| import Mathlib |
| import OAI.Probability.CriticalPercolation.Gluing.CorridorNormalization |
| import OAI.Probability.CriticalPercolation.Gluing.MoveWitness |
| import OAI.Probability.CriticalPercolation.GraphGeometry.EllipseExit |
| import OAI.Probability.CriticalPercolation.GraphGeometry.EllipseLinear |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open Set Function MeasureTheory ProbabilityTheory |
| open scoped Classical ENNReal |
|
|
| namespace CriticalPercolation.BondGraph |
| variable {V E : Type*} (G : BondGraph V E) |
|
|
| def ellipseTime (Psi : V → ℂ) (c d : Circle) (x : V) : ℝ := |
| EllipseLinear.time (c : ℂ).re (c : ℂ).im (d : ℂ).re (d : ℂ).im (Psi x).re (Psi x).im |
|
|
| def ellipseHeight (Psi : V → ℂ) (c d : Circle) (x : V) : ℝ := |
| EllipseLinear.height (c : ℂ).re (c : ℂ).im (d : ℂ).re (d : ℂ).im (Psi x).re (Psi x).im |
|
|
| lemma circle_component_bounds (c : Circle) : |(c : ℂ).re|≤1 ∧ |(c : ℂ).im|≤1 := by |
| constructor |
| · exact (Complex.abs_re_le_norm _).trans_eq (Circle.norm_coe c) |
| · exact (Complex.abs_im_le_norm _).trans_eq (Circle.norm_coe c) |
|
|
| lemma ellipse_coordinates_norm_bound (Psi : V → ℂ) (c d : Circle) |
| (hdet : (1 : ℝ)/4≤|EllipseLinear.det (c : ℂ).re (c : ℂ).im (d : ℂ).re (d : ℂ).im|) |
| (x : V) : |ellipseTime Psi c d x|≤16*‖Psi x‖ ∧ |ellipseHeight Psi c d x|≤16*‖Psi x‖ := by |
| exact EllipseLinear.row_bound (circle_component_bounds c).1 (circle_component_bounds c).2 |
| (circle_component_bounds d).1 (circle_component_bounds d).2 hdet (Psi x).re (Psi x).im |
| ‖Psi x‖ (norm_nonneg _) (Complex.abs_re_le_norm _) (Complex.abs_im_le_norm _) |
|
|
| lemma ellipse_coordinates_sub (Psi : V → ℂ) (c d : Circle) (x y : V) : |
| ellipseTime Psi c d x-ellipseTime Psi c d y= |
| ellipseTime (fun v => Psi v-Psi y) c d x ∧ |
| ellipseHeight Psi c d x-ellipseHeight Psi c d y= |
| ellipseHeight (fun v => Psi v-Psi y) c d x := by |
| constructor |
| · exact EllipseLinear.time_sub _ _ _ _ _ _ _ _ |
| · exact EllipseLinear.height_sub _ _ _ _ _ _ _ _ |
|
|
| lemma ellipse_coordinates_step (Psi : V → ℂ) (c d : Circle) |
| (hdet : (1 : ℝ)/4≤|EllipseLinear.det (c : ℂ).re (c : ℂ).im (d : ℂ).re (d : ℂ).im|) |
| (D : ℝ) (hPsi : ∀ ⦃x y⦄, G.fullGraph.Adj x y → ‖Psi x-Psi y‖≤D) : |
| (∀ ⦃x y⦄, G.fullGraph.Adj x y → |ellipseTime Psi c d x-ellipseTime Psi c d y|≤16*D) ∧ |
| (∀ ⦃x y⦄, G.fullGraph.Adj x y → |ellipseHeight Psi c d x-ellipseHeight Psi c d y|≤16*D) := by |
| constructor |
| · intro x y hxy |
| rw [(ellipse_coordinates_sub Psi c d x y).1] |
| exact (ellipse_coordinates_norm_bound (fun v => Psi v-Psi y) c d hdet x).1.trans (by nlinarith [hPsi hxy]) |
| · intro x y hxy |
| rw [(ellipse_coordinates_sub Psi c d x y).2] |
| exact (ellipse_coordinates_norm_bound (fun v => Psi v-Psi y) c d hdet x).2.trans (by nlinarith [hPsi hxy]) |
|
|
| theorem ellipseExit_boxMove (Psi Psi' : V → ℂ) (o : V) (m : ℕ) (c d : Circle) |
| (hdet : (1 : ℝ)/4≤|EllipseLinear.det (c : ℂ).re (c : ℂ).im (d : ℂ).re (d : ℂ).im|) |
| (eps : ℝ) (heps : 0<eps) (heps' : eps<1/10) (ho : Psi o=0) |
