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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 | import OAI.Probability.FreeIsing.Trees.ClusterFactorMap
import OAI.Probability.FreeIsing.Trees.CodedPercolation
namespace OAI
noncomputable section
open MeasureTheory
namespace Problem367.ClusterFactorMap
variable {d : ℕ}
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)
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
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⟩
end Problem367.ClusterFactorMap
end
end OAI
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