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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 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 | import Mathlib
import OAI.Probability.CriticalPercolation.Model
import OAI.Probability.CriticalPercolation.Clusters.ForcedDisconnectContinuity
import OAI.Probability.CriticalPercolation.GraphGeometry.GraphBalls
import OAI.Probability.CriticalPercolation.Modular.CriticalHeavyIdentity
import OAI.Probability.CriticalPercolation.Modular.SignedDiscrepancyContinuity
namespace OAI
section
noncomputable section
open Set MeasureTheory Filter
open scoped Classical ENNReal Topology BigOperators
namespace CriticalPercolation.BondGraph
open BernoulliProduct
variable {V E : Type*} (G : BondGraph V E) [Countable V] [Countable E]
lemma integrable_disconnectedIndicator (p : unitInterval) (e : E) :
Integrable (fun ω => G.disconnectedIndicator ω e) (G.law p) :=
Integrable.of_bound (G.measurable_disconnectedIndicator e).aestronglyMeasurable 1
(Eventually.of_forall (fun ω => G.norm_disconnectedIndicator_le ω e))
lemma integrable_clusterCountDrop (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(p : unitInterval) (e : E) : Integrable (fun ω => G.clusterCountDrop ω e) (G.law p) := by
apply Integrable.of_bound (G.measurable_clusterCountDrop e).aestronglyMeasurable 1
apply Eventually.of_forall
intro ω
rcases G.clusterCountDrop_mem hc hl ω e with h|h <;> simp [h]
lemma tendsto_expected_signed_discrepancy_law (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) (q : ℕ → unitInterval) (hmono : Monotone q)
(hqp : ∀ n, q n≤p) (hlim : Tendsto q atTop (𝓝 p)) :
Tendsto (fun n => ∑ x∈(G.orbitRepresentatives hq).roots,
∫ ω, G.normalizedSignedDiscrepancy (q n) ω x ∂G.law (q n)) atTop
(𝓝 (∑ x∈(G.orbitRepresentatives hq).roots,
∫ ω, G.normalizedSignedDiscrepancy p ω x ∂G.law p)) := by
apply tendsto_finsetSum _
intro x hx
have h := G.tendsto_expected_normalizedSignedDiscrepancy hc hl hq D K hD hK p q hmono hqp hlim x hx
simpa only [G.integral_atParameter _ _ (G.measurable_normalizedSignedDiscrepancy _ x)] using h
theorem critical_expected_drop_eq_disconnected (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(hq : G.QuasiTransitive) (hp0 : 0<G.criticalProbability) (hp1 : G.criticalProbability<1) :
(∑ y∈(G.orbitRepresentatives hq).roots, ∑ e∈(hl y).toFinset,
∫ ω, G.clusterCountDrop ω e ∂G.law G.criticalProbability) =
(∑ y∈(G.orbitRepresentatives hq).roots, ∑ e∈(hl y).toFinset,
∫ ω, G.disconnectedIndicator ω e ∂G.law G.criticalProbability) := by
obtain ⟨D,hD0,hD⟩ := G.uniform_incident_bound hq
obtain ⟨K,hK⟩ := G.uniform_ball_bound hq 1
obtain ⟨q,hmono,hqmem,hlim⟩ := exists_seq_strictMono_tendsto' hp0
have hqp : ∀ n, q n≤G.criticalProbability := fun n => (hqmem n).2.le
have hsig := G.tendsto_expected_signed_discrepancy_law hc hl hq D K hD hK
G.criticalProbability q hmono.monotone hqp hlim
have hd : Tendsto (fun n => ∑ y∈(G.orbitRepresentatives hq).roots, ∑ e∈(hl y).toFinset,
∫ ω, G.disconnectedIndicator ω e ∂G.law (q n)) atTop