| (hpert : ∀ x, ‖Psi x-Psi' x‖≤(eps/1000)*‖Psi' x‖) (b : Bool) : |
| G.ellipseExit Psi' (G.vertexBall o m) (eps/1000) (if b then c else d) ⊆ |
| G.boxMove (ellipseTime Psi c d) (ellipseHeight Psi c d) m (eps/8) 30 o (Corridor.sign b) := by |
| intro ω hω |
| obtain ⟨a,ha,x,w,hw,hx⟩ := (G.ellipseExit_iff _ _ _ _ ω).mp hω |
| have hzeta : 0<eps/1000 := by positivity |
| have hzeta' : eps/1000<1/10000 := by linarith |
| have hdet' : EllipseLinear.det (c : ℂ).re (c : ℂ).im (d : ℂ).re (d : ℂ).im≠0 := by |
| intro he |
| rw [he,abs_zero] at hdet |
| norm_num at hdet |
| have hto : ellipseTime Psi c d o=0 := by simp [ellipseTime,ho,EllipseLinear.time] |
| have hzo : ellipseHeight Psi c d o=0 := by simp [ellipseHeight,ho,EllipseLinear.height] |
| have hpath : ∀ y∈w.support, ‖Psi y‖≤(1+eps/1000)^2 := by |
| intro y hy |
| have htri : ‖Psi y‖≤‖Psi y-Psi' y‖+‖Psi' y‖ := by |
| simpa only [sub_add_cancel] using norm_add_le (Psi y-Psi' y) (Psi' y) |
| nlinarith [hpert y,hw y hy] |
| have hterminal : ‖Psi x-((if b then c else d) : Circle)‖≤4*(eps/1000) := by |
| have htri : ‖Psi x-((if b then c else d) : Circle)‖≤ |
| ‖Psi x-Psi' x‖+‖Psi' x-((if b then c else d) : Circle)‖ := by |
| simpa only [sub_add_sub_cancel] using norm_add_le (Psi x-Psi' x) |
| (Psi' x-((if b then c else d) : Circle)) |
| nlinarith [hpert x,hw x w.end_mem_support,mul_pos hzeta (by linarith : 0<1-eps/1000)] |
| have hcenter : EllipseLinear.time (c : ℂ).re (c : ℂ).im (d : ℂ).re (d : ℂ).im |
| (((if b then c else d) : Circle) : ℂ).re (((if b then c else d) : Circle) : ℂ).im=1 ∧ |
| EllipseLinear.height (c : ℂ).re (c : ℂ).im (d : ℂ).re (d : ℂ).im |
| (((if b then c else d) : Circle) : ℂ).re (((if b then c else d) : Circle) : ℂ).im=Corridor.sign b := by |
| cases b |
| · exact EllipseLinear.center_right _ _ _ _ hdet' |
| · exact EllipseLinear.center_left _ _ _ _ hdet' |
| have hend := ellipse_coordinates_norm_bound (fun v => Psi v-((if b then c else d) : Circle)) c d hdet x |
| have htend : |ellipseTime Psi c d x-1|≤eps/8 := by |
| rw [←hcenter.1,ellipseTime,EllipseLinear.time_sub] |
| exact hend.1.trans ((mul_le_mul_of_nonneg_left hterminal (by norm_num)).trans |
| (EllipseLinear.source_budgets heps heps').1.le) |
| have hzend : |ellipseHeight Psi c d x-Corridor.sign b|≤eps/8 := by |
| rw [←hcenter.2,ellipseHeight,EllipseLinear.height_sub] |
| exact hend.2.trans ((mul_le_mul_of_nonneg_left hterminal (by norm_num)).trans |
| (EllipseLinear.source_budgets heps heps').1.le) |
| refine ⟨a,ha,x,w,?_,?_,?_⟩ |
| · intro y hy |
| rw [hto,hzo,sub_zero,sub_zero] |
| have hnorm := ellipse_coordinates_norm_bound Psi c d hdet y |
| have hbudget : 16*‖Psi y‖≤30 := (mul_le_mul_of_nonneg_left (hpath y hy) (by norm_num)).trans |
| (EllipseLinear.source_budgets heps heps').2.le |
| exact ⟨hnorm.1.trans hbudget,hnorm.2.trans hbudget⟩ |
| · simpa only [hto,zero_add] using htend |
| · simpa only [hzo,zero_add] using hzend |
|
|
| end CriticalPercolation.BondGraph |
| end |
| end |
| end |
|
|
| end OAI |
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|
