| import Mathlib |
|
|
| namespace OAI |
|
|
| open MeasureTheory ProbabilityTheory Filter Preorder |
| open scoped ENNReal BigOperators Topology |
|
|
| namespace DirectionalZeroOne |
|
|
| abbrev Site (d : ℕ) := Fin d → ℤ |
| abbrev Step (d : ℕ) := Fin d × Bool |
|
|
| def stepVector {d : ℕ} (e : Step d) : Site d := |
| fun i => if i = e.1 then (if e.2 then 1 else -1) else 0 |
|
|
| def Row (d : ℕ) := |
| {p : Step d → ℝ // (∀ e, p e ∈ Set.Icc (0 : ℝ) 1) ∧ ∑ e, p e = 1} |
| |
| instance (d : ℕ) : MeasurableSpace (Row d) := inferInstanceAs |
| (MeasurableSpace {p : Step d → ℝ // |
| (∀ e, p e ∈ Set.Icc (0 : ℝ) 1) ∧ ∑ e, p e = 1}) |
| |
| abbrev Environment (d : ℕ) := Site d → Row d |
| abbrev Path (d : ℕ) := ℕ → Site d |
| abbrev State (d : ℕ) := Environment d × Site d |
| |
| def StrictEllipticity {d : ℕ} (μ : Measure (Row d)) : Prop := |
| ∀ᵐ p ∂μ, ∀ e, 0 < p.val e |
| |
| noncomputable def environmentLaw {d : ℕ} (μ : Measure (Row d)) |
| [IsProbabilityMeasure μ] : Measure (Environment d) := |
| Measure.infinitePi (fun _ => μ) |
| |
| noncomputable def transition (d : ℕ) : Kernel (State d) (State d) where |
| toFun s := ∑ e : Step d, |
| ENNReal.ofReal ((s.1 s.2).val e) • Measure.dirac (s.1, s.2 + stepVector e) |
| measurable' := by |
| apply measurable_from_prod_countable_left |
| intro x |
| apply Finset.measurable_sum |
| intro e he |
| apply Measure.measurable_of_measurable_coe |
| intro t ht |
| simp only [Measure.smul_apply, smul_eq_mul] |
| apply Measurable.mul |
| · exact (measurable_pi_apply e |>.comp |
| (measurable_subtype_coe.comp (measurable_pi_apply x))).ennreal_ofReal |
| · exact (Measure.measurable_coe ht).comp |
| (Measure.measurable_dirac.comp (measurable_id.prodMk measurable_const)) |
| |
| instance (d : ℕ) : IsMarkovKernel (transition d) where |
| isProbabilityMeasure s := by |
| constructor |
| change (∑ e : Step d, |
| ENNReal.ofReal ((s.1 s.2).val e) • |
| Measure.dirac (s.1, s.2 + stepVector e)) Set.univ = 1 |
| simp only [Measure.finsetSum_apply, Measure.smul_apply, |
| Measure.dirac_apply_of_mem (Set.mem_univ _), smul_eq_mul, mul_one] |
| rw [← ENNReal.ofReal_sum_of_nonneg] |
| · rw [(s.1 s.2).property.2, ENNReal.ofReal_one] |
| · intro e he |
| exact ((s.1 s.2).property.1 e).1 |
|
|
| noncomputable def historyKernel (d n : ℕ) : |
| Kernel ((i : Finset.Iic n) → State d) (State d) := |
| (transition d).comap (fun h => h ⟨n, Finset.mem_Iic.mpr le_rfl⟩) |
| (measurable_pi_apply _) |
| |
| instance (d n : ℕ) : IsMarkovKernel (historyKernel d n) := by |
| unfold historyKernel |
| infer_instance |
| |
| noncomputable def initialLaw {d : ℕ} (μ : Measure (Row d)) |
| [IsProbabilityMeasure μ] (x : Site d) : Measure (State d) := |
| (environmentLaw μ).map (fun ω => (ω, x)) |
| |
| noncomputable def annealed {d : ℕ} (μ : Measure (Row d)) |
| [IsProbabilityMeasure μ] (x : Site d) : Measure (Path d) := |
| (Kernel.trajMeasure (X := fun _ => State d) (initialLaw μ x) (historyKernel d)).map |
| (fun z n => (z n).2) |
| |
| def escape {d : ℕ} (ℓ : Fin d → ℝ) : Set (Path d) := |
| {X | Tendsto (fun n => ∑ i, (X n i : ℝ) * ℓ i) atTop atTop} |
| |
| lemma probabilityMeasure_map {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] |
| {μ : Measure α} [IsProbabilityMeasure μ] {f : α → β} |
| (hf : AEMeasurable f μ) : IsProbabilityMeasure (μ.map f) := |
| (Measure.isProbabilityMeasure_map_iff hf).2 inferInstance |
| |
| instance {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ] : |
| IsProbabilityMeasure (environmentLaw μ) := by |
| unfold environmentLaw |
| infer_instance |
