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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 | import OAI.Probability.FreeIsing.Critical.CodedPercolation
import OAI.Probability.FreeIsing.Trees.ClusterFactorEnvironment
namespace OAI
noncomputable section
open MeasureTheory
namespace Problem367.ClusterFactorMap
variable {d : ℕ}
theorem ae_all_finite_environment_clusters_critical (hd : 3 ≤ d) {theta : ℝ}
(h0 : 0 ≤ theta) (h1 : theta ≤ 1) (hc : ((d - 1 : ℕ) : ℝ) * 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_critical hd h0 h1 hc
theorem ae_good_environment_critical (hd : 3 ≤ d) {theta : ℝ}
(h0 : 0 ≤ theta) (h1 : theta ≤ 1) (hc : ((d - 1 : ℕ) : ℝ) * 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_critical hd h0 h1 hc] 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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