| import OAI.Probability.FreeIsing.Estimates.FairSpin |
| import OAI.Probability.FreeIsing.Trees.Cardinality |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory |
| namespace Problem367 |
|
|
| lemma freeCylinderWeight_zero_of_card {d : ℕ} (D : Finset (TreeVertex d)) |
| (hcard : (inducedEdges D).card + 1 = D.card) (eta : TreeVertex d → IsingSpin) : |
| freeCylinderWeight 0 D eta = (1 / 2 : ENNReal) ^ D.card := by |
| simp only [freeCylinderWeight, neg_zero, ite_self, add_zero, Finset.prod_const] |
| rw [← pow_succ', hcard, ENNReal.ofReal_pow (by norm_num : (0 : ℝ) ≤ 1 / 2)] |
| norm_num [ENNReal.ofReal_div_of_pos] |
|
|
| lemma fairSpinFactor_hasFreeIsingLaw_of_card {d : ℕ} |
| (hcard : ∀ D : Finset (TreeVertex d), ConnectedFinset D → |
| (inducedEdges D).card + 1 = D.card) : |
| HasFreeIsingLaw 0 (Measure.map (fairSpinFactor d) (labelMeasure d)) := by |
| intro D hD eta |
| rw [fairSpinFactor_cylinder, freeCylinderWeight_zero_of_card D (hcard D hD)] |
|
|
| lemma isFactorOfIID_zero_of_card (d : ℕ) |
| (hcard : ∀ D : Finset (TreeVertex d), ConnectedFinset D → |
| (inducedEdges D).card + 1 = D.card) : IsFactorOfIID d 0 := by |
| refine ⟨fairSpinFactor d, measurable_fairSpinFactor d, |
| fairSpinFactor_hasFreeIsingLaw_of_card hcard, ?_⟩ |
| intro g |
| exact Filter.Eventually.of_forall (fairSpinFactor_relabel d g) |
|
|
| theorem isFactorOfIID_zero (d : ℕ) : IsFactorOfIID d 0 := by |
| exact isFactorOfIID_zero_of_card d (fun _ hD => hD.inducedEdges_card_add_one) |
|
|
| theorem free_ising_factor_beta_zero (d : ℕ) : IsFactorOfIID d (Real.tanh 0) := by |
| simpa only [Real.tanh_zero] using isFactorOfIID_zero d |
|
|
| end Problem367 |
|
|
| end |
|
|
| end OAI |
|
|