| |
| instance {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ] (x : Site d) : |
| IsProbabilityMeasure (initialLaw μ x) := by |
| apply probabilityMeasure_map |
| exact (measurable_id.prodMk measurable_const).aemeasurable |
| |
| instance {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ] (x : Site d) : |
| IsProbabilityMeasure (annealed μ x) := by |
| apply probabilityMeasure_map |
| exact (Measurable.of_eval (fun n => measurable_snd.comp (measurable_pi_apply n))).aemeasurable |
| |
| lemma measurableSet_escape {d : ℕ} (ℓ : Fin d → ℝ) : MeasurableSet (escape ℓ) := by |
| apply measurableSet_tendsto |
| intro n |
| exact (measurable_of_countable (fun x : Site d => ∑ i, (x i : ℝ) * ℓ i)).comp |
| (measurable_pi_apply n) |
| |
| lemma all_rows_strictly_positive {d : ℕ} (μ : Measure (Row d)) |
| [IsProbabilityMeasure μ] (hμ : StrictEllipticity μ) : |
| ∀ᵐ ω ∂environmentLaw μ, ∀ x e, 0 < (ω x).val e := by |
| rw [ae_all_iff] |
| intro x |
| apply ae_of_ae_map (μ := environmentLaw μ) (f := fun ω : Environment d => ω x) |
| (p := fun p : Row d => ∀ e, 0 < p.val e) (measurable_pi_apply x).aemeasurable |
| simpa only [environmentLaw, Measure.infinitePi_map_eval, StrictEllipticity] using hμ |
| |
| abbrev rowSigma {d : ℕ} (s : Set (Site d)) : MeasurableSpace (Environment d) := |
| ⨆ x ∈ s, MeasurableSpace.comap (fun ω : Environment d => ω x) inferInstance |
| |
| lemma rowSigma_le {d : ℕ} (s : Set (Site d)) : rowSigma s ≤ (inferInstance : MeasurableSpace (Environment d)) := by |
| exact iSup₂_le (fun x _ => (measurable_pi_apply x).comap_le) |
| |
| lemma measurable_row {d : ℕ} {s : Set (Site d)} {x : Site d} (hx : x ∈ s) : |
| @Measurable (Environment d) (Row d) (rowSigma s) _ (fun ω => ω x) := by |
| exact measurable_iff_comap_le.mpr (le_iSup₂_of_le x hx le_rfl) |
| |
| lemma independent_rowSigma {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ] |
| {s t : Set (Site d)} (hst : Disjoint s t) : |
| Indep (rowSigma s) (rowSigma t) (environmentLaw μ) := by |
| apply indep_iSup_of_disjoint (fun x => (measurable_pi_apply x).comap_le) _ hst |
| exact iIndepFun_infinitePi (X := fun _ (p : Row d) => p) (fun _ => measurable_id) |
| |
| noncomputable def quenchedWeight {d n : ℕ} (x : Fin n → Site d) (e : Fin n → Step d) |
| (ω : Environment d) : ℝ≥0∞ := ∏ j, ENNReal.ofReal ((ω (x j)).val (e j)) |
| |
| noncomputable def wordWeight {d n : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ] |
| (x : Fin n → Site d) (e : Fin n → Step d) : ℝ≥0∞ := |
| ∫⁻ ω, quenchedWeight x e ω ∂environmentLaw μ |
| |
| lemma measurable_quenchedWeight_on_rows {d n : ℕ} (x : Fin n → Site d) |
| (e : Fin n → Step d) : |
| @Measurable (Environment d) ℝ≥0∞ (rowSigma (Set.range x)) _ |
| (quenchedWeight x e) := by |
| apply Finset.measurable_prod |
| intro j hj |
| exact ((measurable_pi_apply (e j)).comp |
| (measurable_subtype_coe.comp (measurable_row (s := Set.range x) (x := x j) ⟨j, rfl⟩))).ennreal_ofReal |
| |
| lemma measurable_quenchedWeight {d n : ℕ} (x : Fin n → Site d) |
| (e : Fin n → Step d) : Measurable (quenchedWeight x e) := |
| (measurable_quenchedWeight_on_rows x e).mono (rowSigma_le _) le_rfl |
| |
| lemma departure_transfer {d n : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ] |
| (x : Fin n → Site d) (e : Fin n → Step d) (f : Environment d → ℝ≥0∞) |
| (hf : @Measurable (Environment d) ℝ≥0∞ (rowSigma ((Set.range x)ᶜ)) _ f) : |
| (∫⁻ ω, quenchedWeight x e ω * f ω ∂environmentLaw μ) = |
| wordWeight μ x e * ∫⁻ ω, f ω ∂environmentLaw μ := by |
| exact lintegral_mul_eq_lintegral_mul_lintegral_of_independent_measurableSpace |
