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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 | import Mathlib
import OAI.Probability.CriticalPercolation.GraphGeometry.VertexBall
import OAI.Probability.CriticalPercolation.Criticality.OpenRay
import OAI.Probability.CriticalPercolation.GraphGeometry.InducedCritical
namespace OAI
section
section
noncomputable section
open Set Function MeasureTheory ProbabilityTheory Filter
open scoped Classical ENNReal
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
lemma image_restrict (S : Set V) (ω : Set E) :
Subtype.val '' (Subtype.val ⁻¹' ω : Set (G.inducedEdges S))=ω∩G.inducedEdges S := by
ext e
constructor
· rintro ⟨a,ha,rfl⟩
exact ⟨ha,a.property⟩
· rintro ⟨he,hs⟩
exact ⟨⟨e,hs⟩,he,rfl⟩
def domainComponent (S : Set V) (o : V) : Set V := G.cluster (G.inducedEdges S) o
lemma domainComponent_subset {S : Set V} {o : V} (ho : o∈S) :
G.domainComponent S o⊆S := by
have h := G.induced_cluster_subset S Set.univ (⟨o,ho⟩ : S)
simpa only [Set.image_univ,Subtype.range_coe,domainComponent] using h
lemma domainComponent_self (S : Set V) (o : V) : o∈G.domainComponent S o :=
SimpleGraph.Reachable.refl o
lemma domainComponent_closed (S : Set V) (o : V) {x y : V}
(hx : x∈G.domainComponent S o) (hxy : (G.openGraph (G.inducedEdges S)).Adj x y) :
y∈G.domainComponent S o := hx.trans hxy.reachable
lemma domainComponent_connected (S : Set V) (o : V) :
(G.induce (G.domainComponent S o)).fullGraph.Connected := by
let C := G.domainComponent S o
let a : C := ⟨o,G.domainComponent_self S o⟩
have hreach : ∀ x : C, ((G.induce C).fullGraph).Reachable a x := by
intro x
apply (G.induce_reachable C Set.univ a x).mpr
rw [Set.image_univ,Subtype.range_coe]
have h : (G.openGraph (G.inducedEdges S)).Reachable o x.val := x.property
rw [SimpleGraph.reachable_iff_reflTransGen] at h ⊢
have H : ∀ y, Relation.ReflTransGen (G.openGraph (G.inducedEdges S)).Adj o y →
Relation.ReflTransGen (G.openGraph (G.inducedEdges C)).Adj o y := by
intro y hy
induction hy with
| refl => exact .refl
| @tail b c hr hadj ih =>
have hb : b∈C := (SimpleGraph.reachable_iff_reflTransGen _ _).mpr hr
have hc : c∈C := G.domainComponent_closed S o hb hadj
obtain ⟨⟨e,he,hend⟩,hne⟩ := (SimpleGraph.fromEdgeSet_adj _).mp hadj
have hC : e∈G.inducedEdges C := by
intro v hv
rw [hend] at hv
rcases Sym2.mem_iff.mp hv with rfl|rfl
· exact hb
· exact hc
exact ih.tail ((SimpleGraph.fromEdgeSet_adj _).mpr ⟨⟨e,hC,hend⟩,hne⟩)
exact H _ h
exact (SimpleGraph.connected_iff_exists_forall_reachable _).mpr ⟨a,hreach⟩
lemma OpenRay.reachable_start {ω : Set E} (r : G.OpenRay ω) (n : ℕ) :
(G.openGraph ω).Reachable (r.vertex 0) (r.vertex n) := by
induction n with
| zero => exact .refl _
| succ n ih => exact ih.trans (r.adjacent n).reachable
lemma OpenRay.percolates {ω : Set E} (r : G.OpenRay ω) : ω∈G.percolates := by
refine ⟨r.vertex 0,?_⟩
exact (Set.infinite_range_of_injective r.injective).mono (by
rintro x ⟨n,rfl⟩
exact r.reachable_start G n)
