| import OAI.Probability.FreeIsing.Critical.ClusterFactorEnvironment |
| import OAI.Probability.FreeIsing.Trees.ClusterFactorEnvironment |
| import OAI.Probability.FreeIsing.Trees.ClusterFactorMap |
| import OAI.Probability.FreeIsing.Trees.ClusterInputSampler |
| import OAI.Probability.FreeIsing.Trees.EncodedClusterLaw |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory |
|
|
| namespace Problem367 |
|
|
| theorem isFactorOfIID_of_cluster_input_law {d : ℕ} {theta : ℝ} |
| (hLaw : HasFreeIsingLaw theta |
| (Measure.map (ClusterFactorMap.factor theta) (ClusterInput.law d))) : |
| IsFactorOfIID d theta := by |
| obtain ⟨S, hS, hSlaw, hSeq⟩ := ClusterInput.exists_sampler d |
| refine ⟨ClusterFactorMap.factor theta ∘ S, |
| (ClusterFactorMap.measurable_factor theta).comp hS, ?_, ?_⟩ |
| · rw [← Measure.map_map (ClusterFactorMap.measurable_factor theta) hS, hSlaw] |
| exact hLaw |
| · intro g |
| apply Filter.Eventually.of_forall |
| intro U |
| change ClusterFactorMap.factor theta (S (relabel g U)) = _ |
| rw [hSeq] |
| exact ClusterFactorMap.factor_relabel g theta (S U) |
|
|
| theorem isFactorOfIID_of_ae_finite_coded_clusters {d : ℕ} {theta : ℝ} |
| (h0 : 0 ≤ theta) (h1 : theta ≤ 1) |
| (hfinite : ∀ᵐ q ∂EdgeIIDLaw.codingMeasure d, ∀ v : TreeVertex d, |
| {a | Relation.ReflTransGen |
| (EncodedClusterLaw.environmentRelation theta q) v a}.Finite) : |
| IsFactorOfIID d theta := by |
| apply isFactorOfIID_of_cluster_input_law |
| exact EncodedClusterLaw.hasFreeIsingLaw_factor_of_ae_finite h0 h1 hfinite |
|
|
| theorem isFactorOfIID_of_strict_uniqueness {d : ℕ} {theta : ℝ} |
| (h0 : 0 ≤ theta) (h1 : theta ≤ 1) (hsub : (d - 1 : ℕ) * theta < 1) : |
| IsFactorOfIID d theta := by |
| apply isFactorOfIID_of_ae_finite_coded_clusters h0 h1 |
| exact ClusterFactorMap.ae_all_finite_environment_clusters h0 h1 hsub |
|
|
| theorem isFactorOfIID_of_uniqueness {d : ℕ} (hd : 3 ≤ d) {theta : ℝ} |
| (h0 : 0 ≤ theta) (h1 : theta ≤ 1) (hc : (d - 1 : ℕ) * theta ≤ 1) : |
| IsFactorOfIID d theta := by |
| apply isFactorOfIID_of_ae_finite_coded_clusters h0 h1 |
| exact ClusterFactorMap.ae_all_finite_environment_clusters_critical hd h0 h1 hc |
|
|
| end Problem367 |
|
|
| end |
|
|
| end OAI |
|
|
