| import OAI.Probability.DirectionalWalk.Columns |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory Filter Preorder |
| open scoped ENNReal BigOperators Topology |
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| namespace DirectionalZeroOne |
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| def strictRecord {d : ℕ} (v : Fin d → ℝ) (X : Path d) (n : ℕ) : Prop := |
| 0 < n ∧ ∀ i < n, height v (X i) < height v (X n) |
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| def trueCut {d : ℕ} (v : Fin d → ℝ) (X : Path d) (n : ℕ) : Prop := |
| strictRecord v X n ∧ ∀ j, height v (X n) ≤ height v (X (n+j)) |
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| lemma measurableSet_strictRecord {d : ℕ} (v : Fin d → ℝ) (n : ℕ) : |
| MeasurableSet {X : Path d | strictRecord v X n} := by |
| simp only [strictRecord, Set.ofPred_and, Set.ofPred_forall] |
| exact (MeasurableSet.const _).inter (MeasurableSet.iInter (fun i => |
| MeasurableSet.iInter (fun _ => measurableSet_lt |
| ((measurable_of_countable (height v)).comp (measurable_pi_apply i)) |
| ((measurable_of_countable (height v)).comp (measurable_pi_apply n))))) |
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| lemma measurableSet_trueCut {d : ℕ} (v : Fin d → ℝ) (n : ℕ) : |
| MeasurableSet {X : Path d | trueCut v X n} := by |
| simp only [trueCut, Set.ofPred_and, Set.ofPred_forall] |
| exact (measurableSet_strictRecord v n).inter (MeasurableSet.iInter (fun j => |
| measurableSet_le ((measurable_of_countable (height v)).comp (measurable_pi_apply n)) |
| ((measurable_of_countable (height v)).comp (measurable_pi_apply (n+j))))) |
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| lemma strictRecord_prefix {d : ℕ} (v : Fin d → ℝ) {X Y : Path d} {n : ℕ} |
| (hX : strictRecord v X n) (hY : Y ∈ pathCylinder n X) : strictRecord v Y n := by |
| refine ⟨hX.1,fun i hi => ?_⟩ |
| rw [hY i hi.le, hY n le_rfl] |
| exact hX.2 i hi |
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| lemma unbounded_has_later_strictRecord {d : ℕ} (v : Fin d → ℝ) (X : Path d) |
| (hX : ¬∃ b : ℝ, ∀ n, height v (X n) ≤ b) (N : ℕ) : |
| ∃ n, N < n ∧ strictRecord v X n := by |
| classical |
| let b := (Finset.range (N+1)).sup' (by simp) (fun n => height v (X n)) |
| have hex : ∃ n, b < height v (X n) := by |
| by_contra h |
| push Not at h |
| exact hX ⟨b,h⟩ |
| refine ⟨Nat.find hex, ?_, ?_⟩ |
| · by_contra h |
| have hn : Nat.find hex ∈ Finset.range (N+1) := Finset.mem_range.mpr (by omega) |
| exact (not_le_of_gt (Nat.find_spec hex)) (Finset.le_sup' (fun n => height v (X n)) hn) |
| · refine ⟨?_,fun i hi => ?_⟩ |
| · have hzero : height v (X 0) ≤ b := Finset.le_sup' (fun n => height v (X n)) (Finset.mem_range.mpr (Nat.succ_pos N)) |
| by_contra h |
| have hn : Nat.find hex = 0 := by omega |
| have hh := Nat.find_spec hex |
| rw [hn] at hh |
| linarith |
| · exact (le_of_not_gt (Nat.find_min hex hi)).trans_lt (Nat.find_spec hex) |
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| lemma measurableSet_boundedHeight {d : ℕ} (v : Fin d → ℝ) : |
| MeasurableSet {X : Path d | ∃ b : ℝ, ∀ n, height v (X n) ≤ b} := by |
| have heq : {X : Path d | ∃ b : ℝ, ∀ n, height v (X n) ≤ b} = |
| ⋃ b : ℤ, alwaysBelow v (b : ℝ) := by |
| ext X |
| simp only [Set.mem_ofPred_eq, Set.mem_iUnion] |
| constructor |
| · rintro ⟨b,hb⟩ |
| exact ⟨⌈b⌉,fun n => (hb n).trans (Int.le_ceil b)⟩ |
| · rintro ⟨b,hb⟩; exact ⟨b,hb⟩ |
| rw [heq] |
| exact MeasurableSet.iUnion (fun b => measurableSet_alwaysBelow v b) |
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| lemma no_later_trueCut_null {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ] |
| (y : Site d) (v : Fin d → ℝ) (hp : 0 < annealed μ 0 (nonBacktracking v)) (N : ℕ) : |
| annealed μ y {X | (¬∃ b : ℝ, ∀ n, height v (X n) ≤ b) ∧ |
| ∀ n, N < n → ¬trueCut v X n} = 0 := by |
| let p := (annealed μ 0 (nonBacktracking v)).toReal |
| have hp' : 0 < p := ENNReal.toReal_pos (ne_of_gt hp) (measure_ne_top _ _) |
| have hE : MeasurableSet {X : Path d | (¬∃ b : ℝ, ∀ n, height v (X n) ≤ b) ∧ |
| ∀ n, N < n → ¬trueCut v X n} := by |
| simp only [Set.ofPred_and, Set.ofPred_forall] |
| exact (measurableSet_boundedHeight v).compl.inter (MeasurableSet.iInter (fun n => |
| MeasurableSet.iInter (fun _ => (measurableSet_trueCut v n).compl))) |
| apply null_of_prefix_contraction (annealed μ y) hE (1-p) (by linarith) |
