| import Mathlib |
| import OAI.Probability.CriticalPercolation.ProductMeasure.MonotoneIndicatorUpper |
| import OAI.Probability.CriticalPercolation.Exploration.SliceExploration |
| import OAI.Probability.CriticalPercolation.Criticality.ContractingDomains |
| import OAI.Probability.CriticalPercolation.Criticality.DomainPercolation |
| import OAI.Probability.CriticalPercolation.Gluing.FreshExitContinuity |
| import OAI.Probability.CriticalPercolation.Gluing.FreshExitWitness |
| import OAI.Probability.CriticalPercolation.Gluing.UniformFreshBoundary |
| import OAI.Probability.CriticalPercolation.Modular.ClusterDiscrepancy |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set MeasureTheory ProbabilityTheory |
| open scoped Classical ENNReal |
|
|
| namespace CriticalPercolation.BondGraph |
| open BernoulliProduct |
|
|
| variable {V E : Type*} (G : BondGraph V E) |
|
|
| def domainFreshExitEvent (D S : Set V) (v : V) (N : ℕ) : Set (Set E) := |
| {ω | ω∩G.inducedEdges D ∈ G.freshExitEvent S v N} |
|
|
| lemma domainFreshExitEvent_upper {D S : Set V} {v : V} (hv : v∈S) (N : ℕ) : |
| IsUpperSet (G.domainFreshExitEvent D S v N) := |
| (G.freshExitEvent_upper hv N).preimage (fun _ _ h => Set.inter_subset_inter_left _ h) |
|
|
| lemma domainFreshExitEvent_finiteWitness {D S : Set V} {v : V} (hv : v∈S) (N : ℕ) : |
| HasFiniteOpenWitness (G.domainFreshExitEvent D S v N) := by |
| intro ω hω |
| obtain ⟨F,hF,hFω,hFe⟩ := G.freshExitEvent_finiteWitness hv N _ hω |
| refine ⟨F,hF,hFω.trans Set.inter_subset_left,?_⟩ |
| change F∩G.inducedEdges D ∈ G.freshExitEvent S v N |
| rwa [Set.inter_eq_left.mpr (hFω.trans Set.inter_subset_right)] |
|
|
| lemma measurableSet_domainFreshExitEvent [Countable E] |
| {D S : Set V} {v : V} (hv : v∈S) (N : ℕ) : |
| MeasurableSet (G.domainFreshExitEvent D S v N) := |
| measurableSet_finiteWitness _ (G.domainFreshExitEvent_upper hv N) |
| (G.domainFreshExitEvent_finiteWitness hv N) |
|
|
| lemma freshExit_in_domain_of_mass (D S : Set V) (v : V) (K : ℝ≥0∞) (N : ℕ) |
| (hD : G.weightCeiling v K⊆D) : |
| G.freshExitEvent S v N\G.massTailEvent v v K ⊆ G.domainFreshExitEvent D S v N := by |
| rintro ω ⟨⟨F,hF,hFω⟩,hm⟩ |
| have hmass : G.modularMass v (G.cluster ω v)≤K := le_of_not_gt hm |
| let X := G.inducedEdges S |
| let C := G.inducedEdges (G.weightCeiling v K) |
| have hCD : C⊆G.inducedEdges D := G.inducedEdges_monotone hD |
| have hC : G.cluster ((ω∩G.inducedEdges D)∩X) v=G.cluster (ω∩X) v := by |
| apply Set.Subset.antisymm (G.cluster_mono (Set.inter_subset_inter_left _ Set.inter_subset_left) v) |
| have he : G.cluster ((ω∩X)∩C) v=G.cluster (ω∩X) v := |
| G.cluster_eq_with_weightCeiling |
| ((G.modularMass_mono v (G.cluster_mono Set.inter_subset_left v)).trans hmass) |
| rw [←he] |
| apply G.cluster_mono |
| intro e he |
| exact ⟨⟨he.1.1,hCD he.2⟩,he.1.2⟩ |
| refine ⟨F,hF,fun e he => ?_⟩ |
| obtain ⟨⟨x,y,hexy,hx,hy⟩,heX⟩ := (hFω he).1 |
| have hyC : y∈G.cluster ω v := G.cluster_mono Set.inter_subset_left v hy |
| have hxC : x∈G.cluster ω v := by |
| by_cases heq : y=x |
| · simpa only [heq] using hyC |
| · exact hyC.trans (show (G.openGraph ω).Adj y x from by |
| apply (SimpleGraph.fromEdgeSet_adj _).mpr |
| refine ⟨⟨e,(hFω he).2,?_⟩,heq⟩ |
| rw [hexy,Sym2.eq_swap]).reachable |
| have heD : e∈G.inducedEdges D := by |
| apply hCD |
| intro z hz |
| rw [hexy] at hz |
| rcases Sym2.mem_iff.mp hz with rfl|rfl |
| · exact (G.stabilizerWeight_le_modularMass hxC).trans hmass |
| · exact (G.stabilizerWeight_le_modularMass hyC).trans hmass |
| refine ⟨⟨⟨x,y,hexy,?_,?_⟩,heX⟩,(hFω he).2,heD⟩ |
| · change x∉G.cluster ((ω∩G.inducedEdges D)∩X) v |
| rwa [hC] |
| · change y∈G.cluster ((ω∩G.inducedEdges D)∩X) v |
