openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Harmonic /GraphHarmonicLinear.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Growth.HomogeneousGeometry | |
| import OAI.Probability.CriticalPercolation.Growth.HomogeneousDimension | |
| import OAI.Probability.CriticalPercolation.Growth.HomogeneousPolynomialSpace | |
| import OAI.Probability.CriticalPercolation.GraphGeometry.FiniteActionAverage | |
| import OAI.Probability.CriticalPercolation.Harmonic.GraphHarmonicCompactness | |
| import OAI.Probability.CriticalPercolation.Harmonic.HarmonicFunctionSpace | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function | |
| open scoped Classical BigOperators | |
| namespace CriticalPercolation.HomogeneousAnalysis.Geometry | |
| open HarmonicAnalysis | |
| variable {X : Type*} (W : Geometry X) | |
| lemma harmonic_constant_of_attained_max {u : X → ℝ} (hu : W.IsHarmonic u) | |
| (a : X) (hmax : ∀ x, u x≤u a) : ∀ x, u x=u a := by | |
| have hstep {x y : X} (hx : u x=u a) (hxy : W.graph.Adj x y) : u y=u a := by | |
| have hn (z : X) (hz : z∈W.neighbors x) : 0≤u x-u z := by rw [hx]; exact sub_nonneg.mpr (hmax z) | |
| have hz := (Finset.sum_eq_zero_iff_of_nonneg hn).mp (hu x) y (W.mem_neighbors.mpr hxy) | |
| linarith | |
| have hw : ∀ {x y : X}, W.graph.Walk x y → u x=u a → u y=u a := by | |
| intro x y p | |
| induction p with | |
| | nil => exact id | |
| | x y z h p ih => exact fun hx => ih (hstep hx h) | |
| intro x | |
| obtain ⟨p⟩ := W.connected a x | |
| exact hw p rfl | |
| lemma harmonic_constant_of_finite_range {u : X → ℝ} (hu : W.IsHarmonic u) | |
| (hf : (Set.range u).Finite) : ∀ x, u x=u W.root := by | |
| let : Nonempty X := ⟨W.root⟩ | |
| obtain ⟨m,⟨a,rfl⟩,hm⟩ := Set.exists_max_image (Set.range u) id hf (Set.range_nonempty u) | |
| have he := W.harmonic_constant_of_attained_max hu a (fun x => hm (u x) ⟨x,rfl⟩) | |
| exact fun x => (he x).trans (he W.root).symm | |
| def translateFunction (a : W.Aut) (u : X → ℝ) : X → ℝ := fun x => u (a.symm x) | |
| lemma translate_harmonic (a : W.Aut) {u : X → ℝ} (hu : W.IsHarmonic u) : | |
| W.IsHarmonic (W.translateFunction a u) := by | |
| intro x | |
| have h := hu (a.symm x) | |
| rw [←W.neighbors_image a.symm x,Finset.sum_image] at h | |
| · exact h | |
| · exact fun b hb c hc he => a.symm.injective he | |
| lemma translate_polynomial_growth (a : W.Aut) (k : ℕ) {u : X → ℝ} | |
| (hu : W.HasPolynomialGrowth k u) : W.HasPolynomialGrowth k (W.translateFunction a u) := by | |
| obtain ⟨C,hC,hu⟩ := (W.polynomial_growth_iff_length k u).mp hu | |
| apply (W.polynomial_growth_iff_length k _).mpr | |
| refine ⟨C*((W.length (a.symm W.root):ℝ)+1)^k,by positivity,?_⟩ | |
| intro x | |
| have hl := W.aut_length_le a.symm x | |
| have hl' : (W.length (a.symm x):ℝ)≤(W.length (a.symm W.root):ℝ)+(W.length x:ℝ) := by exact_mod_cast hl | |
| change |u (a.symm x)|≤_ | |
| calc | |
| _ ≤ C*((W.length (a.symm x):ℝ)+1)^k := hu _ | |
| _ ≤ C*(((W.length (a.symm W.root):ℝ)+1)*((W.length x:ℝ)+1))^k := by | |
| gcongr | |
| nlinarith [Nat.cast_nonneg (α:=ℝ) (W.length (a.symm W.root)),Nat.cast_nonneg (α:=ℝ) (W.length x)] | |
| _ = _ := by rw [mul_pow]; ring | |
| def harmonicTranslation (k : ℕ) (a : W.Aut) : | |
| W.polynomialHarmonicSpace k ≃ₗ[ℝ] W.polynomialHarmonicSpace k where | |
| toFun u := ⟨W.translateFunction a u,W.translate_harmonic a u.property.1, | |
| W.translate_polynomial_growth a k u.property.2⟩ | |
| invFun u := ⟨W.translateFunction a.symm u,W.translate_harmonic a.symm u.property.1, | |
