File size: 4,647 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 | import Mathlib
import OAI.Probability.CriticalPercolation.GraphGeometry.EllipseGrid
import OAI.Probability.CriticalPercolation.GraphGeometry.EllipseAngles
import OAI.Probability.CriticalPercolation.GraphGeometry.EllipseLinear
namespace OAI
section
section
noncomputable section
open Set Function
namespace CriticalPercolation.EuclideanEllipse
theorem separated_neighboring_ellipses (W U zeta : ℝ)
(hW : 0<W) (hWU : W≤U) (hzeta : 0<zeta)
(chooseCenter : (ℂ → ℂ) → Circle)
(hboundary : ∀ phi : ℝ,
|((Circle.exp phi : ℂ)⁻¹*(chooseCenter (normalize U W (Circle.exp phi)) : ℂ)).re|<1/8) :
∃ p∈parameterDisk, ∃ q∈parameterDisk,
(∀ z : ℂ, ‖diskNormalize W U p (diskExpand W U q z)-z‖≤zeta*‖z‖) ∧
(1 : ℝ)/4≤|EllipseLinear.det
(chooseCenter (diskNormalize W U p) : ℂ).re (chooseCenter (diskNormalize W U p) : ℂ).im
(chooseCenter (diskNormalize W U q) : ℂ).re (chooseCenter (diskNormalize W U q) : ℂ).im| := by
obtain ⟨N,K,hN,hK,hangle,hcomp⟩ := exists_grid_comparison W U zeta hW hWU hzeta
have hNr : (N : ℝ)≠0 := (Nat.cast_pos.mpr hN).ne'
have hKr : (K : ℝ)≠0 := (Nat.cast_pos.mpr hK).ne'
let phi : ℕ → ℝ := fun k => (k : ℝ)*Real.pi/K
let centers : ℕ → ℕ → Circle := fun i k => chooseCenter (diskNormalize W U (gridPoint N K i k))
let F : ℕ → ℕ → Real.Angle := fun i k => lineAngle (centers i k)
have hp0 : phi 0=0 := by simp [phi]
have hpK : phi K=Real.pi := by dsimp [phi]; field_simp
have hn : ∀ k, diskNormalize W U (gridPoint N K N k)=normalize U W (Circle.exp (phi k)) := by
intro k
simp only [diskNormalize,gridPoint,div_self hNr,diskRadius,mul_one,add_sub_cancel,phi]
have hc : ∀ k≤K, F 0 k=F 0 0 := by
intro k _hk
dsimp [F,centers,gridPoint]
simp only [Nat.cast_zero,zero_div,zero_mul]
rw [diskNormalize_center W U ((k : ℝ)*Real.pi/K)]
have hs : ∀ i≤N, F i K=F i 0 := by
intro i _hi
have he : gridPoint N K i K=((i : ℝ)/N,Real.pi) := by
simp only [gridPoint]
congr 1
have he0 : gridPoint N K i 0=((i : ℝ)/N,0) := by simp [gridPoint]
dsimp [F,centers]
rw [he,he0,diskNormalize_seam]
let err : ℕ → ℝ := fun k => boundaryError (phi k) (centers N k)
let b : ℕ → ℝ := fun k => 2*phi k+Real.pi+err k
have herr : ∀ k≤K, |err k|<Real.pi/8 := by
intro k _hk
apply boundary_error_lt
simpa only [centers,hn] using hboundary (phi k)
have hlift : ∀ k≤K, F N k=(b k : Real.Angle) := by
intro k _hk
exact boundary_lift (phi k) (centers N k)
have hmesh : ∀ k<K, |phi (k+1)-phi k|<Real.pi/32 := by
intro k _hk
have he := gridPoint_angular_step N K k hK
exact he.trans_lt hangle
have hstep := DiskWinding.boundary_mesh_lifts phi err K hmesh herr
have hturn : b K-b 0=2*Real.pi := by
have hcseam : centers N K=centers N 0 := by
dsimp [centers]
rw [hn,hn,hpK,hp0,circle_exp_pi]
congr 1
funext z
exact normalize_neg U W (Circle.exp 0) z
have heseam : err K=err 0 := by
dsimp [err]
rw [hpK,hp0,hcseam,boundary_error_seam]
dsimp [b]
rw [hpK,hp0,heseam]
ring
have hex := DiskWinding.exists_large_edge F N K b hc hs hlift hstep hturn
have edge : ∀ i≤N, ∀ k≤K, ∀ a≤1, ∀ b≤1, i+a≤N → k+b≤K →
Real.pi/4≤|DiskWinding.increment (F i k) (F (i+a) (k+b))| →
∃ p∈parameterDisk, ∃ q∈parameterDisk,
(∀ z : ℂ, ‖diskNormalize W U p (diskExpand W U q z)-z‖≤zeta*‖z‖) ∧
(1 : ℝ)/4≤|EllipseLinear.det
(chooseCenter (diskNormalize W U p) : ℂ).re (chooseCenter (diskNormalize W U p) : ℂ).im
(chooseCenter (diskNormalize W U q) : ℂ).re (chooseCenter (diskNormalize W U q) : ℂ).im| := by
intro i hi k hk a ha b hb hia hkb hsep
refine ⟨gridPoint N K i k,gridPoint_mem hN hK hi hk,
gridPoint N K (i+a) (k+b),gridPoint_mem hN hK hia hkb,hcomp i hi k hk a ha b hb hia hkb,?_⟩
have hd := EllipseLinear.det_lower_of_separation (centers i k : ℂ).arg (centers (i+a) (k+b) : ℂ).arg hsep
simpa only [circle_arg_re,circle_arg_im] using hd
rcases hex with ⟨i,hi,k,hk,hik⟩ | ⟨i,hi,k,hk,hik⟩ | ⟨i,hi,k,hk,hik⟩
· exact edge i hi k (by omega) 0 (by omega) 1 (by omega) (by omega) (by omega) (by simpa using hik)
· exact edge i (by omega) k hk 1 (by omega) 0 (by omega) (by omega) (by omega) (by simpa using hik)
· exact edge i (by omega) k (by omega) 1 (by omega) 1 (by omega) (by omega) (by omega) hik
end CriticalPercolation.EuclideanEllipse
end
end
end
end OAI
|