openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Clusters /InfiniteUniquenessGeometry.lean
| 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 | |