def OpenRay.induce {ω : Set E} (r : G.OpenRay ω) (S : Set V)
(hS : ∀ n, r.vertex n∈S) :
(G.induce S).OpenRay (Subtype.val ⁻¹' ω) where
vertex n := ⟨r.vertex n,hS n⟩
injective := fun _ _ h => r.injective (congrArg Subtype.val h)
adjacent n := by
apply (G.induce_adj S (Subtype.val ⁻¹' ω) _ _).mpr
rw [G.image_restrict]
exact (G.open_induced_adj S ω _ _).mpr ⟨r.adjacent n,hS n,hS (n+1)⟩
lemma OpenRay.mem_domainComponent {ω : Set E} (r : G.OpenRay ω) (S : Set V)
(hS : ∀ n, r.vertex n∈S) (n : ℕ) :
r.vertex n∈G.domainComponent S (r.vertex 0) := by
induction n with
| zero => exact G.domainComponent_self S _
| succ n ih =>
apply G.domainComponent_closed S _ ih
apply G.openGraph_mono Set.inter_subset_right
exact (G.open_induced_adj S ω _ _).mpr ⟨r.adjacent n,hS n,hS (n+1)⟩
lemma ae_restriction_not_percolates [Countable V] [Countable E]
(S : Set V) (p : unitInterval) (h : (G.induce S).law p (G.induce S).percolates=0) :
∀ᵐ ω ∂G.law p, Subtype.val ⁻¹' ω∉(G.induce S).percolates := by
rw [←G.law_induce S p,Measure.map_apply] at h
· apply ae_iff.mpr
convert h using 1
congr 1
ext ω
simp only [Set.mem_ofPred_eq,Set.mem_preimage,not_not]
· exact Measurable.of_eval (fun edge => measurable_set_mem edge.val)
· exact (G.induce S).measurableSet_percolates
theorem bounded_ray_critical_component [Countable V] [Countable E]
(u v : V → ℝ)
(hbad : ¬∀ᵐ ω ∂G.law G.criticalProbability,
¬∃ r : G.OpenRay ω, ∃ U W : ℝ,
∀ n, |u (r.vertex n)|≤U ∧ |v (r.vertex n)|≤W) :
∃ (k : ℕ) (o : V),
let S := {x | |u x|≤(k:ℝ) ∧ |v x|≤(k:ℝ)}
o∈S ∧
let H := G.induce (G.domainComponent S o)
H.criticalProbability=G.criticalProbability ∧
0<H.law H.criticalProbability H.percolates := by
let S : ℕ → Set V := fun k => {x | |u x|≤(k:ℝ) ∧ |v x|≤(k:ℝ)}
have hex : ∃ (k : ℕ) (o : V), o∈S k ∧
0<(G.induce (G.domainComponent (S k) o)).law G.criticalProbability
(G.induce (G.domainComponent (S k) o)).percolates := by
by_contra hh
push Not at hh
have ha : ∀ k o, ∀ᵐ ω ∂G.law G.criticalProbability,
o∈S k → Subtype.val ⁻¹' ω∉(G.induce (G.domainComponent (S k) o)).percolates := by
intro k o
by_cases ho : o∈S k
· exact (G.ae_restriction_not_percolates _ _ (le_antisymm (hh k o ho) bot_le)).mono
(fun _ h _ => h)
· exact Filter.Eventually.of_forall (fun _ h => (ho h).elim)
have hall : ∀ᵐ ω ∂G.law G.criticalProbability, ∀ k o,
o∈S k → Subtype.val ⁻¹' ω∉(G.induce (G.domainComponent (S k) o)).percolates :=
(ae_all_iff.mpr fun k => ae_all_iff.mpr (ha k))
apply hbad
filter_upwards [hall] with ω hω
rintro ⟨r,U,W,hr⟩
obtain ⟨k,hk⟩ := exists_nat_ge (max U W)
have hrS : ∀ n, r.vertex n∈S k := fun n =>
⟨(hr n).1.trans ((le_max_left _ _).trans hk),
(hr n).2.trans ((le_max_right _ _).trans hk)⟩
exact hω k (r.vertex 0) (hrS 0)
((r.induce G _ (r.mem_domainComponent G _ hrS)).percolates _)
obtain ⟨k,o,ho,hp⟩ := hex
have heq := G.induce_criticalProbability_eq_of_pos (G.domainComponent (S k) o) hp
exact ⟨k,o,ho,heq,by rwa [heq]⟩
end CriticalPercolation.BondGraph
end
end
end
end OAI
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