| import Mathlib |
| import OAI.Analysis.HardSphere.Geometry.KilledObservable |
| import OAI.Analysis.HardSphere.PathBounds.VisitPacking |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology BigOperators |
| namespace HardSphere.FreeVisits |
| open HardSphere.BathSlice HardSphere.Lattice HardSphere.CellOrbital HardSphere.Wiener |
|
|
| lemma pathVisitedSites_card_measurable {S : Type*} {n : ℕ} (L : Length) |
| (v : S → Torus L) (A : Finset S) (i : Fin n) : |
| Measurable (fun ω : ConfigurationPath n L ↦ (pathVisitedSites L v A i ω).card) := by |
| classical |
| simp only [pathVisitedSites,Finset.card_filter] |
| apply Finset.measurable_sum |
| intro a _ |
| exact Measurable.ite (pathVisitsBall_isClosed L 12 (v a) i).measurableSet |
| measurable_const measurable_const |
|
|
| def visitCount {n K : ℕ} (L : Length) (A : Finset (DiscreteTorus K)) (i : Fin n) |
| (t : ℝ) (X : Configuration n L) (w : CoordinatePaths n) : ℕ := |
| (pathVisitedSites L (siteCenter L) A i (freeTrajectory n L t w X)).card |
|
|
| lemma visitCount_measurable {n K : ℕ} (L : Length) (A : Finset (DiscreteTorus K)) (i : Fin n) |
| (t : ℝ) (X : Configuration n L) : Measurable (visitCount L A i t X) := |
| (pathVisitedSites_card_measurable L (siteCenter L) A i).comp |
| ((freeTrajectory_continuous n L t).comp (continuous_id.prodMk continuous_const)).measurable |
|
|
| lemma one_segment_visit_tail {c : ℝ} {M : ℝ≥0∞} |
| (htail : ∀ R : ℝ, 0≤R → wiener {w | R≤‖w‖}≤M*ENNReal.ofReal (Real.exp (-c*R^2))) |
| {n K : ℕ} [NeZero K] (L : Length) (hL : L.val=(K:ℝ)) |
| (A : Finset (DiscreteTorus K)) (i : Fin n) {t : ℝ} (ht : 0<t) (X : Configuration n L) : |
| coordinateWiener n {w | 29^3<visitCount L A i t X w} ≤ |
| 3*M*ENNReal.ofReal (Real.exp (-c/(2*t))) := by |
| let R : ℝ := (Real.sqrt (2*t))⁻¹ |
| have hs : 0<Real.sqrt (2*t) := Real.sqrt_pos.mpr (by positivity) |
| have hR : 0≤R := inv_nonneg.mpr hs.le |
| have hsub : {w | 29^3<visitCount L A i t X w} ⊆ |
| ⋃ j : Fin 3, {w | R≤‖w (i,j)‖} := by |
| intro w hw |
| by_contra hnot |
| have hn : ∀ j, ‖w (i,j)‖<R := by simpa only [Set.mem_iUnion,Set.mem_ofPred_eq,not_exists,not_le] using hnot |
| have hb := free_visited_card_le_of_norm L hL A i t w X (fun j ↦ ?_) |
| · exact (not_lt_of_ge hb) hw |
| · calc |
| Real.sqrt (2*t)*‖w (i,j)‖ ≤ Real.sqrt (2*t)*R := mul_le_mul_of_nonneg_left (hn j).le hs.le |
| _ = 1 := mul_inv_cancel₀ (ne_of_gt hs) |
| have he (j : Fin 3) : coordinateWiener n {w | R≤‖w (i,j)‖} = wiener {w | R≤‖w‖} := by |
| exact (measurePreserving_eval (fun _ : Direction n ↦ wiener) (i,j)).measure_preimage |
| (measurableSet_le measurable_const measurable_norm).nullMeasurableSet |
| calc |
| _ ≤ coordinateWiener n (⋃ j : Fin 3, {w | R≤‖w (i,j)‖}) := measure_mono hsub |
| _ ≤ ∑' j : Fin 3, coordinateWiener n {w | R≤‖w (i,j)‖} := measure_iUnion_le _ |
