| import OAI.Probability.FreeIsing.Model | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory | |
| namespace Problem367 | |
| def IsingSpin.flip : IsingSpin → IsingSpin | |
| | .neg => .pos | |
| | .pos => .neg | |
| @[simp] theorem IsingSpin.flip_flip (s : IsingSpin) : s.flip.flip = s := by | |
| cases s <;> rfl | |
| theorem IsingSpin.flip_injective : Function.Injective IsingSpin.flip := | |
| Function.LeftInverse.injective IsingSpin.flip_flip | |
| @[simp] theorem IsingSpin.flip_eq_flip (s t : IsingSpin) : | |
| s.flip = t.flip ↔ s = t := IsingSpin.flip_injective.eq_iff | |
| @[simp] theorem IsingSpin.flip_eq_iff (s t : IsingSpin) : | |
| s.flip = t ↔ s = t.flip := by | |
| cases s <;> cases t <;> simp [IsingSpin.flip] | |
| theorem IsingSpin.measurable_flip : Measurable IsingSpin.flip := by | |
| intro s hs | |
| trivial | |
| def flipConfig {ι : Type*} (x : ι → IsingSpin) : ι → IsingSpin := | |
| fun v => (x v).flip | |
| @[simp] theorem flipConfig_apply {ι : Type*} (x : ι → IsingSpin) (v : ι) : | |
| flipConfig x v = (x v).flip := rfl | |
| @[simp] theorem flipConfig_flipConfig {ι : Type*} (x : ι → IsingSpin) : | |
| flipConfig (flipConfig x) = x := by | |
| funext v | |
| exact IsingSpin.flip_flip (x v) | |
| theorem measurable_flipConfig {ι : Type*} : | |
| Measurable (flipConfig : (ι → IsingSpin) → (ι → IsingSpin)) := by | |
| exact Measurable.of_eval (fun vertex => | |
| IsingSpin.measurable_flip.comp (measurable_pi_apply vertex)) | |
| def flipConfigMeasurableEquiv (ι : Type*) : (ι → IsingSpin) ≃ᵐ (ι → IsingSpin) where | |
| toFun := flipConfig | |
| invFun := flipConfig | |
| left_inv := flipConfig_flipConfig | |
| right_inv := flipConfig_flipConfig | |
| measurable_toFun := measurable_flipConfig | |
| measurable_invFun := measurable_flipConfig | |
| theorem integral_eq_zero_of_flipConfig_odd {ι : Type*} | |
| {mu : Measure (ι → IsingSpin)} (hmu : mu.map flipConfig = mu) | |
| (f : (ι → IsingSpin) → ℝ) (hf : ∀ x, f (flipConfig x) = -f x) : | |
| ∫ x, f x ∂mu = 0 := by | |
| have h := integral_map_equiv (μ := mu) (flipConfigMeasurableEquiv ι) f | |
| change (∫ x, f x ∂mu.map flipConfig) = ∫ x, f (flipConfig x) ∂mu at h | |
| rw [hmu] at h | |
| simp only [hf, integral_neg] at h | |
| linarith | |
| @[simp] theorem flipConfig_preimage_spin {ι : Type*} (v : ι) (s : IsingSpin) : | |
| flipConfig ⁻¹' {x : ι → IsingSpin | x v = s} = | |
| {x : ι → IsingSpin | x v = s.flip} := by | |
| ext x | |
| simp [flipConfig] | |
| theorem measurableSet_spin_event {ι : Type*} (v : ι) (s : IsingSpin) : | |
| MeasurableSet {x : ι → IsingSpin | x v = s} := | |
| (show MeasurableSet ({s} : Set IsingSpin) from trivial).preimage (measurable_pi_apply v) | |
| theorem measure_even_event_inter_spin {ι : Type*} {mu : Measure (ι → IsingSpin)} | |
| (hmu : mu.map flipConfig = mu) {A : Set (ι → IsingSpin)} | |
| (hA : MeasurableSet A) (hAflip : flipConfig ⁻¹' A = A) (v : ι) (s : IsingSpin) : | |
| mu (A ∩ {x | x v = s}) = mu (A ∩ {x | x v = s.flip}) := by | |
| calc | |
| mu (A ∩ {x | x v = s}) = (mu.map flipConfig) (A ∩ {x | x v = s}) := by rw [hmu] | |
| _ = mu (A ∩ {x | x v = s.flip}) := by | |
| rw [Measure.map_apply measurable_flipConfig (hA.inter (measurableSet_spin_event v s)), | |
| Set.preimage_inter, hAflip, flipConfig_preimage_spin] | |
