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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 | import Mathlib
import OAI.Probability.CriticalPercolation.GraphGeometry.VertexBall
import OAI.Probability.CriticalPercolation.GraphGeometry.InducedCritical
import OAI.Probability.CriticalPercolation.Gluing.RelayMoves
import OAI.Probability.CriticalPercolation.Modular.CriticalHeavyNull
import OAI.Probability.CriticalPercolation.Modular.LightClusterNumber
namespace OAI
section
noncomputable section
open Set MeasureTheory
open scoped Classical ENNReal
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
def modularDownSet (v : V) : Set V := {x | G.stabilizerWeight v x≤1}
lemma mem_modularDownSet_self (v : V) : v∈G.modularDownSet v := by
simp [modularDownSet]
lemma cluster_eq_in_modularDownSet {ω : Set E} {v : V}
(hv : G.clusterPeak ω v) :
G.cluster (ω∩G.inducedEdges (G.modularDownSet v)) v=G.cluster ω v := by
apply Set.Subset.antisymm (G.cluster_mono Set.inter_subset_left v)
rintro x ⟨w⟩
apply G.walk_reachable_induced _ ω w
intro y hy
exact hv y (w.takeUntil y hy).reachable
lemma lightInfiniteEvent_subset_some_lightInfinitePeak
(hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (o v : V) :
G.lightInfiniteEvent o v⊆⋃ x : V, {ω | x∈G.lightInfinitePeaks ω} := by
intro ω hω
have hm : G.modularMass v (G.cluster ω v)≠∞ := by
intro h
exact hω.2.ne ((G.modularMass_eq_top_change_base hc hl o v _).mpr h)
obtain ⟨x,hx,hpeak⟩ := G.clusterPeaks_nonempty_of_mass_ne_top hc hl ω v hm
apply Set.mem_iUnion.mpr
refine ⟨x,?_,hpeak⟩
have hself : ω∈G.lightInfiniteEvent v v := by
rwa [G.lightInfiniteEvent_change_base hc hl o v v] at hω
exact G.lightInfiniteEvent_of_reachable hc hl hx hself
lemma exists_positive_lightInfinitePeak [Countable V] [Countable E]
(hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (p : unitInterval) (o v : V)
(hv : 0<G.law p (G.lightInfiniteEvent o v)) :
∃ x, 0<G.law p {ω | x∈G.lightInfinitePeaks ω} := by
by_contra hn
have hz : ∀ x, G.law p {ω | x∈G.lightInfinitePeaks ω}=0 := fun x =>
le_antisymm (le_of_not_gt (fun h => hn ⟨x,h⟩)) bot_le
exact hv.ne' (measure_mono_null (G.lightInfiniteEvent_subset_some_lightInfinitePeak hc hl o v)
(measure_iUnion_null hz))
lemma peakInfinite_subset_induce_percolates (v : V) :
{ω | (G.cluster ω v).Infinite ∧ G.clusterPeak ω v} ⊆
(fun ω : Set E => (Subtype.val ⁻¹' ω : Set (G.inducedEdges (G.modularDownSet v)))) ⁻¹'
(G.induce (G.modularDownSet v)).percolates := by
rintro ω ⟨hi,hp⟩
let v' : G.modularDownSet v := ⟨v,G.mem_modularDownSet_self v⟩
refine ⟨v',?_⟩
have he : Subtype.val '' (Subtype.val ⁻¹' ω : Set (G.inducedEdges (G.modularDownSet v)))=
ω∩G.inducedEdges (G.modularDownSet v) := by
ext e
simp only [Set.mem_image,Set.mem_preimage,Set.mem_inter_iff]
exact ⟨fun ⟨a,ha,he⟩ => he ▸ ⟨ha,a.property⟩,
fun ⟨he,hS⟩ => ⟨⟨e,hS⟩,he,rfl⟩⟩
apply Set.Infinite.of_image (f := Subtype.val)
rw [G.induce_cluster_image,he]
exact (G.cluster_eq_in_modularDownSet hp).symm ▸ hi
theorem positive_light_implies_downward_percolation
(hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (p : unitInterval) (o v : V)
(hv : 0<G.law p (G.lightInfiniteEvent o v)) :
∃ x, 0<(G.induce (G.modularDownSet x)).law p (G.induce (G.modularDownSet x)).percolates := by
let := G.countable_vertices hc hl
let := G.countable_bonds hc hl
obtain ⟨x,hx⟩ := G.exists_positive_lightInfinitePeak hc hl p o v hv
refine ⟨x,?_⟩
rw [←G.law_induce,Measure.map_apply]
· apply hx.trans_le
apply measure_mono
intro ω hω
exact G.peakInfinite_subset_induce_percolates x ⟨hω.1.1,hω.2⟩
· exact Measurable.of_eval (fun edge => measurable_set_mem edge.val)
· exact (G.induce (G.modularDownSet x)).measurableSet_percolates
theorem nonunimodular_counterexample_downward_critical [Infinite V]
(hc : G.fullGraph.Connected) (hl : G.LocallyFinite) (hq : G.QuasiTransitive)
(hpc : G.criticalProbability<1) (hu : ¬G.CombinatoriallyUnimodular)
(o : V) (hp : G.law G.criticalProbability G.percolates≠0) :
∃ x, (G.induce (G.modularDownSet x)).criticalProbability=G.criticalProbability ∧
0<(G.induce (G.modularDownSet x)).law
(G.induce (G.modularDownSet x)).criticalProbability
(G.induce (G.modularDownSet x)).percolates := by
let := G.countable_vertices hc hl
let := G.countable_bonds hc hl
have hmass := G.nonunimodular_critical_mass_finite_ae hc hl hq hpc hu o
have hex : ∃ v, 0<G.law G.criticalProbability (G.lightInfiniteEvent o v) := by
by_contra hn
have hz : ∀ v, G.law G.criticalProbability (G.lightInfiniteEvent o v)=0 := fun v =>
le_antisymm (le_of_not_gt (fun h => hn ⟨v,h⟩)) bot_le
apply hp
apply le_antisymm ?_ bot_le
calc
G.law G.criticalProbability G.percolates ≤
G.law G.criticalProbability (⋃ v, G.lightInfiniteEvent o v) := by
apply measure_mono_ae
filter_upwards [hmass] with ω hω
intro ⟨v,hv⟩
exact Set.mem_iUnion.mpr ⟨v,hv,hω v⟩
_ = 0 := measure_iUnion_null hz
obtain ⟨v,hv⟩ := hex
obtain ⟨x,hx⟩ := G.positive_light_implies_downward_percolation hc hl _ o v hv
have he := G.induce_criticalProbability_eq_of_pos (G.modularDownSet x) hx
exact ⟨x,he,by rwa [he]⟩
end CriticalPercolation.BondGraph
end
end
end OAI
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