| (rowSigma_le _) (rowSigma_le _) |
| (independent_rowSigma μ disjoint_compl_right) |
| (measurable_quenchedWeight_on_rows x e) hf |
| |
| lemma quenchedWeight_le_one {d n : ℕ} (x : Fin n → Site d) (e : Fin n → Step d) |
| (ω : Environment d) : quenchedWeight x e ω ≤ 1 := by |
| apply Finset.prod_le_one |
| intro j hj |
| exact ENNReal.ofReal_le_one.mpr ((ω (x j)).property.1 (e j)).2 |
| |
| lemma wordWeight_le_one {d n : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ] |
| (x : Fin n → Site d) (e : Fin n → Step d) : wordWeight μ x e ≤ 1 := |
| lintegral_le_const (ae_of_all _ (quenchedWeight_le_one x e)) |
| |
| lemma wordWeight_pos {d n : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ] |
| (hμ : StrictEllipticity μ) (x : Fin n → Site d) (e : Fin n → Step d) : |
| 0 < wordWeight μ x e := by |
| apply pos_iff_ne_zero.mpr |
| intro hz |
| have hzero := (lintegral_eq_zero_iff (measurable_quenchedWeight x e)).mp hz |
| obtain ⟨ω, hω, hωzero⟩ := ((all_rows_strictly_positive μ hμ).and hzero).exists |
| have hp : 0 < quenchedWeight x e ω := by |
| apply pos_iff_ne_zero.mpr |
| apply Finset.prod_ne_zero_iff.mpr |
| intro j hj |
| exact ne_of_gt (ENNReal.ofReal_pos.mpr (hω (x j) (e j))) |
| exact (ne_of_gt hp) hωzero |
| |
| noncomputable def meanStep {d : ℕ} (μ : Measure (Row d)) (e : Step d) : ℝ≥0∞ := |
| ∫⁻ p, ENNReal.ofReal (p.val e) ∂μ |
| |
| lemma meanStep_pos {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ] |
| (hμ : StrictEllipticity μ) (e : Step d) : 0 < meanStep μ e := by |
| apply pos_iff_ne_zero.mpr |
| intro hz |
| have hm : Measurable (fun p : Row d => ENNReal.ofReal (p.val e)) := |
| ((measurable_pi_apply e).comp measurable_subtype_coe).ennreal_ofReal |
| have hzero := (lintegral_eq_zero_iff hm).mp hz |
| obtain ⟨p, hp, hpzero⟩ := (hμ.and hzero).exists |
| exact (ne_of_gt (ENNReal.ofReal_pos.mpr (hp e))) hpzero |
| |
| lemma meanStep_le_one {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ] |
| (e : Step d) : meanStep μ e ≤ 1 := by |
| apply lintegral_le_const |
| exact ae_of_all _ (fun p => ENNReal.ofReal_le_one.mpr (p.property.1 e).2) |
| |
| lemma meanStep_at_site {d : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ] |
| (x : Site d) (e : Step d) : |
| (∫⁻ ω, ENNReal.ofReal ((ω x).val e) ∂environmentLaw μ) = meanStep μ e := by |
| have hm : Measurable (fun p : Row d => ENNReal.ofReal (p.val e)) := |
| ((measurable_pi_apply e).comp measurable_subtype_coe).ennreal_ofReal |
| rw [← lintegral_map hm (measurable_pi_apply x)] |
| simp only [environmentLaw, Measure.infinitePi_map_eval, meanStep] |
| |
| lemma fresh_word_weight {d n : ℕ} (μ : Measure (Row d)) [IsProbabilityMeasure μ] |
| (x : Fin n → Site d) (e : Fin n → Step d) (hx : Function.Injective x) : |
| wordWeight μ x e = ∏ j, meanStep μ (e j) := by |
| have hi : iIndepFun (fun j : Fin n => fun ω : Environment d => ω (x j)) |
| (environmentLaw μ) := |
| iIndepFun.precomp hx |
| (iIndepFun_infinitePi (X := fun _ (p : Row d) => p) (fun _ => measurable_id)) |
| let f : Fin n → Row d → ℝ≥0∞ := fun j p => ENNReal.ofReal (p.val (e j)) |
| have hf : ∀ j, Measurable (f j) := fun j => |
| ((measurable_pi_apply (e j)).comp measurable_subtype_coe).ennreal_ofReal |
| have hid := hi.comp f hf |
| unfold wordWeight quenchedWeight |
| rw [lintegral_prod_eq_prod_lintegral_of_indepFun] |
| · exact Finset.prod_congr rfl (fun j hj => meanStep_at_site μ (x j) (e j)) |
| · exact hid |
| · intro j |
| exact (hf j).comp (measurable_pi_apply (x j)) |
| |
| end DirectionalZeroOne |
| |
| end OAI |
| |