(𝓝 (∑ y∈(G.orbitRepresentatives hq).roots, ∑ e∈(hl y).toFinset,
∫ ω, G.disconnectedIndicator ω e ∂G.law G.criticalProbability)) := by
apply tendsto_finsetSum _
intro y hy
apply tendsto_finsetSum _
intro e he
exact G.tendsto_integral_disconnectedIndicator G.criticalProbability q hmono.monotone hqp hlim e
have hqreal : Tendsto (fun n => (q n:ℝ)) atTop (𝓝 (G.criticalProbability:ℝ)) :=
continuous_subtype_val.continuousAt.tendsto.comp hlim
have hprod := ((hqreal.neg.mul (hqreal.const_sub 1)).mul hd)
have heq (n : ℕ) : (∑ x∈(G.orbitRepresentatives hq).roots,
∫ ω, G.normalizedSignedDiscrepancy (q n) ω x ∂G.law (q n)) =
-(q n:ℝ)*(1-(q n:ℝ))*(∑ y∈(G.orbitRepresentatives hq).roots, ∑ e∈(hl y).toFinset,
∫ ω, G.disconnectedIndicator ω e ∂G.law (q n)) := by
rw [G.expected_signed_discrepancy_drop hc hl hq D K hD hK]
congr 1
apply Finset.sum_congr rfl
intro y hy
apply Finset.sum_congr rfl
intro e he
exact integral_congr_ae (G.clusterCountDrop_eq_disconnected_ae_subcritical hc hl (q n) (hqmem n).2 e)
have hv := tendsto_nhds_unique hsig (hprod.congr' (Eventually.of_forall (fun n => (heq n).symm)))
rw [G.expected_signed_discrepancy_drop hc hl hq D K hD hK] at hv
apply mul_left_cancel₀ (a := -(G.criticalProbability:ℝ)*(1-(G.criticalProbability:ℝ))) _ hv
exact mul_ne_zero (neg_ne_zero.mpr (ne_of_gt (show 0<(G.criticalProbability:ℝ) from hp0)))
(sub_ne_zero.mpr (ne_of_gt (show (G.criticalProbability:ℝ)<1 from hp1)))
theorem critical_root_incident_drop_ae (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(hq : G.QuasiTransitive) (hp0 : 0<G.criticalProbability) (hp1 : G.criticalProbability<1)
(y : V) (hy : y∈(G.orbitRepresentatives hq).roots) (e : E) (he : y∈G.ends e) :
(fun ω => G.disconnectedIndicator ω e)=ᵐ[G.law G.criticalProbability]
(fun ω => G.clusterCountDrop ω e) := by
let f := fun (e : E) ω => G.disconnectedIndicator ω e-G.clusterCountDrop ω e
have hnon (e : E) (ω : Set E) : 0≤f e ω := sub_nonneg.mpr (G.clusterCountDrop_le_disconnected hc hl ω e)
have hint (e : E) : Integrable (f e) (G.law G.criticalProbability) :=
(G.integrable_disconnectedIndicator _ e).sub (G.integrable_clusterCountDrop hc hl _ e)
have hsum : (∑ z∈(G.orbitRepresentatives hq).roots, ∑ a∈(hl z).toFinset,
∫ ω, f a ω ∂G.law G.criticalProbability)=0 := by
simp_rw [f,integral_sub (G.integrable_disconnectedIndicator _ _)
(G.integrable_clusterCountDrop hc hl _ _),Finset.sum_sub_distrib]
exact sub_eq_zero.mpr (G.critical_expected_drop_eq_disconnected hc hl hq hp0 hp1).symm
have hout : (∑ a∈(hl y).toFinset, ∫ ω, f a ω ∂G.law G.criticalProbability)=0 :=
(Finset.sum_eq_zero_iff_of_nonneg (fun z hz => Finset.sum_nonneg
(fun a ha => integral_nonneg (hnon a)))).mp hsum y hy
have hin : (∫ ω, f e ω ∂G.law G.criticalProbability)=0 :=
(Finset.sum_eq_zero_iff_of_nonneg (fun a ha => integral_nonneg (hnon a))).mp hout e
((hl y).mem_toFinset.mpr he)
have hae := (integral_eq_zero_iff_of_nonneg (hnon e) (hint e)).mp hin
filter_upwards [hae] with ω hω
exact sub_eq_zero.mp hω
end CriticalPercolation.BondGraph
end
end
end OAI
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