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f61e547 | 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 | 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
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