| import Mathlib |
| import OAI.Probability.CriticalPercolation.Criticality.BlockIndependence |
| import OAI.Probability.CriticalPercolation.ProductMeasure.LevelContactMeasurability |
| import OAI.Probability.CriticalPercolation.Criticality.NestedLayerBlock |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set MeasureTheory ProbabilityTheory |
| open scoped Classical ENNReal |
|
|
| namespace CriticalPercolation.BernoulliProduct |
| variable {E : Type*} |
| lemma measurableSet_block_of_dependsOn (A : Set (Set E)) (L : Set E) |
| (hA : MeasurableSet A) (hdep : DependsOn (fun ω : Set E => ω∈A) L) : |
| MeasurableSet[blockSigma L] A := by |
| have he : A=(fun ω : Set E => ω∩L) ⁻¹' A := by |
| ext ω |
| exact (hdep (x := ω) (y := (show Set E from ω∩L)) |
| (fun e he => propext (show e∈ω ↔ e∈ω∩L from by simp [he]))).to_iff |
| rw [he] |
| exact hA.preimage (measurable_inter_block L) |
| end CriticalPercolation.BernoulliProduct |
| namespace CriticalPercolation.BondGraph |
| open BernoulliProduct |
|
|
| variable {V E : Type*} (G : BondGraph V E) |
|
|
| def entrySeeds (D : Set V) (o : V) (ω : Set E) : Set V := |
| G.cluster (ω∩(G.inducedEdges D)ᶜ) o ∩ D |
|
|
| lemma entrySeeds_subset_cluster (D : Set V) (o : V) (ω : Set E) : |
| G.entrySeeds D o ω⊆G.cluster ω o := |
| fun _ h => G.cluster_mono Set.inter_subset_left o h.1 |
|
|
| lemma measurable_entrySeeds [Countable V] [Countable E] (D : Set V) (o : V) : |
| @Measurable (Set E) (Set V) (blockSigma (G.inducedEdges D)ᶜ) inferInstance |
| (G.entrySeeds D o) := by |
| let : MeasurableSpace (Set E) := blockSigma (G.inducedEdges D)ᶜ |
| change Measurable (G.entrySeeds D o) |
| apply Measurable.of_eval |
| intro x |
| exact (((measurable_pi_apply x).comp ((G.measurable_cluster o).comp |
| (measurable_inter_block (G.inducedEdges D)ᶜ))).and measurable_const) |
|
|
| lemma measurableSet_entrySeed [Countable V] [Countable E] (D : Set V) (o x : V) : |
| MeasurableSet[blockSigma (G.inducedEdges D)ᶜ] {ω | x∈G.entrySeeds D o ω} := by |
| let : MeasurableSpace (Set E) := blockSigma (G.inducedEdges D)ᶜ |
| exact ((measurable_pi_apply x).comp (G.measurable_entrySeeds D o)).setOf |
|
|
| lemma entrySeed_eq_or_adj_exterior {D : Set V} {o x : V} {ω : Set E} |
| (hx : x∈G.entrySeeds D o ω) : |
| x=o ∨ ∃ y, y∉D ∧ G.fullGraph.Adj x y := by |
| by_cases he : x=o |
| · exact Or.inl he |
| right |
| obtain ⟨w⟩ := hx.1.symm |
| obtain ⟨y,hxy,w',hw⟩ := SimpleGraph.Walk.exists_eq_cons_of_ne he w |
| obtain ⟨⟨e,heω,hexy⟩,hne⟩ := (SimpleGraph.fromEdgeSet_adj _).mp hxy |
| refine ⟨y,?_,G.openGraph_mono (Set.subset_univ _) hxy⟩ |
| intro hy |
| apply heω.2 |
| intro z hz |
| rw [hexy] at hz |
| exact (Sym2.mem_iff.mp hz).elim (fun h => h ▸ hx.2) (fun h => h ▸ hy) |
|
|
| lemma entrySeeds_weight_band (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) |
| (hq : G.QuasiTransitive) (R : ℕ) : |
| ∃ a : ℝ, 0<a ∧ a≤1 ∧ ∀ o T, a*T≤1 → ∀ ω x, |
| x∈G.entrySeeds (G.safeBallDomain o T R) o ω → |
| a*T≤G.modularRatio x o ∧ G.modularRatio x o≤T := by |
| obtain ⟨C,hC,hlow⟩ := G.safeBallDomain_low_inclusion hc hl hq R |
