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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Criticality.SubexponentialVolumes
import OAI.Probability.CriticalPercolation.GraphGeometry.InfiniteAdjacent
import OAI.Probability.CriticalPercolation.GraphGeometry.VertexBall
import OAI.Probability.CriticalPercolation.Modular.ClusterDiscrepancy
import OAI.Probability.CriticalPercolation.ProductMeasure.FiniteInterpolationMeasure
namespace OAI
section
section
noncomputable section
open Set MeasureTheory ProbabilityTheory Filter
open scoped Classical Topology
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
def distinctInfiniteEvent (x y : V) : Set (Set E) :=
{ω | (G.cluster ω x).Infinite ∧ (G.cluster ω y).Infinite ∧
¬ (G.openGraph ω).Reachable x y}
lemma measurableSet_distinctInfiniteEvent [Countable V] [Countable E] (x y : V) :
MeasurableSet (G.distinctInfiniteEvent x y) :=
((G.measurable_infinite_cluster x).and
((G.measurable_infinite_cluster y).and (G.measurable_reachable x y).not)).setOf
lemma distinctInfiniteEvent_subset_twoClusterEvent (x y : V) (s : ℕ) :
G.distinctInfiniteEvent x y ⊆ G.twoClusterEvent x y s := by
rintro ω ⟨hx,hy,hxy⟩
exact ⟨⟨hxy,G.infiniteEvent_subset_clusterTail x s hx⟩,
G.infiniteEvent_subset_clusterTail y s hy⟩
lemma cluster_insert_edge_unchanged (ω : Set E) (e : E) (o : V)
(hends : ∀ v ∈ G.ends e, v ∉ G.cluster ω o) :
G.cluster (insert e ω) o = G.cluster ω o := by
apply Set.Subset.antisymm
· intro y hy
have hr := ((G.openGraph (insert e ω)).reachable_iff_reflTransGen o y).mp hy
clear hy
induction hr with
| refl => exact .refl _
| @tail x y hr hxy ih =>
obtain ⟨⟨f,hf,hfe⟩,hne⟩ := (SimpleGraph.fromEdgeSet_adj _).mp hxy
rcases hf with rfl | hf
· exact False.elim (hends x (by simp [hfe]) ih)
· have hadj : (G.openGraph ω).Adj x y :=
(SimpleGraph.fromEdgeSet_adj _).mpr ⟨⟨f,hf,hfe⟩,hne⟩
exact ih.trans hadj.reachable
· exact G.cluster_mono (Set.subset_insert e ω) o
lemma distinctInfinite_force_adjacent_length (n : ℕ) :
∀ (ω : Set E) (x y : V), lengthReach G.fullGraph n x y →
ω ∈ G.distinctInfiniteEvent x y →
∃ S : Set E, S.Finite ∧ ∃ a b : V, G.fullGraph.Adj a b ∧
S ∪ ω ∈ G.distinctInfiniteEvent a b := by
induction n with
| zero =>
intro ω x y hxy h
exact False.elim (h.2.2 (hxy ▸ SimpleGraph.Reachable.refl x))
| succ n ih =>
rintro ω x y ⟨u,hxu,huy⟩ ⟨hx,hy,hxy⟩
by_cases hxuω : (G.openGraph ω).Reachable x u
· refine ⟨∅,Set.finite_empty,u,y,huy,?_⟩
rw [Set.empty_union]
refine ⟨?_,hy,fun huy => hxy (hxuω.trans huy)⟩
rwa [← G.cluster_eq_of_reachable hxuω]
· obtain ⟨⟨e,_,he⟩,hne⟩ := (SimpleGraph.fromEdgeSet_adj _).mp huy
have hends : ∀ v ∈ G.ends e, v ∉ G.cluster ω x := by
intro v hv
rw [he,Sym2.mem_iff] at hv
rcases hv with rfl | rfl
· exact hxuω
· exact hxy
have hc := G.cluster_insert_edge_unchanged ω e x hends
have hnewx : (G.cluster (insert e ω) x).Infinite := hc ▸ hx
have hay : (G.openGraph (insert e ω)).Adj u y :=
(SimpleGraph.fromEdgeSet_adj _).mpr ⟨⟨e,Set.mem_insert e ω,he⟩,hne⟩
have hnewu : (G.cluster (insert e ω) u).Infinite := by
have hh := hy.mono (G.cluster_mono (Set.subset_insert e ω) y)
rwa [← G.cluster_eq_of_reachable hay.reachable] at hh
have hnot : ¬ (G.openGraph (insert e ω)).Reachable x u := by
change u ∉ G.cluster (insert e ω) x
rwa [hc]
obtain ⟨S,hS,a,b,hab,hhab⟩ := ih (insert e ω) x u hxu ⟨hnewx,hnewu,hnot⟩
refine ⟨insert e S,hS.insert e,a,b,hab,?_⟩
simpa only [Set.insert_union,Set.union_insert] using hhab
lemma distinctInfinite_force_adjacent (hconnected : G.fullGraph.Connected)
(ω : Set E) (x y : V) (h : ω ∈ G.distinctInfiniteEvent x y) :
∃ S : Set E, S.Finite ∧ ∃ a b : V, G.fullGraph.Adj a b ∧
S ∪ ω ∈ G.distinctInfiniteEvent a b := by
obtain ⟨n,hn⟩ := (reachable_iff_exists_lengthReach G.fullGraph x y).mp (hconnected x y)
exact G.distinctInfinite_force_adjacent_length n ω x y hn h
theorem adjacent_distinctInfinite_null [Countable V] [Countable E]
(hconnected : G.fullGraph.Connected) (hlocal : G.LocallyFinite)
(hquasi : G.QuasiTransitive) (hg : G.SubexponentialGrowth)
(p : unitInterval) (hp : (p:ℝ) ∈ Set.Ioo 0 1) (x y : V)
(hxy : G.fullGraph.Adj x y) : G.law p (G.distinctInfiniteEvent x y) = 0 := by
obtain ⟨A,_,hA⟩ := G.subexponential_adjacent_bound hconnected hlocal hquasi hg
obtain ⟨⟨e,_,he⟩,_⟩ := (SimpleGraph.fromEdgeSet_adj _).mp hxy
have hbound : ∀ K : ℕ, (G.law p).real (G.distinctInfiniteEvent x y) ≤
A/((p:ℝ)*(2:ℝ)^K) := by
intro K
have hsub := measureReal_mono (G.distinctInfiniteEvent_subset_twoClusterEvent x y (4^K))
(μ := G.law p)
have heq : s(x,y) = s((G.ends e).out.1,(G.ends e).out.2) := by
exact he.symm.trans (G.ends e).out_eq.symm
rw [G.twoClusterEvent_sym2 heq (4^K)] at hsub
exact hsub.trans (hA p hp e K)
have ht : Tendsto (fun K : ℕ => A/((p:ℝ)*(2:ℝ)^K)) atTop (𝓝 0) :=
tendsto_const_nhds.div_atTop
((tendsto_pow_atTop_atTop_of_one_lt (by norm_num : (1:ℝ)<2)).const_mul_atTop hp.1)
have hz : (G.law p).real (G.distinctInfiniteEvent x y) = 0 :=
le_antisymm (ge_of_tendsto ht (Eventually.of_forall hbound)) measureReal_nonneg
exact (measureReal_eq_zero_iff (measure_ne_top _ _)).mp hz
end CriticalPercolation.BondGraph
end
end
end
end OAI
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