| import OAI.Probability.LatticePassage.CalibrationShift |
| import OAI.Probability.LatticePassage.SecondMoment |
|
|
| namespace OAI |
|
|
| namespace CompanionFPP |
| open MeasureTheory Real Finset Filter Topology |
|
|
| noncomputable def rectangleTargetCocycle (z a b : Vertex) (N M : ℕ) |
| (ω : Environment) (x y : Vertex) : ℝ := |
| (∑ i ∈ range N, ∑ j ∈ range M, |
| targetCocycle (z+scaleVertex i a+scaleVertex j b) ω x y) / ((N : ℝ)*M) |
|
|
| @[fun_prop] lemma measurable_rectangleTargetCocycle (z a b x y : Vertex) (N M : ℕ) : |
| Measurable (fun ω => rectangleTargetCocycle z a b N M ω x y) := by |
| unfold rectangleTargetCocycle |
| fun_prop |
|
|
| lemma integrable_rectangleTargetCocycle (z a b x y : Vertex) (N M : ℕ) : |
| Integrable (fun ω => rectangleTargetCocycle z a b N M ω x y) environmentLaw := |
| (integrable_finsetSum _ (fun _i _ => integrable_finsetSum _ |
| (fun _j _ => integrable_targetCocycle _ _ _))).div_const _ |
|
|
| theorem rectangleTargetCocycle_add (z a b : Vertex) (N M : ℕ) |
| (ω : Environment) (x y w : Vertex) : |
| rectangleTargetCocycle z a b N M ω x w = |
| rectangleTargetCocycle z a b N M ω x y + rectangleTargetCocycle z a b N M ω y w := by |
| unfold rectangleTargetCocycle |
| simp only [targetCocycle, sum_sub_distrib] |
| ring |
|
|
| lemma rectangleTargetCocycle_swap (z a b : Vertex) (N M : ℕ) |
| (ω : Environment) (x y : Vertex) : |
| rectangleTargetCocycle z a b N M ω x y = rectangleTargetCocycle z b a M N ω x y := by |
| unfold rectangleTargetCocycle |
| rw [sum_comm, mul_comm (N : ℝ)] |
| congr 1 |
| apply sum_congr rfl |
| intro j _ |
| apply sum_congr rfl |
| intro i _ |
| rw [show z+scaleVertex i a+scaleVertex j b = z+scaleVertex j b+scaleVertex i a by abel] |
|
|
| theorem rectangleTargetCocycle_dominated (z a b : Vertex) (N M : ℕ) |
| (hN : 1 ≤ N) (hM : 1 ≤ M) : |
| ∀ᵐ ω ∂environmentLaw, ∀ x y, |
| |rectangleTargetCocycle z a b N M ω x y| ≤ passageTime ω x y := by |
| filter_upwards [targetCocycle_dominated] with ω hω |
| intro x y |
| unfold rectangleTargetCocycle |
| rw [abs_div, abs_of_nonneg (by positivity : 0 ≤ (N : ℝ)*M)] |
| have hb : |∑ i ∈ range N, ∑ j ∈ range M, |
| targetCocycle (z+scaleVertex i a+scaleVertex j b) ω x y| ≤ |
| (N : ℝ)*M*passageTime ω x y := by |
| apply (abs_sum_le_sum_abs _ _).trans |
| calc |
| _ ≤ ∑ i ∈ range N, ∑ j ∈ range M, passageTime ω x y := by |
| apply sum_le_sum |
| intro i _ |
| exact (abs_sum_le_sum_abs _ _).trans (sum_le_sum (fun j _ => hω _ x y)) |
| _ = _ := by simp; ring |
| have hp : 0 < (N : ℝ)*M := mul_pos (by exact_mod_cast hN) (by exact_mod_cast hM) |
| exact (div_le_iff₀ hp).mpr (by simpa only [mul_comm (passageTime ω x y)] using hb) |
|
|
| theorem rectangleTargetCocycle_shift_identity (z a b : Vertex) (N M : ℕ) |
| (ω : Environment) (x y : Vertex) : |
