File size: 6,692 Bytes
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 | 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
| @cons 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
|