| 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 |
|
|