| theorem twice_measure_even_event_inter_spin {ι : Type*} {mu : Measure (ι → IsingSpin)} | |
| (hmu : mu.map flipConfig = mu) {A : Set (ι → IsingSpin)} | |
| (hA : MeasurableSet A) (hAflip : flipConfig ⁻¹' A = A) (v : ι) (s : IsingSpin) : | |
| 2 * mu (A ∩ {x | x v = s}) = mu A := by | |
| have hunion : A = (A ∩ {x | x v = s}) ∪ (A ∩ {x | x v = s.flip}) := by | |
| ext x | |
| cases hs : x v <;> cases s <;> simp [hs, IsingSpin.flip] | |
| have hdisjoint : Disjoint (A ∩ {x | x v = s}) (A ∩ {x | x v = s.flip}) := by | |
| apply Set.disjoint_left.mpr | |
| intro x hx hy | |
| have heq : s = s.flip := hx.2.symm.trans hy.2 | |
| cases s <;> cases heq | |
| calc | |
| 2 * mu (A ∩ {x | x v = s}) = | |
| mu (A ∩ {x | x v = s}) + mu (A ∩ {x | x v = s.flip}) := by | |
| rw [← measure_even_event_inter_spin hmu hA hAflip v s, two_mul] | |
| _ = mu A := by | |
| rw [← measure_union hdisjoint (hA.inter (measurableSet_spin_event v s.flip)), ← hunion] | |
| theorem setIntegral_spin_eq_half_of_flipConfig_even {ι : Type*} | |
| {mu : Measure (ι → IsingSpin)} (hmu : mu.map flipConfig = mu) | |
| {f : (ι → IsingSpin) → ℝ} (hf : Integrable f mu) | |
| (heven : ∀ x, f (flipConfig x) = f x) (v : ι) (s : IsingSpin) : | |
| ∫ x in {x | x v = s}, f x ∂mu = (1 / 2 : ℝ) * ∫ x, f x ∂mu := by | |
| classical | |
| let A : Set (ι → IsingSpin) := {x | x v = s} | |
| have hA : MeasurableSet A := measurableSet_spin_event v s | |
| have htransport := integral_map_equiv (μ := mu) (flipConfigMeasurableEquiv ι) (A.indicator f) | |
| change (∫ x, A.indicator f x ∂mu.map flipConfig) = | |
| ∫ x, A.indicator f (flipConfig x) ∂mu at htransport | |
| rw [hmu] at htransport | |
| have hindicator : (fun x => A.indicator f (flipConfig x)) = Aᶜ.indicator f := by | |
| funext x | |
| cases hx : x v <;> cases s <;> | |
| simp [A, Set.indicator, flipConfig, hx, IsingSpin.flip, heven x] | |
| rw [hindicator, integral_indicator hA, integral_indicator hA.compl] at htransport | |
| have hsum := integral_add_compl hA hf | |
| change (∫ x in A, f x ∂mu) = (1 / 2 : ℝ) * ∫ x, f x ∂mu | |
| linarith | |
| @[simp] theorem freeCylinderWeight_flipConfig {d : ℕ} (theta : ℝ) | |
| (D : Finset (TreeVertex d)) (eta : TreeVertex d → IsingSpin) : | |
| freeCylinderWeight theta D (flipConfig eta) = freeCylinderWeight theta D eta := by | |
| simp [freeCylinderWeight, flipConfig] | |
| theorem HasFreeIsingLaw.map_flipConfig {d : ℕ} {theta : ℝ} | |
| {mu : Measure (TreeVertex d → IsingSpin)} (hmu : HasFreeIsingLaw theta mu) : | |
| HasFreeIsingLaw theta (mu.map flipConfig) := by | |
| intro D hD eta | |
| have hmeas : MeasurableSet {x : TreeVertex d → IsingSpin | ∀ v ∈ D, x v = eta v} := by | |
| simp only [Set.ofPred_forall] | |
| apply D.measurableSet_biInter | |
| intro v hv | |
| exact (show MeasurableSet ({eta v} : Set IsingSpin) from trivial).preimage | |
| (measurable_pi_apply v) | |
| rw [Measure.map_apply measurable_flipConfig hmeas] | |
| have hpre : flipConfig ⁻¹' {x : TreeVertex d → IsingSpin | ∀ v ∈ D, x v = eta v} = | |
| {x : TreeVertex d → IsingSpin | ∀ v ∈ D, x v = flipConfig eta v} := by | |
| ext x | |
| simp [flipConfig] | |
| rw [hpre, hmu D hD (flipConfig eta), freeCylinderWeight_flipConfig] | |
| end Problem367 | |
| end | |
| end OAI | |