| rectangleTargetCocycle z a b N M (environmentTranslation a ω) x y - |
| rectangleTargetCocycle z a b N M ω (a+x) (a+y) = |
| (∑ j ∈ range M, |
| (targetCocycle (z+scaleVertex N a+scaleVertex j b) ω (a+x) (a+y) - |
| targetCocycle (z+scaleVertex j b) ω (a+x) (a+y))) / ((N : ℝ)*M) := by |
| unfold rectangleTargetCocycle |
| rw [← sub_div, ← sum_sub_distrib] |
| have he : ∀ i j : ℕ, |
| targetCocycle (z+scaleVertex i a+scaleVertex j b) (environmentTranslation a ω) x y = |
| targetCocycle (z+scaleVertex (i+1) a+scaleVertex j b) ω (a+x) (a+y) := by |
| intro i j |
| rw [← targetCocycle_translation] |
| congr 1 |
| rw [scaleVertex_succ] |
| abel |
| simp only [he, ← sum_sub_distrib] |
| rw [sum_comm] |
| have htel (j : ℕ) := sum_range_sub |
| (fun i : ℕ => targetCocycle (z+scaleVertex i a+scaleVertex j b) ω (a+x) (a+y)) N |
| simp_rw [htel] |
| have hz : scaleVertex 0 a = (0 : Vertex) := by ext <;> simp [scaleVertex] |
| simp only [hz, add_zero] |
|
|
| theorem rectangleTargetCocycle_shift_bound (z a b : Vertex) (N M : ℕ) |
| (hN : 1 ≤ N) (hM : 1 ≤ M) (x y : Vertex) : |
| ∀ᵐ ω ∂environmentLaw, |
| |rectangleTargetCocycle z a b N M (environmentTranslation a ω) x y - |
| rectangleTargetCocycle z a b N M ω (a+x) (a+y)| ≤ |
| 2*passageTime ω (a+x) (a+y)/N := by |
| filter_upwards [targetCocycle_dominated] with ω hω |
| rw [rectangleTargetCocycle_shift_identity, abs_div, |
| abs_of_nonneg (by positivity : 0 ≤ (N : ℝ)*M)] |
| have hb : |∑ j ∈ range M, |
| (targetCocycle (z+scaleVertex N a+scaleVertex j b) ω (a+x) (a+y) - |
| targetCocycle (z+scaleVertex j b) ω (a+x) (a+y))| ≤ |
| (M : ℝ)*(2*passageTime ω (a+x) (a+y)) := by |
| apply (abs_sum_le_sum_abs _ _).trans |
| calc |
| _ ≤ ∑ j ∈ range M, 2*passageTime ω (a+x) (a+y) := by |
| apply sum_le_sum |
| intro j _ |
| apply (abs_sub _ _).trans |
| linarith [hω (z+scaleVertex N a+scaleVertex j b) (a+x) (a+y), |
| hω (z+scaleVertex j b) (a+x) (a+y)] |
| _ = _ := by simp |
| apply (div_le_div_of_nonneg_right hb (by positivity)).trans_eq |
| have hn : (N : ℝ) ≠ 0 := by exact_mod_cast (show N ≠ 0 by omega) |
| have hm : (M : ℝ) ≠ 0 := by exact_mod_cast (show M ≠ 0 by omega) |
| field_simp |
|
|
| theorem rectangleTargetCocycle_shift_integral (z a b x y : Vertex) (N M : ℕ) |
| (hN : 1 ≤ N) (hM : 1 ≤ M) : |
| (∫ ω, |rectangleTargetCocycle z a b N M (environmentTranslation a ω) x y - |
| rectangleTargetCocycle z a b N M ω (a+x) (a+y)| ∂environmentLaw) ≤ |
| 2*(latticeDistance x y : ℝ)/N := by |
| have hi := ((environmentTranslation_preserving a).integrable_comp_of_integrable |
| (integrable_rectangleTargetCocycle z a b x y N M)).sub |
| (integrable_rectangleTargetCocycle z a b (a+x) (a+y) N M) |
| apply (integral_mono_ae hi.abs |
| (((integrable_passageTime (a+x) (a+y)).const_mul 2).div_const N) |
