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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.Growth.HomogeneousGeometry
import OAI.Probability.CriticalPercolation.GraphGeometry.FiniteActionAverage
import OAI.Probability.CriticalPercolation.Growth.HomogeneousTransport
namespace OAI
section
noncomputable section
open Set Function
open scoped Classical BigOperators
namespace CriticalPercolation.HomogeneousAnalysis.Geometry
open ActionOrbitTransport
variable {X : Type*} (W : Geometry X)
abbrev Stab := MulAction.stabilizer W.Aut W.root
def frame (x : X) : W.Aut := (W.transitive W.root x).choose
lemma frame_root (x : X) : W.frame x W.root=x := (W.transitive W.root x).choose_spec
lemma frame_smul_root (x : X) : W.frame x • W.root=x := W.frame_root x
lemma action_transitive : ∀ x y : X, ∃ g : W.Aut, g • x=y := W.transitive
lemma suborbit_finite (x y : X) : (stabilizerOrbit W.Aut X x y).Finite := by
obtain ⟨p,_,_⟩ := W.connected.exists_path_of_dist x y
apply (W.around x p.length).finite_toSet.subset
intro z hz
obtain ⟨g,hgx,rfl⟩ := (mem_stabilizerOrbit W.Aut X).mp hz
change g x=x at hgx
change g y∈W.around x p.length
apply W.mem_around.mpr
simpa only [hgx] using GraphTopology.iso_reachBall W.graph g (show y∈GraphTopology.reachBall W.graph x p.length from ⟨p,le_rfl⟩)
lemma suborbit_card_pos (x y : X) : 0<suborbitCard W.Aut X x y := by
apply Set.ncard_pos (W.suborbit_finite x y) |>.mpr
exact ⟨y,(mem_stabilizerOrbit W.Aut X).mpr ⟨1,one_smul _ _,one_smul _ _⟩⟩
lemma stab_ball_preserved (N : ℕ) (k : W.Stab) (x : X) (hx : x∈W.ball N) : k • x∈W.ball N := by
have hk : k.val W.root=W.root := k.property
change k.val x∈W.around W.root N
rw [←hk,←W.aut_around_image]
exact Finset.mem_image.mpr ⟨x,hx,rfl⟩
lemma row_image (b x : X) :
(W.suborbit_finite W.root b).toFinset.image (W.frame x)=
(pair_row_finite W.action_transitive W.suborbit_finite W.root b x).toFinset := by
ext y
simp only [Finset.mem_image,Set.Finite.mem_toFinset]
constructor
· rintro ⟨z,hz,rfl⟩
obtain ⟨k,hk,hz⟩ := (mem_stabilizerOrbit W.Aut X).mp hz
apply (pairOrbit_eq W.Aut X).mpr
refine ⟨(W.frame x*k)⁻¹,?_,?_⟩
· have he : (W.frame x*k) • W.root=x := by
rw [mul_smul,hk]; exact W.frame_root x
calc
_ = (W.frame x*k)⁻¹ • ((W.frame x*k) • W.root) := congrArg _ he.symm
_ = _ := inv_smul_smul _ _
· have he : (W.frame x*k) • b=W.frame x z := by rw [mul_smul,hz]; rfl
rw [←he,inv_smul_smul]
· intro hy
obtain ⟨g,hx,hb⟩ := (pairOrbit_eq W.Aut X).mp hy
refine ⟨(W.frame x)⁻¹ • y,?_,smul_inv_smul (W.frame x) y⟩
apply (mem_stabilizerOrbit W.Aut X).mpr
refine ⟨(g*W.frame x)⁻¹,?_,?_⟩
· have he : (g*W.frame x) • W.root=W.root := by rw [mul_smul,W.frame_smul_root,hx]
calc
_ = (g*W.frame x)⁻¹ • ((g*W.frame x) • W.root) := congrArg _ he.symm
_ = _ := inv_smul_smul _ _
· rw [mul_inv_rev,mul_smul,←hb,inv_smul_smul]
lemma average_row {H : Type*} [Group H] [Fintype H]
(N : ℕ) (r : H → W.Stab)
(hm : ∀ a b x, x∈W.ball N → r (a*b) • x=r a • (r b • x))
(hs : ∀ k : W.Stab, ∃ h : H, ∀ x∈W.ball N, r h • x=k • x)
(b : X) (hb : b∈W.ball N) (x : X) (f : X → ℝ) :
(suborbitCard W.Aut X W.root b : ℝ)*(∑ h : H, f (W.frame x (r h • b)))=
(Fintype.card H : ℝ)*∑ y∈(pair_row_finite W.action_transitive W.suborbit_finite W.root b x).toFinset, f y := by
have hh := restricted_orbit_average (G:=W.Stab) (W.ball N : Set X) r hm hs b hb
(W.suborbit_finite W.root b) (fun z => f (W.frame x z))
change (suborbitCard W.Aut X W.root b : ℝ)*_= _ at hh
rw [←W.row_image,Finset.sum_image]
· exact hh
· intro a _ b _ he; exact (W.frame x).injective he
lemma average_sum_bound
(hu : ∀ x y, suborbitCard W.Aut X x y=suborbitCard W.Aut X y x)
{H : Type*} [Group H] [Fintype H] (N : ℕ) (r : H → W.Stab)
(hm : ∀ a b x, x∈W.ball N → r (a*b) • x=r a • (r b • x))
(hs : ∀ k : W.Stab, ∃ h : H, ∀ x∈W.ball N, r h • x=k • x)
(b : X) (hb : b∈W.ball N) (R : ℕ) (f : X → ℝ) (hf : ∀ x, 0≤f x) :
(∑ x∈W.ball R, ∑ h : H, f (W.frame x (r h • b)))≤
(Fintype.card H : ℝ)*∑ y∈W.ball (R+N), f y := by
have hrow := finite_row_sum_bound W.action_transitive W.suborbit_finite hu W.root b
(W.ball R) (W.ball (R+N)) f (fun y _ => hf y) (by
intro x hx y hy
obtain ⟨g,hgx,hgy⟩ := (pairOrbit_eq W.Aut X).mp hy
change g x=W.root at hgx
change g y=b at hgy
have hyx : y∈W.around x N := by
have hbx : g y∈W.around (g x) N := by rw [hgx,hgy]; exact hb
rw [←W.aut_around_image] at hbx
obtain ⟨z,hz,hze⟩ := Finset.mem_image.mp hbx
exact (g.injective hze) ▸ hz
exact W.mem_around_add hx hyx)
have he : (suborbitCard W.Aut X W.root b : ℝ)*
(∑ x∈W.ball R, ∑ h : H, f (W.frame x (r h • b)))=
(Fintype.card H : ℝ)*∑ x∈W.ball R,
∑ y∈(pair_row_finite W.action_transitive W.suborbit_finite W.root b x).toFinset, f y := by
simp only [Finset.mul_sum]
apply Finset.sum_congr rfl
intro x _
simpa only [Finset.mul_sum] using W.average_row N r hm hs b hb x f
have hi := mul_le_mul_of_nonneg_left hrow (Nat.cast_nonneg (Fintype.card H))
rw [←he] at hi
have hp : (0:ℝ)<suborbitCard W.Aut X W.root b := by exact_mod_cast W.suborbit_card_pos W.root b
nlinarith
end CriticalPercolation.HomogeneousAnalysis.Geometry
end
end
end OAI
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