| import OAI.Probability.FreeIsing.Model | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory | |
| namespace Problem367 | |
| def fairSpin (u : ℝ) : IsingSpin := | |
| if u ≤ 1 / 2 then IsingSpin.neg else IsingSpin.pos | |
| lemma measurable_fairSpin : Measurable fairSpin := by | |
| exact measurable_const.ite measurableSet_Iic measurable_const | |
| lemma fairSpin_neg_preimage : fairSpin ⁻¹' {IsingSpin.neg} = Set.Iic (1 / 2 : ℝ) := by | |
| ext u | |
| simp [fairSpin] | |
| lemma fairSpin_pos_preimage : fairSpin ⁻¹' {IsingSpin.pos} = Set.Ioi (1 / 2 : ℝ) := by | |
| ext u | |
| simp [fairSpin, not_le] | |
| lemma fairSpin_neg_mass : uniform01 (fairSpin ⁻¹' {IsingSpin.neg}) = 1 / 2 := by | |
| rw [fairSpin_neg_preimage, uniform01, Measure.restrict_apply measurableSet_Iic] | |
| have h : Set.Iic (1 / 2 : ℝ) ∩ Set.Icc 0 1 = Set.Icc 0 (1 / 2) := by | |
| ext u | |
| simp only [Set.mem_inter_iff, Set.mem_Iic, Set.mem_Icc] | |
| constructor | |
| · rintro ⟨hu, h0, h1⟩ | |
| exact ⟨h0, hu⟩ | |
| · rintro ⟨h0, hu⟩ | |
| exact ⟨hu, h0, by linarith⟩ | |
| rw [h, Real.volume_Icc] | |
| norm_num [ENNReal.ofReal_div_of_pos] | |
| lemma fairSpin_pos_mass : uniform01 (fairSpin ⁻¹' {IsingSpin.pos}) = 1 / 2 := by | |
| rw [fairSpin_pos_preimage, uniform01, Measure.restrict_apply measurableSet_Ioi] | |
| have h : Set.Ioi (1 / 2 : ℝ) ∩ Set.Icc 0 1 = Set.Ioc (1 / 2) 1 := by | |
| ext u | |
| simp only [Set.mem_inter_iff, Set.mem_Ioi, Set.mem_Icc, Set.mem_Ioc] | |
| constructor | |
| · rintro ⟨hu, h0, h1⟩ | |
| exact ⟨hu, h1⟩ | |
| · rintro ⟨hu, h1⟩ | |
| exact ⟨hu, by linarith, h1⟩ | |
| rw [h, Real.volume_Ioc] | |
| norm_num [ENNReal.ofReal_div_of_pos] | |
| lemma fairSpin_mass (a : IsingSpin) : uniform01 (fairSpin ⁻¹' {a}) = 1 / 2 := by | |
| cases a | |
| · exact fairSpin_neg_mass | |
| · exact fairSpin_pos_mass | |
| def fairSpinFactor (d : ℕ) (U : TreeVertex d → ℝ) : TreeVertex d → IsingSpin := | |
| fun v => fairSpin (U v) | |
| lemma measurable_fairSpinFactor (d : ℕ) : Measurable (fairSpinFactor d) := by | |
| exact Measurable.of_eval (fun vertex => measurable_fairSpin.comp (measurable_pi_apply vertex)) | |
| lemma fairSpinFactor_relabel (d : ℕ) (g : TreeAut d) (U : TreeVertex d → ℝ) : | |
| fairSpinFactor d (relabel g U) = relabel g (fairSpinFactor d U) := rfl | |
| private lemma fairSpin_measurable_singleton (a : IsingSpin) : MeasurableSet ({a} : Set IsingSpin) := by | |
| trivial | |
| lemma fairSpinFactor_cylinder (d : ℕ) (D : Finset (TreeVertex d)) | |
| (eta : TreeVertex d → IsingSpin) : | |
| (Measure.map (fairSpinFactor d) (labelMeasure d)) | |
| {x | ∀ v ∈ D, x v = eta v} = (1 / 2 : ENNReal) ^ D.card := by | |
| classical | |
| have hmeas : MeasurableSet {x : TreeVertex d → IsingSpin | ∀ v ∈ D, x v = eta v} := by | |
| exact MeasurableSet.pi D.countable_toSet (fun v hv => fairSpin_measurable_singleton (eta v)) | |
| rw [Measure.map_apply (measurable_fairSpinFactor d) hmeas] | |
| change (Measure.infinitePi (fun _ : TreeVertex d => uniform01)) | |
| (Set.pi (D : Set (TreeVertex d)) (fun v => fairSpin ⁻¹' {eta v})) = _ | |
| rw [Measure.infinitePi_pi _ (fun v _ => measurable_fairSpin (fairSpin_measurable_singleton (eta v)))] | |
| simp [fairSpin_mass] | |
| end Problem367 | |
| end | |
| end OAI | |