| _ = 3 * wiener {w | R≤‖w‖} := by simp [he,tsum_fintype] |
| _ ≤ 3*(M*ENNReal.ofReal (Real.exp (-c*R^2))) := mul_le_mul_right (htail R hR) _ |
| _ = _ := by |
| dsimp [R] |
| rw [inv_pow,Real.sq_sqrt (by positivity : 0≤2*t)] |
| simp only [div_eq_mul_inv,mul_assoc] |
|
|
| lemma visit_count_split {n K : ℕ} (L : Length) (A : Finset (DiscreteTorus K)) (i : Fin n) |
| {s t : ℝ} (hs : 0<s) (ht : 0<t) (w : CoordinatePaths n) (X : Configuration n L) : |
| visitCount L A i (s+t) X w ≤ |
| visitCount L A i s X (fun p ↦ normalizedPrefix (splitTime s t hs ht) (w p)) + |
| visitCount L A i t |
| (freeTrajectory n L s (fun p ↦ normalizedPrefix (splitTime s t hs ht) (w p)) X 1) |
| (fun p ↦ normalizedSuffix (splitTime s t hs ht) (w p)) := by |
| have hle := visited_card_split_le L (siteCenter L) A i (freeTrajectory n L (s+t) w X) |
| (splitTime s t hs ht) (splitTime_pos s t hs ht) (splitTime_lt_one s t hs ht) |
| have hp := congrArg Prod.fst (split_fixedTrajectory n L s t hs ht w X) |
| have hq := congrArg Prod.snd (split_fixedTrajectory n L s t hs ht w X) |
| dsimp only [Prod.fst,Prod.snd] at hp hq |
| rw [hp,hq] at hle |
| exact hle |
|
|
| attribute [local irreducible] freeTrajectory pathVisitedSites |
|
|
| lemma visit_tail_add {n K : ℕ} (L : Length) (A : Finset (DiscreteTorus K)) (i : Fin n) |
| {s t : ℝ} (hs : 0<s) (ht : 0<t) (a b : ℕ) (p q : ℝ≥0∞) |
| (hsbound : ∀ X, coordinateWiener n {w | a<visitCount L A i s X w}≤p) |
| (htbound : ∀ X, coordinateWiener n {w | b<visitCount L A i t X w}≤q) |
| (X : Configuration n L) : |
| coordinateWiener n {w | a+b<visitCount L A i (s+t) X w}≤p+q := by |
| let u := splitTime s t hs ht |
| let split : CoordinatePaths n → CoordinatePaths n × CoordinatePaths n := fun w ↦ |
| (fun p ↦ normalizedPrefix u (w p),fun p ↦ normalizedSuffix u (w p)) |
| let E₁ : Set (CoordinatePaths n × CoordinatePaths n) := |
| {p | a<visitCount L A i s X p.1} |
| let E₂ : Set (CoordinatePaths n × CoordinatePaths n) := |
| {p | b<visitCount L A i t (freeTrajectory n L s p.1 X 1) p.2} |
| have hE₁ : MeasurableSet E₁ := measurableSet_lt measurable_const |
| ((visitCount_measurable L A i s X).comp measurable_fst) |
| have hE₂ : MeasurableSet E₂ := by |
| apply measurableSet_lt measurable_const |
| exact (pathVisitedSites_card_measurable L (siteCenter L) A i).comp |
| (measurable_snd.comp (split_fixedTrajectory_measurable n L s t X)) |
| have hsub : {w | a+b<visitCount L A i (s+t) X w} ⊆ split ⁻¹' (E₁∪E₂) := by |
| intro w hw |
| have hle := visit_count_split L A i hs ht w X |
| change a<visitCount L A i s X (split w).1 ∨ |
| b<visitCount L A i t (freeTrajectory n L s (split w).1 X 1) (split w).2 |
| change a+b<visitCount L A i (s+t) X w at hw |
| change visitCount L A i (s+t) X w ≤ visitCount L A i s X (split w).1 + |
| visitCount L A i t (freeTrajectory n L s (split w).1 X 1) (split w).2 at hle |
