| import OAI.Probability.FirstPassage.CrossingChain |
| import OAI.Probability.FirstPassage.RareFastGeometry |
|
|
| namespace OAI |
|
|
| namespace PlanarFPP |
| open Filter |
| open scoped Topology |
|
|
| theorem crossing_endpoints_escape_linear (b σ S : ℝ) (n d : ℤ) |
| (hb : |b| ≤ 1) (hσ : |σ|=1) (hS : 0 < S) (hn : 0 < n) (hd : 0 ≤ d) |
| (v z : ℕ → Vertex) (hv0 : v 0=0) |
| (hstep : ∀ t, v (t+1)=v t+(1,0) ∨ v (t+1)=v t+(0,1)) |
| (hx : ∀ t, (z t).1=blockX n d (v t)+n) |
| (hr : ∀ t, |transverse b (z t)-blockY σ S (v t)| ≤ S/4) : |
| Tendsto (fun t => ‖latticeEmbed (z t)‖) atTop atTop ∧ |
| ∀ t, ‖latticeEmbed (z t)‖ ≤ (2*((n : ℝ)+d)+3*S)*((t : ℝ)+1) := by |
| have hn' : (1 : ℝ) ≤ n := by exact_mod_cast hn |
| have hd' : (0 : ℝ) ≤ d := by exact_mod_cast hd |
| have hcoords (t : ℕ) := oriented_path_coordinates v hv0 hstep t |
| have hxt (t : ℕ) : ((z t).1 : ℝ) = (t : ℝ)*((n : ℝ)+d)+n := by |
| rw [hx] |
| simp only [blockX,blockLayer,Int.cast_add,Int.cast_mul] |
| have hh := congrArg (fun a : ℤ => (a : ℝ)) (hcoords t).1 |
| push_cast at hh |
| rw [hh] |
| have hxpos (t : ℕ) : (t : ℝ)+1 ≤ ((z t).1 : ℝ) := by |
| rw [hxt] |
| nlinarith [Nat.cast_nonneg (α := ℝ) t] |
| have hnlow (t : ℕ) : (t : ℝ) ≤ ‖latticeEmbed (z t)‖ := by |
| have hh := plane_coordinate_le_norm (latticeEmbed (z t)) 0 |
| change |((z t).1 : ℝ)| ≤ _ at hh |
| have hp : (0 : ℝ) ≤ (z t).1 := by linarith [Nat.cast_nonneg (α := ℝ) t,hxpos t] |
| rw [abs_of_nonneg hp] at hh |
| linarith [hxpos t] |
| refine ⟨tendsto_atTop_mono hnlow tendsto_natCast_atTop_atTop,?_⟩ |
| intro t |
| have hlevel : (0 : ℝ) ≤ blockLevel (v t) ∧ (blockLevel (v t) : ℝ) ≤ 2*t := by |
| have hh := hcoords t |
| constructor <;> dsimp [blockLevel] <;> exact_mod_cast (by omega : _) |
| have hy : |blockY σ S (v t)| ≤ 2*(t : ℝ)*S := by |
| unfold blockY |
| rw [abs_mul,abs_mul,hσ,one_mul,abs_of_nonneg hlevel.1,abs_of_pos hS] |
| exact mul_le_mul_of_nonneg_right hlevel.2 hS.le |
| have hrbound : |transverse b (z t)| ≤ 2*(t : ℝ)*S+S/4 := by |
| have hh := abs_add_le (transverse b (z t)-blockY σ S (v t)) (blockY σ S (v t)) |
| rw [sub_add_cancel] at hh |
| linarith [hr t] |
| have hpos : (0 : ℝ) ≤ (z t).1 := by linarith [hxpos t,Nat.cast_nonneg (α := ℝ) t] |
| have hyy : |((z t).2 : ℝ)| ≤ |transverse b (z t)|+((z t).1 : ℝ) := by |
| have hh := abs_add_le (transverse b (z t)) (b*((z t).1 : ℝ)) |
| have he : transverse b (z t)+b*((z t).1 : ℝ)=((z t).2 : ℝ) := by unfold transverse; ring |
| rw [he,abs_mul,abs_of_nonneg hpos] at hh |
| have hm := mul_le_mul_of_nonneg_right hb hpos |
| linarith |
| have hnorm := plane_norm_le_coordinates (latticeEmbed (z t)) |
| change ‖latticeEmbed (z t)‖ ≤ |((z t).1 : ℝ)|+|((z t).2 : ℝ)| at hnorm |
| rw [abs_of_nonneg hpos] at hnorm |
| have ht : (0 : ℝ) ≤ t := Nat.cast_nonneg _ |
| rw [hxt] at hyy hnorm |
| nlinarith |
|
|
| end PlanarFPP |
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| end OAI |
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