| import OAI.Probability.BenjaminiSchramm.PeakComparison |
| import OAI.Probability.CriticalPercolation.ProductMeasure.MonotoneBernoulli |
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|
| namespace OAI |
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|
| noncomputable section |
| open Set MeasureTheory Filter |
| open scoped Classical ENNReal Topology |
|
|
| namespace Percolation.BenjaminiSchramm |
| variable {E : Type*} |
|
|
| lemma finset_subset_monotone_iUnion (s : Finset E) (W : ℕ → Set E) (hW : Monotone W) |
| (hs : (s : Set E) ⊆ ⋃ n, W n) : ∃ n, (s : Set E) ⊆ W n := by |
| induction s using Finset.induction_on with |
| | empty => exact ⟨0,by simp⟩ |
| | @insert e s he ih => |
| obtain ⟨n,hn⟩ := Set.mem_iUnion.mp (hs (Finset.mem_insert_self e s)) |
| obtain ⟨m,hm⟩ := ih (fun _ h => hs (Finset.mem_insert_of_mem h)) |
| refine ⟨max n m,?_⟩ |
| intro f hf |
| rcases Finset.mem_insert.mp hf with rfl | hf |
| · exact hW (le_max_left _ _) hn |
| · exact hW (le_max_right _ _) (hm hf) |
|
|
| lemma finiteWitness_mem_iUnion_iff (A : Set (Set E)) (hA : IsUpperSet A) (hfA : FiniteWitness A) |
| (W : ℕ → Set E) (hW : Monotone W) : (⋃ n, W n) ∈ A ↔ ∃ n, W n ∈ A := by |
| constructor |
| · intro h |
| obtain ⟨s,hs,hsa⟩ := hfA _ h |
| obtain ⟨n,hn⟩ := finset_subset_monotone_iUnion s W hW hs |
| exact ⟨n,hA hn hsa⟩ |
| · rintro ⟨n,hn⟩ |
| exact hA (Set.subset_iUnion W n) hn |
|
|
| theorem law_finiteWitness_left_tendsto [Countable E] |
| (A : Set (Set E)) (hA : IsUpperSet A) (hfA : FiniteWitness A) |
| (p : unitInterval) (q : ℕ → unitInterval) (hmono : Monotone q) |
| (hqp : ∀ n, q n ≤ p) (hq : Tendsto q atTop (𝓝 p)) : |
| Tendsto (fun n => law (q n) A) atTop (𝓝 (law p A)) := by |
| let B := fun n => CriticalPercolation.BernoulliProduct.atParameter (q n) ⁻¹' A |
| have hmA := measurableSet_finiteWitness A hA hfA |
| have hB : Monotone B := by |
| intro n m hnm u hu |
| exact hA (CriticalPercolation.BernoulliProduct.atParameter_mono u (hmono hnm)) hu |
| have heq : (⋃ n, B n) = CriticalPercolation.BernoulliProduct.atParameter p ⁻¹' A := by |
| ext u |
| simp only [Set.mem_iUnion,Set.mem_preimage] |
| rw [← CriticalPercolation.BernoulliProduct.atParameter_iUnion p q hqp hq u] |
| exact (finiteWitness_mem_iUnion_iff A hA hfA _ |
| (fun n m hnm => CriticalPercolation.BernoulliProduct.atParameter_mono u (hmono hnm))).symm |
| have hμ (r : unitInterval) : CriticalPercolation.BernoulliProduct.markLaw |
| (CriticalPercolation.BernoulliProduct.atParameter r ⁻¹' A) = law r A := by |
| rw [← Measure.map_apply (CriticalPercolation.BernoulliProduct.measurable_atParameter r) hmA, |
| CriticalPercolation.BernoulliProduct.markLaw_map_atParameter, |
| CriticalPercolation.BernoulliProduct.law_const] |
| rfl |
| have h := tendsto_measure_iUnion_atTop (μ := CriticalPercolation.BernoulliProduct.markLaw) |
| hB |
| rw [heq,hμ] at h |
| simpa only [Function.comp_def,B,hμ] using h |
|
|
| end Percolation.BenjaminiSchramm |
|
|
| end |
|
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| end OAI |
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