| (rectangleTargetCocycle_shift_bound z a b N M hN hM x y)).trans |
| rw [integral_div, integral_const_mul, mean_passageTime_translation] |
| apply div_le_div_of_nonneg_right _ (Nat.cast_nonneg N) |
| exact mul_le_mul_of_nonneg_left (mean_passageTime_le_distance x y) (by norm_num) |
|
|
| theorem rectangleTargetCocycle_second_shift_integral (z a b x y : Vertex) (N M : ℕ) |
| (hN : 1 ≤ N) (hM : 1 ≤ M) : |
| (∫ ω, |rectangleTargetCocycle z a b N M (environmentTranslation b ω) x y - |
| rectangleTargetCocycle z a b N M ω (b+x) (b+y)| ∂environmentLaw) ≤ |
| 2*(latticeDistance x y : ℝ)/M := by |
| simp_rw [rectangleTargetCocycle_swap z a b N M] |
| exact rectangleTargetCocycle_shift_integral z b a x y M N hM hN |
|
|
| theorem rectangleTargetCocycle_memLp_two (z a b x y : Vertex) (N M : ℕ) |
| (hN : 1 ≤ N) (hM : 1 ≤ M) : |
| MemLp (fun ω => rectangleTargetCocycle z a b N M ω x y) 2 environmentLaw := by |
| apply (passageTime_memLp_two x y).mono' (by fun_prop) |
| filter_upwards [rectangleTargetCocycle_dominated z a b N M hN hM] with ω hω |
| simpa only [Real.norm_eq_abs] using hω x y |
| |
| theorem rectangleTargetCocycle_second_moment (z a b x y : Vertex) (N M : ℕ) |
| (hN : 1 ≤ N) (hM : 1 ≤ M) : |
| (∫ ω, (rectangleTargetCocycle z a b N M ω x y)^2 ∂environmentLaw) ≤ |
| (latticeDistance x y : ℝ)*(latticeDistance x y+1) := by |
| apply (integral_mono_ae (rectangleTargetCocycle_memLp_two z a b x y N M hN hM).integrable_sq |
| (passageTime_memLp_two x y).integrable_sq (by |
| filter_upwards [rectangleTargetCocycle_dominated z a b N M hN hM] with ω hω |
| have h := pow_le_pow_left₀ (abs_nonneg _) (hω x y) 2 |
| simpa only [sq_abs] using h)).trans (passageTime_second_moment x y) |
| |
| theorem rectangleTargetCocycle_shift_tendsto (z : ℕ → Vertex) (a b x y : Vertex) : |
| Tendsto (fun n : ℕ => ∫ ω, |
| |rectangleTargetCocycle (z n) a b n n (environmentTranslation a ω) x y - |
| rectangleTargetCocycle (z n) a b n n ω (a+x) (a+y)| ∂environmentLaw) |
| atTop (𝓝 0) := by |
| apply squeeze_zero (fun _ => integral_nonneg (fun _ => abs_nonneg _)) |
| (fun n => ?_) (tendsto_const_div_atTop_nhds_zero_nat (2*(latticeDistance x y : ℝ))) |
| by_cases hn : n = 0 |
| · subst n |
| simp [rectangleTargetCocycle] |
| · exact rectangleTargetCocycle_shift_integral (z n) a b x y n n (by omega) (by omega) |
| |
| theorem rectangleTargetCocycle_second_shift_tendsto (z : ℕ → Vertex) (a b x y : Vertex) : |
| Tendsto (fun n : ℕ => ∫ ω, |
| |rectangleTargetCocycle (z n) a b n n (environmentTranslation b ω) x y - |
| rectangleTargetCocycle (z n) a b n n ω (b+x) (b+y)| ∂environmentLaw) |
| atTop (𝓝 0) := by |
| simp_rw [rectangleTargetCocycle_swap (z _) a b] |
| exact rectangleTargetCocycle_shift_tendsto z b a x y |
| |
| end CompanionFPP |
| |
| end OAI |
| |
