| import Mathlib |
| import OAI.Analysis.HardSphere.Paths.LevelList |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal BigOperators Topology |
| namespace HardSphere.DyadicSkeleton |
|
|
| lemma occurrenceHit_le (K n : ℕ) (hn : 0 < n) (hnK : 8*n ≤ K) |
| (d : Fin 2 → ℤ) (hd : ∀j, |d j| ≤ n) (ω ω' : Noise) (R : ℝ) : |
| occurrenceHit K R (occurrenceList n d ω) (occurrenceList n d ω') ≤ |
| 20*∑ u ∈ Finset.range (n/2+1), |
| nearPotential u (fun j ↦ position ω' j u) (3*R+5) ω := by |
| have h := oneSideHit_le K n hn hnK d hd ω ω' R |
| have h' := oneSideHit_le K n hn hnK d hd ω' ω R |
| simp_rw [nearPotential_swap _ _ ω ω'] at h' |
| unfold occurrenceHit |
| linarith |
|
|
| lemma occurrenceSoft_le (K n : ℕ) (hn : 0 < n) (hnK : 8*n ≤ K) |
| (d : Fin 2 → ℤ) (hd : ∀j, |d j| ≤ n) (ω ω' : Noise) (B : ℝ) (hB : 0 < B) : |
| occurrenceSoft K B (occurrenceList n d ω) (occurrenceList n d ω') ≤ |
| (10*Real.exp (5/(3*B)))*∑ u ∈ Finset.range (n/2+1), |
| Real.exp (-twoCopyDistance u (ω,ω')/(3*B)) := by |
| have h := level_sum_le n hn d hd ω |
| (fun x ↦ Real.exp (-distanceToList K x (occurrenceList n d ω')/B)) |
| (fun t ↦ Real.exp (5/(3*B))*Real.exp (-twoCopyDistance (min t (n-t)) (ω,ω')/(3*B))) |
| (fun _ ↦ by positivity) (by |
| intro t ht x hx |
| rw [← Real.exp_add] |
| apply Real.exp_le_exp.mpr |
| have hdev := distanceToList_deviation_le K n hn hnK d hd ω ω' t ht x hx |
| have hBp : 0 < 3*B := by positivity |
| rw [← add_div] |
| apply (le_div_iff₀ hBp).mpr |
| have he : (-distanceToList K x (occurrenceList n d ω')/B)*(3*B) = |
| -3*distanceToList K x (occurrenceList n d ω') := by field_simp |
| rw [he] |
| linarith) |
| have hf := folded_sum_le n |
| (fun u ↦ Real.exp (-twoCopyDistance u (ω,ω')/(3*B))) (fun _ ↦ (Real.exp_pos _).le) |
| change occurrenceSoft K B _ _ ≤ _ at h |
| simp_rw [← Finset.mul_sum] at h |
| have hpos : 0 < Real.exp (5/(3*B)) := Real.exp_pos _ |
| nlinarith |
| |
| def skeletonHitConstant : ℝ := hitConstant/720 |
| lemma skeletonHitConstant_pos : 0 < skeletonHitConstant := div_pos hitConstant_pos (by norm_num) |
| |
| theorem skeleton_hit_moment (K n : ℕ) (hn : 0 < n) (hnK : 8*n ≤ K) |
| (d : Fin 2 → ℤ) (hd : ∀j, |d j| ≤ n) (R : ℕ) : |
| (∫⁻ z : Noise × Noise, ENNReal.ofReal (Real.exp |
| ((skeletonHitConstant/(1+(R:ℝ))^2) * |
| occurrenceHit K (R:ℝ) (occurrenceList n d z.1) (occurrenceList n d z.2))) |
| ∂law.prod law) ≤ 2 := by |
| have hcoeff : skeletonHitConstant/(1+(R:ℝ))^2*20 ≤ |
| hitConstant/(1+((3*R+5:ℕ):ℝ))^2 := by |
| have hR : 0 ≤ (R:ℝ) := by positivity |
| have hden : (1+((3*R+5:ℕ):ℝ))^2 ≤ 36*(1+(R:ℝ))^2 := by |
| push_cast |
| nlinarith |
| calc |
| _ = hitConstant/(36*(1+(R:ℝ))^2) := by |
| unfold skeletonHitConstant |
| field_simp |
| ring |
| _ ≤ _ := div_le_div_of_nonneg_left hitConstant_pos.le (by positivity) hden |
| refine le_trans (lintegral_mono (fun z ↦ ?_)) (two_copy_hit_moment (n/2) (3*R+5)) |
| apply ENNReal.ofReal_le_ofReal |
| apply Real.exp_le_exp.mpr |
| have h := occurrenceHit_le K n hn hnK d hd z.1 z.2 (R:ℝ) |