| obtain ⟨B,hB,hball⟩ := G.modularRatio_ball_weight hc hl hq 1 |
| have hC0 : 0<C := lt_of_lt_of_le zero_lt_one hC |
| have hB0 : 0<B := lt_of_lt_of_le zero_lt_one hB |
| refine ⟨1/(C*B),by positivity,?_,fun o T hroot ω x hx => ⟨?_,G.safeBallDomain_subset o T R hx.2⟩⟩ |
| · exact (div_le_one (by positivity)).mpr (by nlinarith : (1:ℝ)≤C*B) |
| rcases G.entrySeed_eq_or_adj_exterior hx with rfl|⟨y,hy,hxy⟩ |
| · simpa only [G.modularRatio_self] using hroot |
| have hlower : T/C<G.modularRatio y o := lt_of_not_ge (fun hh => hy (hlow o T hh)) |
| have hbound : G.modularRatio y o≤B*G.modularRatio x o := |
| hball o y x (G.vertexBall_comm.mp (G.vertexBall_step (G.mem_vertexBall_self x 0) hxy)) |
| have : T<(G.modularRatio x o)*(C*B) := by |
| have hh := (div_lt_iff₀ hC0).mp (hlower.trans_le hbound) |
| nlinarith |
| have hh := (div_lt_iff₀ (mul_pos hC0 hB0)).mpr this |
| simpa only [one_div,inv_mul_eq_div] using hh.le |
|
|
| lemma entrySeeds_weight_band_start (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) |
| (hq : G.QuasiTransitive) (R : ℕ) : |
| ∃ a U : ℝ, 0<a ∧ 0<U ∧ a*U≤1 ∧ (∀ o, o∈G.safeBallDomain o U R) ∧ |
| ∀ o T, a*T≤1 → ∀ ω x, |
| x∈G.entrySeeds (G.safeBallDomain o T R) o ω → |
| a*T≤G.modularRatio x o ∧ G.modularRatio x o≤T := by |
| obtain ⟨C,hC,hlow⟩ := G.safeBallDomain_low_inclusion hc hl hq R |
| obtain ⟨B,hB,hball⟩ := G.modularRatio_ball_weight hc hl hq 1 |
| have hC0 : 0<C := lt_of_lt_of_le zero_lt_one hC |
| have hB0 : 0<B := lt_of_lt_of_le zero_lt_one hB |
| refine ⟨1/(C*B),C,by positivity,hC0,?_,?_,fun o T hroot ω x hx => ⟨?_,G.safeBallDomain_subset o T R hx.2⟩⟩ |
| · have he : (1/(C*B))*C=1/B := by field_simp |
| rw [he] |
| exact (div_le_one hB0).mpr hB |
| · intro o |
| apply hlow o C |
| change G.modularRatio o o≤C/C |
| rw [G.modularRatio_self,div_self hC0.ne'] |
| rcases G.entrySeed_eq_or_adj_exterior hx with rfl|⟨y,hy,hxy⟩ |
| · simpa only [G.modularRatio_self] using hroot |
| have hlower : T/C<G.modularRatio y o := lt_of_not_ge (fun hh => hy (hlow o T hh)) |
| have hbound : G.modularRatio y o≤B*G.modularRatio x o := |
| hball o y x (G.vertexBall_comm.mp (G.vertexBall_step (G.mem_vertexBall_self x 0) hxy)) |
| have : T<(G.modularRatio x o)*(C*B) := by |
| have hh := (div_lt_iff₀ hC0).mp (hlower.trans_le hbound) |
| nlinarith |
| have hh := (div_lt_iff₀ (mul_pos hC0 hB0)).mpr this |
| simpa only [one_div,inv_mul_eq_div] using hh.le |
|
|
| lemma layerStopped_eq (D H : Set V) (ω : Set E) : |
| ((ω∩(G.inducedEdges D\G.inducedEdges H))∩G.inducedEdges D)∩G.inducedEdges Hᶜ = |
| (ω∩G.inducedEdges D)∩G.inducedEdges Hᶜ := by |
| ext e |
| constructor |
| · exact fun h => ⟨⟨h.1.1.1,h.1.2⟩,h.2⟩ |
| · intro h |
| refine ⟨⟨⟨h.1.1,h.1.2,?_⟩,h.1.2⟩,h.2⟩ |
| intro he |
| obtain ⟨x,y,hxy⟩ : ∃ x y, G.ends e=s(x,y) := Sym2.exists.mp ⟨G.ends e,rfl⟩ |
| have hx : x∈G.ends e := by rw [hxy]; simp |
| exact h.2 x hx (he x hx) |
|
|
| lemma domainLayer_restrict (D H : Set V) (x : V) (hx : x∉H) (N : ℕ) |
| (ω : Set E) : |
| ω∩(G.inducedEdges D\G.inducedEdges H)∈G.domainFreshExitEvent D Hᶜ x N ↔ |
