| import Mathlib |
| import OAI.Probability.CriticalPercolation.Modular.ActionOrbitTransport |
| import OAI.Probability.CriticalPercolation.Growth.HomogeneousGeometry |
| import OAI.Probability.CriticalPercolation.Harmonic.HarmonicFunctionSpace |
| import OAI.Probability.CriticalPercolation.Harmonic.CayleyPoincare |
| import OAI.Probability.CriticalPercolation.GraphGeometry.FiniteCoverSums |
| import OAI.Probability.CriticalPercolation.Growth.HomogeneousAverage |
| import OAI.Probability.CriticalPercolation.Harmonic.HomogeneousPoincare |
| import OAI.Probability.CriticalPercolation.Growth.HomogeneousBallPacking |
| import OAI.Probability.CriticalPercolation.Harmonic.HomogeneousCaccioppoli |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set Function |
| open scoped Classical BigOperators |
|
|
| namespace CriticalPercolation.HomogeneousAnalysis.Geometry |
| open HarmonicAnalysis ActionOrbitTransport |
|
|
| variable {A : Type*} (W : Geometry A) |
|
|
| lemma squareMass_around (a : A) (r : ℕ) (u : A → ℝ) : |
| squareMass (W.around a r) u=squareMass (W.ball r) (fun x => u (W.frame a x)) := by |
| unfold squareMass |
| have him : W.around a r=(W.ball r).image (W.frame a) := by |
| rw [W.aut_around_image,W.frame_root] |
| rw [him] |
| rw [Finset.sum_image] |
| intro x _ y _ h |
| exact (W.frame a).injective h |
|
|
| lemma sum_around (a : A) (r : ℕ) (u : A → ℝ) : |
| (∑ x∈W.around a r, u x)=∑ y∈W.ball r, u (W.frame a y) := by |
| have him : W.around a r=(W.ball r).image (W.frame a) := by |
| rw [W.aut_around_image,W.frame_root] |
| rw [him] |
| rw [Finset.sum_image] |
| intro x _ y _ h |
| exact (W.frame a).injective h |
|
|
| lemma energy_around (a : A) (r : ℕ) (u : A → ℝ) : |
| W.energy (W.around a r) u=W.energy (W.ball r) (fun x => u (W.frame a x)) := by |
| unfold energy |
| have him : W.around a r=(W.ball r).image (W.frame a) := by |
| rw [W.aut_around_image,W.frame_root] |
| rw [him,Finset.sum_image] |
| · apply Finset.sum_congr rfl |
| intro x hx |
| rw [←W.neighbors_image,Finset.sum_image] |
| intro x _ y _ h; exact (W.frame a).injective h |
| · intro x _ y _ h; exact (W.frame a).injective h |
| |
| variable (hun : ∀ x y, suborbitCard W.Aut A x y=suborbitCard W.Aut A y x) |
| include hun |
|
|
| lemma translated_poincare (a : A) (r K : ℕ) (u : A → ℝ) |
| (hvol : (W.ball (7*r)).card≤K*(W.ball r).card) |
| (hu : (∑ x∈W.around a (2*r), u x)=0) : |
| squareMass (W.around a (2*r)) u ≤ |
| 8*(r:ℝ)^2*(K:ℝ)*W.energy (W.around a (6*r)) u := by |
| rw [W.sum_around] at hu |
| have hp := W.ball_poincare hun |
| (fun x => u (W.frame a x)) (2*r) hu |
| change ((W.ball (2*r)).card:ℝ)*squareMass (W.ball (2*r)) (fun x => u (W.frame a x)) ≤ _ at hp |
| rw [←W.squareMass_around] at hp |
| have he : W.energy (W.ball (3*(2*r))) (fun x => u (W.frame a x)) = |
| W.energy (W.around a (6*r)) u := by |
| rw [W.energy_around a (6*r) u] |
| rw [show 3*(2*r)=6*r by omega] |
| rw [he] at hp |
| have hv : (W.ball (2*(2*r))).card≤K*(W.ball (2*r)).card := by |
| calc |
| _ ≤ (W.ball (7*r)).card := Finset.card_le_card (W.ball_mono (by omega)) |
| _ ≤ _ := hvol.trans (Nat.mul_le_mul_left _ (Finset.card_le_card (W.ball_mono (by omega)))) |
| have hv' : ((W.ball (2*(2*r))).card:ℝ)≤(K:ℝ)*((W.ball (2*r)).card:ℝ) := by exact_mod_cast hv |