| W.translate_polynomial_growth a.symm k u.property.2⟩ | |
| left_inv u := by apply Subtype.ext; funext x; simp [translateFunction] | |
| right_inv u := by apply Subtype.ext; funext x; simp [translateFunction] | |
| map_add' u v := rfl | |
| map_smul' c u := rfl | |
| @[simp] lemma harmonicTranslation_apply (k : ℕ) (a : W.Aut) (u : W.polynomialHarmonicSpace k) (x : X) : | |
| (W.harmonicTranslation k a u:X → ℝ) x=(u:X → ℝ) (a.symm x) := rfl | |
| def harmonicRepresentation (k : ℕ) : W.Aut →* | |
| (W.polynomialHarmonicSpace k ≃ₗ[ℝ] W.polynomialHarmonicSpace k) where | |
| toFun := W.harmonicTranslation k | |
| map_one' := by ext u x; rfl | |
| map_mul' a b := by ext u x; rfl | |
| lemma harmonicTranslation_eq_of_agree (k r : ℕ) | |
| (hdet : ∀ u : W.polynomialHarmonicSpace k, (∀ x∈W.ball r, (u:X → ℝ) x=0) → u=0) | |
| (a b : W.Aut) (hab : ∀ x∈W.ball r, a.symm x=b.symm x) : | |
| W.harmonicRepresentation k a=W.harmonicRepresentation k b := by | |
| apply LinearEquiv.ext | |
| intro u | |
| apply sub_eq_zero.mp | |
| apply hdet | |
| intro x hx | |
| change (u:X → ℝ) (a.symm x)-(u:X → ℝ) (b.symm x)=0 | |
| rw [hab x hx,sub_self] | |
| lemma finite_stabilizer_harmonic_image (d k : ℕ) (hg : W.HasDegree d) : | |
| (Set.range (fun a : W.Stab => W.harmonicRepresentation k a.val)).Finite := by | |
| let _ := W.polynomialHarmonicSpace_finiteDimensional d k hg | |
| obtain ⟨r,hr⟩ := exists_detecting_ball (fun r => (W.ball r:Set X)) W.ball_mono W.exists_mem_ball | |
| (W.polynomialHarmonicSpace k) | |
| obtain ⟨H,iH,fH,t,ht,hs⟩ := exists_finite_restriction (G:=W.Stab) (W.ball r:Set X) | |
| (W.ball r).finite_toSet (W.stab_ball_preserved r) | |
| let _ := iH | |
| let _ := fH | |
| apply (Set.finite_range (fun h : H => W.harmonicRepresentation k (t h).val⁻¹)).subset | |
| rintro _ ⟨a,rfl⟩ | |
| obtain ⟨h,hh⟩ := hs a⁻¹ | |
| refine ⟨h,?_⟩ | |
| apply W.harmonicTranslation_eq_of_agree k r hr | |
| intro x hx | |
| exact hh x hx | |
| variable {G : Type*} [Group G] (τ : G →* W.Aut) | |
| lemma finite_range_of_finite_harmonic_image (k : ℕ) | |
| (htrans : ∀ x, ∃ a : G, τ a W.root=x) | |
| (hf : (Set.range ((W.harmonicRepresentation k).comp τ)).Finite) (u : W.polynomialHarmonicSpace k) : | |
| (Set.range (u:X → ℝ)).Finite := by | |
| apply (hf.image (fun e => (e u:X → ℝ) W.root)).subset | |
| rintro y ⟨x,rfl⟩ | |
| obtain ⟨a,ha⟩ := htrans x | |
| refine ⟨(W.harmonicRepresentation k).comp τ a⁻¹,⟨a⁻¹,rfl⟩,?_⟩ | |
| change (u:X → ℝ) ((τ a⁻¹).symm W.root)=(u:X → ℝ) x | |
| rw [map_inv] | |
| change (u:X → ℝ) (τ a W.root)=(u:X → ℝ) x | |
| rw [ha] | |
| lemma infinite_harmonic_linear_image [Infinite X] (d : ℕ) (hg : W.HasDegree d) | |
| (htrans : ∀ x, ∃ a : G, τ a W.root=x) : | |
| (Set.range ((W.harmonicRepresentation 1).comp τ)).Infinite := by | |
| obtain ⟨u,hu,hu1,hbound,hedge,s,hs,hne⟩ := W.exists_nonconstant_lipschitz_harmonic d hg | |
| have hupol : W.HasPolynomialGrowth 1 u := by | |
| apply (W.polynomial_growth_iff_length 1 u).mpr | |
| refine ⟨1,by norm_num,?_⟩ | |
| intro x | |
| simpa only [pow_one,one_mul] using (hbound x).trans (by linarith : (W.length x:ℝ)≤(W.length x:ℝ)+1) | |
| let v : W.polynomialHarmonicSpace 1 := ⟨u,hu,hupol⟩ | |
| intro hf | |
| have hconst := W.harmonic_constant_of_finite_range hu | |
| (W.finite_range_of_finite_harmonic_image τ 1 htrans hf v) | |
| exact hne ((hconst s).trans hu1) | |
| end CriticalPercolation.HomogeneousAnalysis.Geometry | |
| end | |
| end | |
| end OAI | |