| import Mathlib |
| import OAI.Probability.SKRatio.Matrices.NormalizedIncrement |
| import OAI.Probability.SKRatio.Matrices.FiniteTimeMesh |
| import OAI.Probability.SKRatio.Matrices.DetMultilinear |
| import OAI.Probability.SKRatio.Matrices.RealSymmetricResolventUnit |
| import OAI.Probability.SKRatio.Matrices.ScaledDiagonalExpectedMargin |
| import OAI.Probability.SKRatio.Gaussian.EmpiricalIntervals |
| import OAI.Probability.SKRatio.Variational.ScalarTransfer |
| import OAI.Probability.SKRatio.Certificates.FinalScalar |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open scoped Topology |
| open MeasureTheory Filter |
| namespace SKRatio |
|
|
| theorem ratio_cutoff (β ε η : ℝ) |
| (hβ0 : 0 ≤ β) (hβhalf : β < 1 / 2) |
| (hε0 : 0 < ε) (hεhalf : ε < 1 / 2) (hη : 0 < η) : |
| Tendsto |
| (fun n : ℕ => disorderLaw β n |
| {g : Disorder n | 1 + η < |
| (mixingTime g ε : ℝ) / (mixingTime g (1 - ε) : ℝ)}) |
| atTop (𝓝 0) ∧ |
| (∀ᶠ n : ℕ in atTop, ∀ g : Disorder n, 0 < mixingTime g (1 - ε)) := by |
| by_cases hsmall : β ≤ 7/20 |
| · exact ratio_cutoff_small_temperature β ε η hβ0 hsmall hε0 hεhalf hη |
| · exact Planted.ratio_cutoff_of_scalar β ε η (9969/10000) |
| (by linarith) hβhalf hε0 hεhalf hη (by norm_num) |
| (fun _ _ hb hr hn => Certificate.variational_bound |
| (by linarith) hβhalf.le hb hr hn) |
|
|
| end SKRatio |
|
|
| end |
| end |
|
|
| end OAI |
|
|
