| import OAI.Probability.FreeIsing.Trees.ClusterFactorMap |
| import OAI.Probability.FreeIsing.Trees.CodedPercolation |
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| namespace OAI |
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| noncomputable section |
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|
| open MeasureTheory |
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| namespace Problem367.ClusterFactorMap |
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|
| variable {d : ℕ} |
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|
| lemma measurableSet_all_finite_environment (theta : ℝ) : |
| MeasurableSet {e : (TreeVertex d → ℝ) × (TreeVertex d × ℕ → ℝ) | ∀ o : TreeVertex d, |
| {v | connected theta (e.1, (e.2, fun _ => 0)) o v}.Finite} := by |
| apply measurableSet_all_finite_reachable |
| intro u v |
| have h : Measurable (fun e : (TreeVertex d → ℝ) × (TreeVertex d × ℕ → ℝ) => |
| (e.1, (e.2, fun _ : TreeVertex d => (0 : ℝ)))) := by fun_prop |
| exact h (measurableSet_openAdj theta u v) |
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|
| theorem ae_all_finite_environment_clusters {theta : ℝ} |
| (h0 : 0 ≤ theta) (h1 : theta ≤ 1) (hsub : (d - 1 : ℕ) * theta < 1) : |
| ∀ᵐ e ∂EdgeIIDLaw.codingMeasure d, ∀ o : TreeVertex d, |
| {v | connected theta (e.1, (e.2, fun _ => 0)) o v}.Finite := by |
| have hmp := EdgeIIDLaw.measurePreserving_splitVertexInputs d |
| rw [← hmp.map_eq] |
| apply (ae_map_iff hmp.measurable.aemeasurable |
| (measurableSet_all_finite_environment theta)).mpr |
| exact CodedPercolation.ae_all_finite_coded_clusters h0 h1 hsub |
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|
| theorem ae_good_environment {theta : ℝ} |
| (h0 : 0 ≤ theta) (h1 : theta ≤ 1) (hsub : (d - 1 : ℕ) * theta < 1) : |
| ∀ᵐ e ∂EdgeIIDLaw.codingMeasure d, |
| Function.Injective e.1 ∧ Equivalence (connected theta (e.1, (e.2, fun _ => 0))) ∧ |
| ∀ o : TreeVertex d, {v | connected theta (e.1, (e.2, fun _ => 0)) o v}.Finite := by |
| let : IsProbabilityMeasure (labelMeasure d) := by |
| unfold labelMeasure |
| infer_instance |
| have hm := (measurePreserving_fst (μ := labelMeasure d) |
| (ν := EdgeIIDLaw.rowMeasure d)).quasiMeasurePreserving.ae |
| (FiniteCluster.ae_uniform_markers_injective d) |
| filter_upwards [hm, ae_all_finite_environment_clusters h0 h1 hsub] with e he hfin |
| exact ⟨he, connected_equivalence theta (p := (e.1, (e.2, fun _ => 0))) he, hfin⟩ |
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| end Problem367.ClusterFactorMap |
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| end |
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| end OAI |
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|
