openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Classification /StructuredCoordinates.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Model | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.EquivariantCoordinates | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.GraphActionCoordinates | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.ActionCoordinates | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.CriticalCoordinates | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open Set Function MeasureTheory ProbabilityTheory | |
| open scoped Classical ENNReal | |
| namespace CriticalPercolation.BondGraph | |
| variable {V E Λ N : Type*} (G : BondGraph V E) | |
| [Group Λ] [Group N] [MulAction Λ V] [MulAction Λ E] | |
| def integerPairComplex (z : ℤ×ℤ) : ℂ := (z.1 : ℂ)+(z.2 : ℂ)*Complex.I | |
| lemma integerPairComplex_add (a b : ℤ×ℤ) : | |
| integerPairComplex (a+b)=integerPairComplex a+integerPairComplex b := by | |
| simp only [integerPairComplex,Prod.fst_add,Prod.snd_add,Int.cast_add] | |
| ring | |
| lemma integerPairComplex_re (z : ℤ×ℤ) : (integerPairComplex z).re=z.1 := by | |
| simp [integerPairComplex] | |
| lemma integerPairComplex_im (z : ℤ×ℤ) : (integerPairComplex z).im=z.2 := by | |
| simp [integerPairComplex] | |
| theorem exists_integer_pair_coordinates | |
| (ρ : Λ →* N) (hρ : Surjective ρ) | |
| (φ : N →* Multiplicative (ℤ×ℤ)) (hφ : Surjective φ) | |
| (hstab : ∀ v : V, ((MulAction.stabilizer Λ v).map ρ : Set N).Finite) | |
| [Nonempty V] : | |
| ∃ z : V → ℤ×ℤ, Surjective z ∧ | |
| ∀ g v, z (g • v)=(φ (ρ g)).toAdd+z v := by | |
| obtain ⟨π,hπ⟩ := EquivariantCoordinates.exists_equivariant (φ.comp ρ) | |
| (EquivariantCoordinates.stabilizer_killed ρ φ hstab) | |
| refine ⟨fun v => (π v).toAdd,?_,?_⟩ | |
| · intro t | |
| obtain ⟨v,hv⟩ := EquivariantCoordinates.surjective_of_equivariant | |
| (φ.comp ρ) (hφ.comp hρ) π hπ (Multiplicative.ofAdd t) | |
| exact ⟨v,congrArg Multiplicative.toAdd hv⟩ | |
| · intro g v | |
| exact congrArg Multiplicative.toAdd (hπ g v) | |
| lemma integer_pair_coordinate_geometry | |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) | |
| (ha : G.PreservesEnds (Λ := Λ)) | |
| (S : Set V) (hS : S.Finite) (hcover : ∀ v, ∃ s∈S, v∈MulAction.orbit Λ s) | |
| (a : Λ → ℤ×ℤ) (z : V → ℤ×ℤ) (honto : Surjective z) | |
| (hz : ∀ g v, z (g • v)=a g+z v) : | |
| ∃ (o : V) (c : ℕ) (D : ℝ), 0≤D ∧ integerPairComplex (z o)=0 ∧ | |
| (∀ ⦃x y⦄, G.fullGraph.Adj x y → | |
| ‖integerPairComplex (z x)-integerPairComplex (z y)‖≤D) ∧ | |
| (∀ m n : ℤ, ∃ x, (integerPairComplex (z x)).re=m ∧ | |
| (integerPairComplex (z x)).im=n) ∧ | |
| (∀ y, ∃ psi : G.Aut, psi.vertex o∈G.vertexBall y c ∧ | |
| ∀ x y, integerPairComplex (z (psi.vertex x))-integerPairComplex (z (psi.vertex y))= | |
| integerPairComplex (z x)-integerPairComplex (z y)) := by | |
| let pi : V → ℂ := fun v => integerPairComplex (z v) | |
| have hpi : ∀ g v, pi (g • v)=integerPairComplex (a g)+pi v := by | |
| intro g v | |
| exact (congrArg integerPairComplex (hz g v)).trans (integerPairComplex_add _ _) | |
| obtain ⟨o,ho⟩ := honto 0 | |
| obtain ⟨c,hc'⟩ := G.action_orbit_dense (fun g => G.smulAut ha g) (fun _ _ => rfl) | |
| hc S hS hcover o | |
| obtain ⟨D,hD,hj⟩ := G.bounded_coordinate_jumps hl ha S hS hcover | |
| (fun g => integerPairComplex (a g)) pi hpi | |
| refine ⟨o,c,D,hD,?_,?_,?_,?_⟩ | |
| · simp [ho,integerPairComplex] | |
| · intro x y hxy | |
| obtain ⟨⟨e,_,he⟩,_⟩ := (SimpleGraph.fromEdgeSet_adj _).mp hxy | |
| exact hj e x y he | |
| · intro m n | |
| obtain ⟨x,hx⟩ := honto (m,n) | |
| exact ⟨x,by rw [hx,integerPairComplex_re],by rw [hx,integerPairComplex_im]⟩ | |
| · intro y | |
| obtain ⟨g,hg⟩ := hc' y | |
| refine ⟨G.smulAut ha g,hg,?_⟩ | |
| intro x y | |
| change pi (g • x)-pi (g • y)=pi x-pi y | |
| rw [hpi,hpi] | |
| abel | |
| theorem critical_null_of_action_coordinates | |
| (hc : G.fullGraph.Connected) (hl : G.LocallyFinite) | |
| (ha : G.PreservesEnds (Λ := Λ)) | |
| (S : Set V) (hS : S.Finite) (hcover : ∀ v, ∃ s∈S, v∈MulAction.orbit Λ s) | |
| (hg : G.SubexponentialGrowth) (hpc : G.criticalProbability<1) | |
| (a : Λ → ℤ×ℤ) (z : V → ℤ×ℤ) (honto : Surjective z) | |
| (hz : ∀ g v, z (g • v)=a g+z v) | |
| (hnoray : ∀ᵐ ω ∂G.law G.criticalProbability, | |
| ¬∃ r : G.OpenRay ω, ∃ U W : ℝ, ∀ n, | |
| |(z (r.vertex n)).1|≤U ∧ |(z (r.vertex n)).2|≤W) : | |
| G.law G.criticalProbability G.percolates=0 := by | |
| obtain ⟨o,c,D,hD,ho,hjump,honto',hdense⟩ := | |
| G.integer_pair_coordinate_geometry hc hl ha S hS hcover a z honto hz | |
| apply G.critical_null_of_unbounded_projection hc hl | |
| (G.quasiTransitive_of_action ha S hS hcover) hg hpc | |
| (fun v => integerPairComplex (z v)) o c D hD ho hjump honto' hdense | |
| simpa only [integerPairComplex_re,integerPairComplex_im,Int.cast_abs] using hnoray | |
| end CriticalPercolation.BondGraph | |
| end | |
| end | |
| end | |
| end OAI | |