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b317718 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 | 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
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