| have hbpos : (0:ℝ)<(W.ball (2*r)).card := by exact_mod_cast W.ball_card_pos (2*r) |
| have hepos := W.energy_nonneg (W.around a (6*r)) u |
| have hm := mul_le_mul_of_nonneg_right hv' (mul_nonneg (sq_nonneg (r:ℝ)) hepos) |
| push_cast at hp |
| nlinarith |
|
|
| lemma cover_mass_energy (R r K : ℕ) (J : Finset A) |
| (hJR : J⊆W.ball R) |
| (hJ : (J:Set A).PairwiseDisjoint (fun a => W.around a r)) |
| (hcover : ∀ x∈W.ball R, ∃ a∈J, x∈W.around a (2*r)) |
| (hvol : (W.ball (7*r)).card≤K*(W.ball r).card) |
| (u : A → ℝ) (hu : ∀ a∈J, (∑ x∈W.around a (2*r), u x)=0) : |
| squareMass (W.ball R) u ≤ |
| 8*(r:ℝ)^2*(K:ℝ)^2*W.energy (W.ball (R+6*r)) u := by |
| have hcover' := sum_le_sum_of_finite_cover J (fun a => W.around a (2*r)) |
| (W.ball R) hcover (fun x => (u x)^2) (fun x => sq_nonneg _) |
| have hpsum := Finset.sum_le_sum (fun a (ha : a∈J) => W.translated_poincare hun a r K u hvol (hu a ha)) |
| have hsub : ∀ a∈J, W.around a (6*r)⊆W.ball (R+6*r) := |
| fun a ha => W.around_subset_ball (hJR ha) |
| have hover : ∀ x∈W.ball (R+6*r), (J.filter (fun a => x∈W.around a (6*r))).card≤K := by |
| intro x _ |
| apply W.packing_overlap J r (6*r) K hJ |
| simpa only [show 6*r+r=7*r by omega] using hvol |
| have he := sum_le_sum_of_finite_overlap J (fun a => W.around a (6*r)) |
| (W.ball (R+6*r)) hsub K hover |
| (fun x => ∑ y∈W.neighbors x, (u y-u x)^2) |
| (fun x => Finset.sum_nonneg (fun s _ => sq_nonneg _)) |
| change (∑ a∈J, W.energy (W.around a (6*r)) u)≤ |
| (K:ℝ)*W.energy (W.ball (R+6*r)) u at he |
| simp only [←Finset.mul_sum] at hpsum |
| change squareMass (W.ball R) u≤∑ a∈J, squareMass (W.around a (2*r)) u at hcover' |
| have hm := mul_le_mul_of_nonneg_left he (by positivity : 0≤8*(r:ℝ)^2*(K:ℝ)) |
| calc |
| _ ≤ _ := hcover'.trans hpsum |
| _ ≤ _ := by nlinarith |
|
|
| theorem harmonic_cover_kernel_bound (R r K : ℕ) (hR : 1≤R) (hrR : 6*r≤R) |
| (J : Finset A) (hJR : J⊆W.ball R) |
| (hJ : (J:Set A).PairwiseDisjoint (fun a => W.around a r)) |
| (hcover : ∀ x∈W.ball R, ∃ a∈J, x∈W.around a (2*r)) |
| (hvol : (W.ball (7*r)).card≤K*(W.ball r).card) |
| (u : A → ℝ) (hharm : W.IsHarmonic u) |
| (hu : ∀ a∈J, (∑ x∈W.around a (2*r), u x)=0) : |
| (R:ℝ)^2*squareMass (W.ball R) u ≤ |
| 8*(K:ℝ)^2*(W.degree:ℝ)*(r:ℝ)^2*squareMass (W.ball (5*R)) u := by |
| have hp := W.cover_mass_energy hun R r K J hJR hJ hcover hvol u hu |
| have hem := W.energy_mono (W.ball_mono (show R+6*r≤2*R by omega)) u |
| have hm := mul_le_mul_of_nonneg_left hem (by positivity : 0≤8*(r:ℝ)^2*(K:ℝ)^2) |
| have hp' : squareMass (W.ball R) u≤8*(r:ℝ)^2*(K:ℝ)^2*W.energy (W.ball (2*R)) u := hp.trans hm |
| have hc := W.caccioppoli u hharm (2*R) |
| have hqm := squareMass_mono (W.ball_mono (show 2*(2*R)+1≤5*R by omega)) u |
| have hq := mul_le_mul_of_nonneg_left hqm (by positivity : 0≤4*(W.degree:ℝ)) |
| have hc' : (R:ℝ)^2*W.energy (W.ball (2*R)) u≤ |
| (W.degree:ℝ)*squareMass (W.ball (5*R)) u := by |
| push_cast at hc |
| nlinarith |
| have hm' := mul_le_mul_of_nonneg_left hp' (sq_nonneg (R:ℝ)) |
| have hm'' := mul_le_mul_of_nonneg_left hc' (by positivity : 0≤8*(r:ℝ)^2*(K:ℝ)^2) |
| nlinarith |
|
|
| end CriticalPercolation.HomogeneousAnalysis.Geometry |
| end |
| end |
|
|
| end OAI |
|
|