| omega |
| have hp₁ : ((coordinateWiener n).prod (coordinateWiener n)) E₁≤p := by |
| have he : E₁={w | a<visitCount L A i s X w} ×ˢ Set.univ := by ext v; simp [E₁] |
| rw [he,Measure.prod_prod,measure_univ,mul_one] |
| exact hsbound X |
| have hp₂ : ((coordinateWiener n).prod (coordinateWiener n)) E₂≤q := by |
| rw [Measure.prod_apply hE₂] |
| calc |
| _ ≤ ∫⁻ _w : CoordinatePaths n, q ∂coordinateWiener n := by |
| apply lintegral_mono |
| intro w |
| exact htbound (freeTrajectory n L s w X 1) |
| _ = q := by simp |
| calc |
| _ ≤ coordinateWiener n (split ⁻¹' (E₁∪E₂)) := measure_mono hsub |
| _ = ((coordinateWiener n).prod (coordinateWiener n)) (E₁∪E₂) := |
| (coordinateSplit_preserving n u (splitTime_pos s t hs ht) (splitTime_lt_one s t hs ht)).measure_preimage |
| (hE₁.union hE₂).nullMeasurableSet |
| _ ≤ ((coordinateWiener n).prod (coordinateWiener n)) E₁ + |
| ((coordinateWiener n).prod (coordinateWiener n)) E₂ := measure_union_le _ _ |
| _ ≤ p+q := add_le_add hp₁ hp₂ |
|
|
| lemma n_segment_visit_tail {c : ℝ} {M : ℝ≥0∞} |
| (htail : ∀ R : ℝ, 0≤R → wiener {w | R≤‖w‖}≤M*ENNReal.ofReal (Real.exp (-c*R^2))) |
| {n K : ℕ} [NeZero K] (L : Length) (hL : L.val=(K:ℝ)) |
| (A : Finset (DiscreteTorus K)) (i : Fin n) {t : ℝ} (ht : 0<t) : |
| ∀ k : ℕ, 0<k → ∀ X : Configuration n L, |
| coordinateWiener n {w | 29^3*k<visitCount L A i ((k:ℝ)*t) X w} ≤ |
| (k:ℝ≥0∞)*(3*M*ENNReal.ofReal (Real.exp (-c/(2*t)))) := by |
| intro k hk |
| induction k, hk using Nat.le_induction with |
| | base => |
| intro X |
| simpa only [Nat.succ_eq_add_one,Nat.zero_add,Nat.cast_one,one_mul,mul_one] using one_segment_visit_tail htail L hL A i ht X |
| | succ k hk ih => |
| intro X |
| have hkt : 0<(k:ℝ)*t := mul_pos (by exact_mod_cast hk) ht |
| have h := visit_tail_add L A i hkt ht (29^3*k) (29^3) |
| ((k:ℝ≥0∞)*(3*M*ENNReal.ofReal (Real.exp (-c/(2*t))))) |
| (3*M*ENNReal.ofReal (Real.exp (-c/(2*t)))) ih |
| (fun X ↦ one_segment_visit_tail htail L hL A i ht X) X |
| convert h using 1 <;> push_cast <;> ring_nf |
|
|
| lemma whole_time_visit_tail {c : ℝ} {M : ℝ≥0∞} |
| (htail : ∀ R : ℝ, 0≤R → wiener {w | R≤‖w‖}≤M*ENNReal.ofReal (Real.exp (-c*R^2))) |
| {n K : ℕ} [NeZero K] (L : Length) (hL : L.val=(K:ℝ)) |
| (A : Finset (DiscreteTorus K)) (i : Fin n) {t : ℝ} (ht : 0<t) |
| (k : ℕ) (hk : 0<k) (X : Configuration n L) : |
| coordinateWiener n {w | 29^3*k<visitCount L A i t X w} ≤ |
| (k:ℝ≥0∞)*(3*M*ENNReal.ofReal (Real.exp (-c*(k:ℝ)/(2*t)))) := by |
| have hk' : (0:ℝ)<k := by exact_mod_cast hk |
| have h := n_segment_visit_tail htail L hL A i (div_pos ht hk') k hk X |
| have he : (k:ℝ)*(t/k)=t := mul_div_cancel₀ t (ne_of_gt hk') |
| rw [he] at h |
| convert h using 1 |
| congr 3 |
| field_simp |
|
|
| end HardSphere.FreeVisits |
|
|
| end |
| end |
|
|
| end |
|
|
| end OAI |
|
|
