| import Mathlib |
| import OAI.Probability.Ballisticity.Estimates.OutwardCanonical |
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| namespace OAI |
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| section |
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| section |
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| open MeasureTheory ProbabilityTheory Filter Function |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| namespace DirectionalTransience |
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| lemma wordCylinder_signed_up_disjoint_stay {d : ℕ} (e : Direction d) |
| (x y : Lattice d) (H : ℕ) : |
| Disjoint (wordCylinder y (List.replicate H e)) |
| (StayUntil (Strip (realPosition (step e)) x H) H) := by |
| apply Set.disjoint_left.mpr |
| intro X hw hS |
| have hb := (hS 0 (Nat.zero_le H)).1 |
| have ht := (hS H le_rfl).2 |
| have hzero := hw 0 (by simp) |
| have hend := hw H (by simp) |
| rw [wordPath_replicate] at hend |
| rw [hzero,wordPath_zero] at hb |
| rw [hend,dot_realPosition_add,dot_realPosition_nsmul,dot_signed_direction e (step e),signedCoordinate_step_self,mul_one] at ht |
| linarith |
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| lemma quenched_signed_strip_block {d : ℕ} (ω : Environment d) {κ : ℝ≥0} |
| (hκ : ∀ y e, κ ≤ (ω y).1 e) (e : Direction d) (x y : Lattice d) (H : ℕ) : |
| quenchedKernel (ω,y) (StayUntil (Strip (realPosition (step e)) x H) H) ≤ |
| 1-(κ:ℝ≥0∞)^H := by |
| have hsub := (wordCylinder_signed_up_disjoint_stay e x y H).subset_compl_left |
| calc |
| _ ≤ quenchedKernel (ω,y) (wordCylinder y (List.replicate H e))ᶜ := measure_mono hsub |
| _ = 1-ENNReal.ofReal (wordWeight ω y (List.replicate H e)) := by |
| rw [measure_compl (measurableSet_wordCylinder _ _) (measure_ne_top _ _),measure_univ,quenched_wordCylinder] |
| _ ≤ 1-(κ:ℝ≥0∞)^H := by |
| apply tsub_le_tsub_left |
| have hs := ENNReal.ofReal_le_ofReal (wordWeight_lower ω hκ y (List.replicate H e)) |
| simpa only [List.length_replicate,ENNReal.ofReal_pow κ.coe_nonneg,ENNReal.ofReal_coe_nnreal] using hs |
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| lemma quenched_signed_strip_geometric {d : ℕ} (ω : Environment d) {κ : ℝ≥0} |
| (hκ : ∀ y e, κ ≤ (ω y).1 e) (e : Direction d) (x y : Lattice d) (H k : ℕ) : |
| quenchedKernel (ω,y) (StayUntil (Strip (realPosition (step e)) x H) (k*H)) ≤ |
| (1-(κ:ℝ≥0∞)^H)^k := |
| quenched_stayUntil_block_bound ω _ H _ (fun y => quenched_signed_strip_block ω hκ e x y H) y k |
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| lemma truncatedEndpointLaw_mass_error {d : ℕ} (ω : Environment d) {κ : ℝ≥0} |
| (hκ : ∀ y e, κ ≤ (ω y).1 e) (e : Direction d) (x : Lattice d) |
| {H : ℕ} (hH : 0 < H) (k : ℕ) : |
| hitKernel (Strip (realPosition (step e)) x H) (Upper (realPosition (step e)) x H) (ω,x) Set.univ ≤ |
| truncatedEndpointLaw e H (k*H) ω x Set.univ+(1-(κ:ℝ≥0∞)^H)^k := by |
| rw [truncatedEndpointLaw_apply e hH,Set.inter_univ,hitKernel_total (disjoint_strip_upper _ _ _)] |
| exact (quenched_hit_truncation ω x _ _ _).trans |
| (add_le_add le_rfl (quenched_signed_strip_geometric ω hκ e x x H k)) |
| end DirectionalTransience |
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| end |
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| end |
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| end OAI |
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