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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Exploration.Score
import OAI.Probability.CriticalPercolation.Modular.CriticalHeavyDrop
import OAI.Probability.CriticalPercolation.Modular.HeavyFiniteGeometry
namespace OAI
section
noncomputable section
open Set MeasureTheory Filter
open scoped Classical ENNReal
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
lemma critical_root_adjacent_heavy_null [Countable V] [Countable E]
(hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive)
(hp0 : 0<G.criticalProbability) (hp1 : G.criticalProbability<1) (o x y : V)
(hx : x∈(G.orbitRepresentatives hq).roots) (hxy : G.fullGraph.Adj x y) :
G.law G.criticalProbability (G.distinctHeavyEvent o x y)=0 := by
obtain ⟨⟨e,_,he⟩,_⟩ := (SimpleGraph.fromEdgeSet_adj _).mp hxy
have hae := G.critical_root_incident_drop_ae hc hl hq hp0 hp1 x hx e (by simp [he])
have han : ∀ᵐ ω ∂G.law G.criticalProbability, ω∉G.distinctHeavyEvent o x y := by
filter_upwards [hae] with ω hω hbad
obtain ⟨hxH,hyH,hR⟩ := hbad
have hclosed : e∉ω := fun h => hR (LazyDFS.prop_openEdge_reachable G ω h he)
have hs : ω\{e}=ω := by ext f; by_cases hf : f=e <;> simp_all
have hx' : G.modularMass x (G.cluster ω x)=∞ :=
(G.modularMass_eq_top_change_base hc hl o x _).mp hxH
have hy' : G.modularMass y (G.cluster ω y)=∞ :=
(G.modularMass_eq_top_change_base hc hl o y _).mp hyH
rw [G.disconnectedIndicator_of_ends ω e x y he,G.clusterCountDrop_of_ends hc hl ω e x y he,hs] at hω
norm_num [hR,hx',hy'] at hω
rw [ae_iff] at han
simpa only [not_not,Set.ofPred_mem_eq] using han
lemma critical_adjacent_heavy_null [Countable V] [Countable E]
(hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive)
(hp0 : 0<G.criticalProbability) (hp1 : G.criticalProbability<1) (o x y : V)
(hxy : G.fullGraph.Adj x y) :
G.law G.criticalProbability (G.distinctHeavyEvent o x y)=0 := by
obtain ⟨r,hr,φ,hφ⟩ := (G.orbitRepresentatives hq).cover x
obtain ⟨s,hs⟩ := φ.vertex.surjective y
rw [←hφ,←hs,φ.distinctHeavyEvent_measure G hc hl]
exact G.critical_root_adjacent_heavy_null hc hl hq hp0 hp1 o r s hr
((φ.fullGraph_adj G r s).mp (by simpa only [hφ,hs] using hxy))
theorem atMostOneHeavy_critical_ae (hc : G.fullGraph.Connected) (hl : G.LocallyFinite)
(hq : G.QuasiTransitive) (hp1 : G.criticalProbability<1) (o : V) :
∀ᵐ ω ∂G.law G.criticalProbability, ω∈G.atMostOneHeavy o := by
let := G.countable_vertices hc hl
let := G.countable_bonds hc hl
by_cases hp0 : 0<G.criticalProbability
· exact G.atMostOneHeavy_ae_of_adjacent_null hc G.criticalProbability hp0 o
(fun x y hxy => G.critical_adjacent_heavy_null hc hl hq hp0 hp1 o x y hxy)
· have he : G.criticalProbability=0 := le_antisymm (le_of_not_gt hp0) (unitInterval.nonneg _)
rw [he]
have hnull := G.law_zero_percolates
have hnp : ∀ᵐ ω ∂G.law 0, ω∉G.percolates := by
rw [ae_iff]
simpa only [not_not,Set.ofPred_mem_eq] using hnull
filter_upwards [hnp] with ω hω
intro x y hx hy
exact False.elim (hω ⟨x,G.infinite_of_modularMass_top o hx⟩)
end CriticalPercolation.BondGraph
end
end
end OAI
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