| import OAI.Probability.FreeIsing.Estimates.ThresholdArithmetic |
| import OAI.Probability.FreeIsing.Factors.ObservableWord |
| import OAI.Probability.FreeIsing.Gaussian.BernoulliSpectral |
| import OAI.Probability.FreeIsing.Ising.WalkObstruction |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace Problem367 |
|
|
| theorem IsFactorOfIID.threshold_of_centeredAverage_norm {d : ℕ} {theta : ℝ} |
| (hd : 3 ≤ d) (hfactor : IsFactorOfIID d theta) |
| (hlo : -1 ≤ theta) (hhi : theta ≤ 1) |
| (hnorm : ‖TreeKoopman.centeredAverage d‖ ≤ RadialWalk.rate d) : |
| theta ≤ 1 / Real.sqrt ((d : ℝ) - 1) := by |
| obtain ⟨f, _, hcov⟩ := hfactor.exists_centered_word_observable hlo hhi |
| apply IsingWalk.threshold_of_operatorAverage_norm hd hhi |
| TreeKoopman.centeredGenerator TreeKoopman.centeredGenerator_sq f |
| · simpa only [TreeKoopman.centered_operatorAverage_eq] using hnorm |
| · exact hcov |
|
|
| theorem IsFactorOfIID.le_threshold {d : ℕ} {theta : ℝ} |
| (hd : 3 ≤ d) (hfactor : IsFactorOfIID d theta) |
| (hlo : -1 ≤ theta) (hhi : theta ≤ 1) : |
| theta ≤ 1 / Real.sqrt ((d : ℝ) - 1) := by |
| apply hfactor.threshold_of_centeredAverage_norm hd hlo hhi |
| exact TreeKoopman.centeredAverage_norm_le hd |
|
|
| theorem free_ising_factor_only_if_threshold (d : ℕ) (beta : ℝ) |
| (hd : 3 ≤ d) (hbeta : 0 ≤ beta) |
| (hfactor : IsFactorOfIID d (Real.tanh beta)) : |
| Real.tanh beta ≤ 1 / Real.sqrt ((d : ℝ) - 1) := by |
| apply hfactor.le_threshold hd |
| · exact le_trans (by norm_num) (tanh_nonneg_of_beta_nonneg hbeta) |
| · exact (Real.tanh_lt_one beta).le |
|
|
| end Problem367 |
|
|
| end |
|
|
| end OAI |
|
|