| have ha : 0 ≤ skeletonHitConstant/(1+(R:ℝ))^2 := by |
| exact div_nonneg skeletonHitConstant_pos.le (sq_nonneg _) |
| calc |
| _ ≤ (skeletonHitConstant/(1+(R:ℝ))^2) * |
| (20*∑ u ∈ Finset.range (n/2+1), |
| nearPotential u (fun j ↦ position z.2 j u) (3*(R:ℝ)+5) z.1) := |
| mul_le_mul_of_nonneg_left h ha |
| _ = (skeletonHitConstant/(1+(R:ℝ))^2*20) * |
| ∑ u ∈ Finset.range (n/2+1), |
| nearPotential u (fun j ↦ position z.2 j u) ((3*R+5:ℕ):ℝ) z.1 := by |
| push_cast |
| ring |
| _ ≤ _ := mul_le_mul_of_nonneg_right hcoeff |
| (Finset.sum_nonneg (fun u _ ↦ nearPotential_nonneg _ _ _ _)) |
| |
| theorem skeleton_soft_moment (B : ℝ) (hB : 0 < B) : |
| ∃ ξ : ℝ, 0 < ξ ∧ ∀ K n : ℕ, 0 < n → 8*n ≤ K → |
| ∀ d : Fin 2 → ℤ, (∀j, |d j| ≤ n) → |
| (∫⁻ z : Noise × Noise, ENNReal.ofReal (Real.exp |
| (ξ*occurrenceSoft K B (occurrenceList n d z.1) (occurrenceList n d z.2))) |
| ∂law.prod law) ≤ 2 := by |
| obtain ⟨η,hη,hm⟩ := two_copy_soft_moment (3*B) (by positivity) |
| let C : ℝ := 10*Real.exp (5/(3*B)) |
| have hC : 0 < C := by dsimp [C]; positivity |
| refine ⟨η/C,div_pos hη hC,fun K n hn hnK d hd ↦ ?_⟩ |
| refine le_trans (lintegral_mono (fun z ↦ ?_)) (hm (n/2)) |
| apply ENNReal.ofReal_le_ofReal |
| apply Real.exp_le_exp.mpr |
| calc |
| _ ≤ (η/C)*(C*∑ u ∈ Finset.range (n/2+1), |
| Real.exp (-twoCopyDistance u z/(3*B))) := |
| mul_le_mul_of_nonneg_left (occurrenceSoft_le K n hn hnK d hd z.1 z.2 B hB) |
| (div_pos hη hC).le |
| _ = _ := by field_simp |
| |
| instance listVertexMeasurableSpace : MeasurableSpace (List Vertex) := ⊤ |
| instance listVertexDiscreteMeasurableSpace : DiscreteMeasurableSpace (List Vertex) := ⟨fun _ ↦ trivial⟩ |
| |
| @[fun_prop] lemma measurable_levelList (n : ℕ) (d : Fin 2 → ℤ) (t : ℕ) : |
| Measurable (fun ω ↦ levelList n d ω t) := by |
| cases t with |
| | zero => |
| exact (measurable_of_countable (fun a : Vertex ↦ [a])).comp |
| (measurable_baseVertex n d 0) |
| | succ t => |
| exact (measurable_of_countable (fun a : Vertex × Vertex ↦ refineStep a.1 a.2)).comp |
| ((measurable_baseVertex n d t).prodMk (measurable_baseVertex n d (t+1))) |
| |
| @[fun_prop] lemma measurable_occurrenceList (n : ℕ) (d : Fin 2 → ℤ) : |
| Measurable (occurrenceList n d) := by |
| have hl (ts : List ℕ) : Measurable (fun ω ↦ ts.flatMap (levelList n d ω)) := by |
| induction ts with |
| | nil => exact measurable_const |
| | cons t ts ih => |
| simpa only [List.flatMap_cons,Function.comp_def] using |
| (measurable_of_countable (fun p : List Vertex × List Vertex ↦ p.1 ++ p.2)).comp |
| ((measurable_levelList n d t).prodMk ih) |
| exact hl (List.range (n+1)) |
| |
| def skeletonLaw (n : ℕ) (d : Fin 2 → ℤ) : Measure (List Vertex) := |
| law.map (occurrenceList n d) |
| |
| instance skeletonLaw_isProbability (n : ℕ) (d : Fin 2 → ℤ) : |
| IsProbabilityMeasure (skeletonLaw n d) := |
| (Measure.isProbabilityMeasure_map_iff |
| (measurable_occurrenceList n d).aemeasurable).mpr inferInstance |
| |
| end HardSphere.DyadicSkeleton |
| end |
| end |
| |
| end OAI |
| |
