| import OAI.Probability.FreeIsing.Ising.ActualVarianceAnalysis | |
| import OAI.Probability.FreeIsing.Ising.SharpVarianceRecovery | |
| import OAI.Probability.FreeIsing.Support | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory | |
| namespace Problem367 | |
| theorem free_ising_factor_below_threshold : | |
| ∀ (d : ℕ) (beta : ℝ), 3 ≤ d → 0 ≤ beta → Real.tanh beta ≤ 1 / Real.sqrt ((d : ℝ) - 1) → IsFactorOfIID d (Real.tanh beta) := by | |
| apply free_ising_factor_below_threshold_of_actual_variance_identity | |
| intro d beta hd hb hcrit T | |
| exact IsingObservationFactor.actual_variance_analysis hd | |
| (freeIsingMeasure_hasFreeIsingLaw d (Real.neg_one_lt_tanh beta).le | |
| (Real.tanh_lt_one beta).le) hb.le hcrit T | |
| theorem free_ising_factor_iff_threshold : | |
| ∀ (d : ℕ) (beta : ℝ), 3 ≤ d → 0 ≤ beta → (IsFactorOfIID d (Real.tanh beta) ↔ Real.tanh beta ≤ 1 / Real.sqrt ((d : ℝ) - 1)) := by | |
| intro d beta hd hbeta | |
| exact ⟨free_ising_factor_only_if_threshold d beta hd hbeta, | |
| free_ising_factor_below_threshold d beta hd hbeta⟩ | |
| end Problem367 | |
| end | |
| end OAI | |