| intro X hX |
| have hf : ∃ᶠ n in atTop, N < n ∧ strictRecord v X n := by |
| apply frequently_atTop.mpr |
| intro k |
| obtain ⟨n,hn,hr⟩ := unbounded_has_later_strictRecord v X hX.1 (max k N) |
| exact ⟨n,(le_max_left k N).trans hn.le,(le_max_right k N).trans_lt hn,hr⟩ |
| apply hf.mono |
| rintro n ⟨hn,hr⟩ |
| refine real_contraction_of_disjoint_success (annealed μ y) (measurableSet_pathCylinder n X) |
| ((measurable_tailPath n) (measurableSet_neverBelow v (height v (X n)))) ?_ p ?_ |
| · apply Set.disjoint_left.mpr |
| rintro Y ⟨hY,hYC⟩ ⟨_,hS⟩ |
| apply hY.2 n hn |
| refine ⟨strictRecord_prefix v hr hYC,fun j => ?_⟩ |
| rw [hYC n le_rfl] |
| exact hS j |
| · change (annealed μ y (pathCylinder n X ∩ tailPath n ⁻¹' |
| neverBelow v (height v (X n)))).toReal = _ |
| rw [annealed_record_tail μ v y n X hr.2 (measurableSet_neverBelow v (height v (X n))) |
| (fun _ h => h), annealed_neverBelow_start, ENNReal.toReal_mul, mul_comm] |
| rfl |
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| lemma ae_infinitely_trueCuts {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ] |
| (y : Site d) (v : Fin d → ℝ) (hp : 0 < annealed μ 0 (nonBacktracking v)) : |
| ∀ᵐ X ∂annealed μ y, (¬∃ b : ℝ, ∀ n, height v (X n) ≤ b) → |
| ∀ N, ∃ n, N < n ∧ trueCut v X n := by |
| have hN (N : ℕ) : ∀ᵐ X ∂annealed μ y, |
| ¬((¬∃ b : ℝ, ∀ n, height v (X n) ≤ b) ∧ ∀ n, N < n → ¬trueCut v X n) := |
| (measure_eq_zero_iff_ae_notMem).mp (no_later_trueCut_null μ y v hp N) |
| filter_upwards [ae_all_iff.mpr hN] with X hX |
| intro hu N |
| by_contra h |
| push Not at h |
| exact hX N ⟨hu,h⟩ |
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| lemma escape_of_infinitely_trueCuts {d : ℕ} (v : Fin d → ℝ) (X : Path d) |
| (hu : ¬∃ b : ℝ, ∀ n, height v (X n) ≤ b) |
| (hcuts : ∀ N, ∃ n, N < n ∧ trueCut v X n) : X ∈ escape v := by |
| apply tendsto_atTop.mpr |
| intro b |
| have hex : ∃ k, b < height v (X k) := by |
| by_contra h; push Not at h; exact hu ⟨b,h⟩ |
| obtain ⟨k,hk⟩ := hex |
| obtain ⟨n,hn,hr,ht⟩ := hcuts k |
| apply eventually_atTop.mpr |
| refine ⟨n,fun m hm => ?_⟩ |
| have ht' := ht (m-n) |
| rw [Nat.add_sub_of_le hm] at ht' |
| exact (hk.trans (hr.2 k hn)).le.trans ht' |
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| lemma height_neg_direction {d : ℕ} (v : Fin d → ℝ) (x : Site d) : |
| height (-v) x = -height v x := by simp [height, Finset.sum_neg_distrib] |
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| lemma ae_escape_or_opposite {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ] |
| (hell : StrictEllipticity μ) (y : Site d) (v : Fin d → ℝ) (hv : v ≠ 0) |
| (hp : 0 < annealed μ 0 (nonBacktracking v)) : |
| ∀ᵐ X ∂annealed μ y, X ∈ escape v ∨ X ∈ escape (-v) := by |
| filter_upwards [finite_supremum μ hell y v hv, ae_infinitely_trueCuts μ y v hp] with X hfin hcut |
| by_cases hb : ∃ b : ℝ, ∀ n, height v (X n) ≤ b |
| · right |
| have ht := tendsto_neg_atBot_atTop.comp (hfin hb) |
| change Tendsto (fun n => height (-v) (X n)) atTop atTop |
| simpa only [height_neg_direction, Function.comp_def] using ht |
| · exact Or.inl (escape_of_infinitely_trueCuts v X hb (hcut hb)) |
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| def nearestNeighbour {d : ℕ} (X : Path d) : Prop := |
| ∀ n, ∃ e : Step d, X (n+1) = X n + stepVector e |
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| lemma ae_start_and_nearestNeighbour {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ] |
| (y : Site d) : ∀ᵐ X ∂annealed μ y, X 0 = y ∧ nearestNeighbour X := by |
| classical |
| filter_upwards [ae_pathCylinder_ne_zero (annealed μ y)] with X hX |
| constructor |
| · by_contra h |
| have hh := hX 0 |
| rw [annealed_cylinder, ite_eq_right (Ne.symm h)] at hh |
| exact hh rfl |
| · intro n |
| by_contra h |
| have hw (ω : Environment d) : entryWeight ω (X n) (X (n+1)) = 0 := by |
| unfold entryWeight |
| apply Finset.sum_eq_zero |
| intro e _ |
| have he : X (n+1) ≠ X n + stepVector e := fun he => h ⟨e,he⟩ |
| simp [Ne.symm he] |
| have hh := hX (n+1) |
| rw [annealed_cylinder] at hh |
| split_ifs at hh |
| · simp only [pathWeight_succ, hw, mul_zero, lintegral_zero] at hh |
| exact hh rfl |
| · exact hh rfl |
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| end DirectionalZeroOne |
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| end OAI |
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