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f205175 | 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 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 | import Mathlib
import OAI.Probability.CriticalPercolation.Modular.WeightedDiscrepancy
import OAI.Probability.CriticalPercolation.Clusters.ClusterCountFlip
import OAI.Probability.CriticalPercolation.ProductMeasure.BernoulliSingletonIntegration
import OAI.Probability.CriticalPercolation.Criticality.SignedCountIdentity
namespace OAI
section
noncomputable section
open Set MeasureTheory Filter
open scoped Classical ENNReal BigOperators
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
def clusterCountDrop (ω : Set E) (e : E) : ℝ :=
(G.incidentClusterCount (ω\{e}) e).toReal -
(G.incidentClusterCount (insert e ω) e).toReal
lemma clusterCountDrop_of_ends (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(ω : Set E) (e : E) (a b : V) (he : G.ends e=s(a,b)) :
G.clusterCountDrop ω e =
if ¬(G.openGraph (ω\{e})).Reachable a b ∧
¬(G.modularMass a (G.cluster (ω\{e}) a)=∞ ∧
G.modularMass b (G.cluster (ω\{e}) b)=∞) then 1 else 0 := by
have hi : insert e (ω\{e})=insert e ω := by ext f; by_cases h : f=e <;> simp [h]
simpa only [clusterCountDrop,hi] using G.incidentClusterCount_flip_of_ends hc hl (ω\{e}) e a b he
lemma clusterCountDrop_mem (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(ω : Set E) (e : E) : G.clusterCountDrop ω e=0 ∨ G.clusterCountDrop ω e=1 := by
obtain ⟨⟨a,b⟩,he⟩ := Quot.exists_rep (G.ends e)
rw [G.clusterCountDrop_of_ends hc hl ω e a b he.symm]
split_ifs <;> simp
lemma measurable_clusterCountDrop [Countable V] [Countable E] (e : E) :
Measurable (fun ω => G.clusterCountDrop ω e) := by
unfold clusterCountDrop
exact ((G.measurable_incidentClusterCount e).comp
(by have h := BernoulliProduct.measurable_forceOn ({e}:Set E) ∅
change Measurable (fun ω => BernoulliProduct.forceOn {e} ∅ ω) at h
simpa only [forceOn_empty_singleton] using h)).ennreal_toReal.sub
((G.measurable_incidentClusterCount e).comp
(by have h := BernoulliProduct.measurable_forceOn ({e}:Set E) Set.univ
change Measurable (fun ω => BernoulliProduct.forceOn {e} Set.univ ω) at h
simpa only [forceOn_univ_singleton] using h)).ennreal_toReal
lemma integral_centered_incidentClusterCount [Countable V] [Countable E]
(hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (p : unitInterval) (e : E) :
(∫ ω, centeredCoefficient p ω e*(G.incidentClusterCount ω e).toReal ∂G.law p) =
-(p:ℝ)*(1-(p:ℝ))*(∫ ω, G.clusterCountDrop ω e ∂G.law p) := by
have h := BernoulliProduct.integral_centered_singleton (fun _ : E => p) e
(fun ω => (G.incidentClusterCount ω e).toReal)
(G.measurable_incidentClusterCount e).ennreal_toReal 2 (by norm_num) (fun ω => by
simpa only [Function.comp_apply,Real.norm_of_nonneg ENNReal.toReal_nonneg] using
G.incidentClusterCount_toReal_le_two hc hl ω e)
rw [BernoulliProduct.law_const, ←show G.law p=ProbabilityTheory.setBernoulli Set.univ p from rfl] at h
change (∫ ω, (if e∈ω then 1-(p:ℝ) else -(p:ℝ))*(G.incidentClusterCount ω e).toReal ∂G.law p) = _
rw [h]
simp only [forceOn_empty_singleton,forceOn_univ_singleton]
have hi (σ : Set E) : Integrable (fun ω => (G.incidentClusterCount
(BernoulliProduct.forceOn {e} σ ω) e).toReal) (G.law p) := by
apply Integrable.of_bound ((G.measurable_incidentClusterCount e).ennreal_toReal.comp
(BernoulliProduct.measurable_forceOn _ _)).aestronglyMeasurable 2
exact Eventually.of_forall (fun ω => by
simpa only [Function.comp_apply,Real.norm_of_nonneg ENNReal.toReal_nonneg] using
G.incidentClusterCount_toReal_le_two hc hl _ e)
have hio := hi Set.univ
have hic := hi ∅
simp only [forceOn_empty_singleton,forceOn_univ_singleton] at hio hic
rw [show (∫ ω, G.clusterCountDrop ω e ∂G.law p) =
(∫ ω, (G.incidentClusterCount (ω\{e}) e).toReal ∂G.law p)-
(∫ ω, (G.incidentClusterCount (insert e ω) e).toReal ∂G.law p) from integral_sub hic hio]
ring
theorem expected_signed_discrepancy_drop [Countable V] [Countable E]
(hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive)
(D K : ℕ) (hD : ∀ v, (G.incidentEdges {v}).ncard≤D)
(hK : ∀ v, (G.vertexBall v 1).ncard≤K) (p : unitInterval) :
(∑ x∈(G.orbitRepresentatives hq).roots, ∫ ω, G.normalizedSignedDiscrepancy p ω x ∂G.law p)=
-(p:ℝ)*(1-(p:ℝ))*(∑ y∈(G.orbitRepresentatives hq).roots, ∑ e∈(hl y).toFinset,
∫ ω, G.clusterCountDrop ω e ∂G.law p) := by
rw [G.expected_signed_discrepancy_count hc hl hq D K hD hK]
simp_rw [G.integral_centered_incidentClusterCount hc hl,Finset.mul_sum]
end CriticalPercolation.BondGraph
end
end
end OAI
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