| ω∈G.domainFreshExitEvent D Hᶜ x N := by |
| constructor |
| · exact G.domainFreshExitEvent_upper hx N Set.inter_subset_left |
| · rintro ⟨F,hF,hFe⟩ |
| refine ⟨F,hF,?_⟩ |
| intro e he |
| obtain ⟨⟨⟨u,v,hev,hu,hv⟩,heH⟩,heω,heD⟩ := hFe he |
| have hvH := G.cluster_induced_subset (ω∩G.inducedEdges D) Hᶜ hx hv |
| have hn : e∉G.inducedEdges H := by |
| intro hh |
| exact hvH (hh v (by rw [hev]; simp)) |
| have hu' : u∉G.cluster (((ω∩(G.inducedEdges D\G.inducedEdges H))∩G.inducedEdges D)∩ |
| G.inducedEdges Hᶜ) x := by |
| rw [G.layerStopped_eq] |
| exact hu |
| have hv' : v∈G.cluster (((ω∩(G.inducedEdges D\G.inducedEdges H))∩G.inducedEdges D)∩ |
| G.inducedEdges Hᶜ) x := by |
| rw [G.layerStopped_eq] |
| exact hv |
| exact ⟨⟨⟨u,v,hev,hu',hv'⟩,heH⟩,⟨heω,heD,hn⟩,heD⟩ |
|
|
| lemma boundedLayer_restrict (D H : Set V) (x : V) (hx : x∉H) (N : ℕ) (M : ℝ≥0∞) |
| (ω : Set E) : |
| ω∩(G.inducedEdges D\G.inducedEdges H)∈G.boundedDomainFreshExitEvent D Hᶜ x N M ↔ |
| ω∈G.boundedDomainFreshExitEvent D Hᶜ x N M := by |
| simp only [boundedDomainFreshExitEvent,Set.mem_inter_iff,Set.mem_ofPred_eq, |
| G.layerStopped_eq,G.domainLayer_restrict D H x hx N ω] |
|
|
| lemma measurableSet_boundedDomainFreshExit [Countable V] [Countable E] |
| (D S : Set V) (x : V) (hx : x∈S) (N : ℕ) (M : ℝ≥0∞) : |
| MeasurableSet (G.boundedDomainFreshExitEvent D S x N M) := by |
| apply (G.measurableSet_domainFreshExitEvent hx N).inter |
| apply measurableSet_le _ measurable_const |
| exact (G.measurable_modularClusterMass x x).comp |
| (((measurable_inter_block (G.inducedEdges S)).mono (blockSigma_le _) le_rfl).comp |
| ((measurable_inter_block (G.inducedEdges D)).mono (blockSigma_le _) le_rfl)) |
|
|
| lemma measurableSet_boundedLayer [Countable V] [Countable E] |
| (D H : Set V) (x : V) (hx : x∉H) (N : ℕ) (M : ℝ≥0∞) : |
| MeasurableSet[blockSigma (G.inducedEdges D\G.inducedEdges H)] |
| (G.boundedDomainFreshExitEvent D Hᶜ x N M) := by |
| have he : G.boundedDomainFreshExitEvent D Hᶜ x N M= |
| (fun ω : Set E => ω∩(G.inducedEdges D\G.inducedEdges H)) ⁻¹' |
| G.boundedDomainFreshExitEvent D Hᶜ x N M := by |
| ext ω |
| exact (G.boundedLayer_restrict D H x hx N M ω).symm |
| rw [he] |
| exact (G.measurableSet_boundedDomainFreshExit D Hᶜ x hx N M).preimage (measurable_inter_block _) |
|
|
| lemma entry_good_independent [Countable V] [Countable E] |
| (q : unitInterval) (D H : Set V) (o x : V) (hx : x∉H) (N : ℕ) (M : ℝ≥0∞) : |
| (G.law q) ({ω | x∈G.entrySeeds D o ω}∩G.boundedDomainFreshExitEvent D Hᶜ x N M)= |
| (G.law q) {ω | x∈G.entrySeeds D o ω}* |
| (G.law q) (G.boundedDomainFreshExitEvent D Hᶜ x N M) := by |
| have hdis : Disjoint (G.inducedEdges D)ᶜ (G.inducedEdges D\G.inducedEdges H) := |
| Set.disjoint_left.mpr (fun e he hf => he hf.1) |
| simpa only [law,←law_const] using ((independent_disjoint_blocks (fun _ : E => q) _ _ hdis).indepSet_of_measurableSet |
| (G.measurableSet_entrySeed D o x) (G.measurableSet_boundedLayer D H x hx N M)).measure_inter_eq_mul |
|
|
| end CriticalPercolation.BondGraph |
| end |
| end |
|
|
| end OAI |
|
|
