ n : ℕ, survives p n) = 0 := by
have hlim : Tendsto (fun n => wordMeasure b (survives p n)) atTop (𝓝 0) := by
have h := (ENNReal.continuous_ofReal.tendsto 0).comp
(probability_tendsto_zero hb hp hcritical)
simpa [Function.comp_def, probability, Measure.real, ENNReal.ofReal_toReal (measure_ne_top _ _)] using h
have hcontinuity := tendsto_measure_iInter_atTop
(fun n => (measurableSet_survives b p n).nullMeasurableSet)
(antitone_survives b p) (show ∃ n, wordMeasure b (survives p n) ≠ ⊤ from
⟨0, measure_ne_top _ _⟩)
exact tendsto_nhds_unique hcontinuity hlim
theorem ae_exists_extinction_depth {b : ℕ} (hb : 2 ≤ b) {p : ℝ}
(hp : p ∈ Set.Icc (0 : ℝ) 1) (hcritical : (b : ℝ) * p ≤ 1) :
∀ᵐ U ∂wordMeasure b, ∃ n : ℕ, U ∉ survives p n := by
apply ae_iff.mpr
have hs : {U : Labels b | ¬ ∃ n : ℕ, U ∉ survives p n} =
⋂ n : ℕ, survives p n := by
ext U
simp
rw [hs]
exact measure_survives_forever hb hp hcritical
theorem measurePreserving_wordShift (b : ℕ) (w : List (Fin b)) :
MeasurePreserving (fun U : Labels b => fun v => U (w ++ v))
(wordMeasure b) (wordMeasure b) := by
refine ⟨by fun_prop, ?_⟩
exact Measure.map_infinitePi_infinitePi_of_inj
(fun x y h => List.append_cancel_left h)
theorem ae_all_extinction_depth {b : ℕ} (hb : 2 ≤ b) {p : ℝ}
(hp : p ∈ Set.Icc (0 : ℝ) 1) (hcritical : (b : ℝ) * p ≤ 1) :
∀ᵐ U ∂wordMeasure b, ∀ w : List (Fin b),
∃ n : ℕ, (fun v => U (w ++ v)) ∉ survives p n := by
apply ae_all_iff.mpr
intro w
exact (measurePreserving_wordShift b w).quasiMeasurePreserving.ae
(ae_exists_extinction_depth hb hp hcritical)
end Problem367.BranchingSurvival
end
end OAI
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