| rwa [hC] |
|
|
| lemma freshExit_same_cluster {S : Set V} {x y : V} {N : ℕ} {ω : Set E} |
| (hxy : y∈G.cluster (ω∩G.inducedEdges S) x) : |
| ω∈G.freshExitEvent S x N ↔ ω∈G.freshExitEvent S y N := by |
| simp only [freshExitEvent,Set.mem_ofPred_eq,freshClusterBoundary] |
| rw [G.cluster_eq_of_reachable hxy] |
|
|
| lemma InsideWalk.domainFreshExit_transfer [Countable V] [Countable E] |
| (p : unitInterval) {D S : Set V} {x y : V} {L : ℕ} (_ : x∈S) (hy : y∈S) |
| (h : G.InsideWalk (D∩S) x y L) (N : ℕ) : |
| (p:ℝ)^L*(G.law p).real (G.domainFreshExitEvent D S y N) ≤ |
| (G.law p).real (G.domainFreshExitEvent D S x N) := by |
| obtain ⟨w,hw,hs⟩ := h |
| obtain ⟨F,hF,hFS,hcard,hconn⟩ := G.walk_finite_bond_witness w (D∩S) hs |
| let A : Set (Set E) := {ω | F⊆ω} |
| have hA : IsUpperSet A := fun _ _ hωη hω => hω.trans hωη |
| have hAw : HasFiniteOpenWitness A := fun ω hω => ⟨F,hF,hω,Set.Subset.rfl⟩ |
| have hH := harris_finiteWitness (fun _ : E => p) A (G.domainFreshExitEvent D S y N) |
| hA (G.domainFreshExitEvent_upper hy N) hAw (G.domainFreshExitEvent_finiteWitness hy N) |
| simp only [law_const] at hH |
| have hsub : A∩G.domainFreshExitEvent D S y N ⊆ G.domainFreshExitEvent D S x N := by |
| intro ω hω |
| apply (G.freshExit_same_cluster (S := S) (x := x) (y := y) (N := N) (ω := ω∩G.inducedEdges D) ?_).mpr hω.2 |
| apply G.cluster_mono (ω := F) _ x hconn |
| intro e he |
| refine ⟨⟨hω.1 he,fun z hz => (hFS he z hz).1⟩,fun z hz => (hFS he z hz).2⟩ |
| have hprob : (p:ℝ)^L≤(G.law p).real A := by |
| rw [law,←law_const,real_law_subset p F hF] |
| exact pow_le_pow_of_le_one p.2.1 p.2.2 (hcard.trans hw) |
| exact (mul_le_mul_of_nonneg_right hprob measureReal_nonneg).trans |
| (hH.trans (measureReal_mono hsub)) |
|
|
| theorem uniform_domainFreshExit_of_mass |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive) |
| (p : unitInterval) (hp : 0<(p:ℝ)) (hp1 : (p:ℝ)<1) |
| (hper : 0<G.law p G.percolates) |
| (hm : ∀ v, ∀ᵐ ω ∂G.law p, G.modularMass v (G.cluster ω v)≠∞) : |
| ∃ θ : ℝ, 0<θ ∧ ∃ K : ℕ, 1≤K ∧ ∀ N : ℕ, ∃ R : ℕ, |
| ∀ (D S : Set V) (v : V), G.weightCeiling v K⊆D → G.vertexBall v R⊆S → |
| (∀ᵐ ω ∂G.law p, (G.cluster (ω∩G.inducedEdges S) v).Finite) → |
| θ<(G.law p).real (G.domainFreshExitEvent D S v N) := by |
| let := G.countable_vertices hc hl |
| let := G.countable_bonds hc hl |
| obtain ⟨θ,hθ,hroot⟩ := G.uniform_rooted_probability hc hl hq p hper |
| obtain ⟨K,hK,htail⟩ := G.uniform_massTail_small hq p hm (θ/4) (by positivity) |
| refine ⟨θ/2,by positivity,K,hK,fun N => ?_⟩ |
| obtain ⟨R,hR⟩ := G.fewOpenFreshBoundary_uniform_ball hc hl hq p hp hp1 N (θ/4) (by positivity) |
| refine ⟨R,fun D S v hD hS hf => ?_⟩ |
| have hfail : (G.law p).real (G.infiniteEvent v\G.freshExitEvent S v N)<θ/4 := by |
| exact lt_of_le_of_lt |
| (ENNReal.toReal_mono (measure_ne_top _ _) |
| (G.infinite_sdiff_freshExit_bound hl p S v N hf)) |
| (hR v _ (G.inducedEdges_monotone hS)) |
| have hcover : G.infiniteEvent v⊆ |
| (G.infiniteEvent v\G.freshExitEvent S v N)∪ |
| (G.massTailEvent v v K∪G.domainFreshExitEvent D S v N) := by |
| intro ω hω |
| by_cases he : ω∈G.freshExitEvent S v N |
| · right |
| by_cases hm : ω∈G.massTailEvent v v K |
| · exact Or.inl hm |
| · exact Or.inr (G.freshExit_in_domain_of_mass D S v K N hD ⟨he,hm⟩) |
| · exact Or.inl ⟨hω,he⟩ |
| have hprob := (measureReal_mono hcover (μ := G.law p)).trans (measureReal_union_le _ _) |
| have hu := measureReal_union_le (G.massTailEvent v v K) (G.domainFreshExitEvent D S v N) (μ := G.law p) |
| have hr := hroot v |
| have ht := htail v |
| linarith |
|
|
| end CriticalPercolation.BondGraph |
| end |
| end |
|
|
| end OAI |
|
|
