| import Mathlib |
| import OAI.Probability.SKRatio.Dynamics.Assembly |
| import OAI.Probability.SKRatio.Entropy.BadAvoidance |
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|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open scoped BigOperators Topology Matrix |
| open MeasureTheory Filter |
| namespace SKRatio.Calculus |
| attribute [local instance] Classical.propDecidable |
|
|
| theorem ratio_cutoff_of_static_inputs (β ε η : ℝ) |
| (hε : 0 < ε) (hε1 : ε < 1) (hη : 0 < η) |
| (G : ∀ n : ℕ, Set (Disorder n)) |
| (bad : ∀ n, Disorder n → Set (Spin n)) {κ A χ γ K : ℝ} |
| (hκ : 0 < κ) (hA : 0 ≤ A) (hχ : 0 < χ) (hγ : 0 < γ) (hK : 0 < K) |
| (hG : Tendsto (fun n => disorderLaw β n (G n)ᶜ) atTop (𝓝 0)) |
| (hnorm : ∀ n, ∀ g ∈ G n, euclideanOpNorm (coupling g) ≤ K) |
| (hgap : ∀ n, ∀ g ∈ G n, ∀ f : Observables n, |
| γ*FiniteLaw.variance (mass g 0) f ≤ stationaryEnergy g f) |
| (hmlsi : ∀ n, ∀ g ∈ G n, ∀ F : Observables n, |
| (∀ y, 0 < F y) → FiniteLaw.mean (mass g 0) F = 1 → |
| χ*FiniteLaw.entropy (mass g 0) F ≤ |
| -stationaryInner g (generator (coupling g) F) (fun y => Real.log (F y))) |
| (hsmall : ∀ n, ∀ g ∈ G n, FiniteLaw.mean (mass g 0) |
| (fun y => if y ∈ bad n g then 1 else 0) ≤ Real.exp (-κ*n)) |
| (hgradient : ∀ n, ∀ g ∈ G n, ∀ f : Observables n, ∀ y, |
| (∑ i, 2*halfDiff i f y*halfDiff i (generator (coupling g) f) y) - |
| generator (coupling g) (unweightedGradient f) y ≤ |
| (-κ + if y ∈ bad n g then A else 0)*unweightedGradient f y) : |
| 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 |
| have hdrift (n : ℕ) (g : Disorder n) (hg : g ∈ G n) (f : Observables n) |
| (r : ℝ) (_hr : 0 ≤ r) (y : Spin n) : |
| deriv (fun v => unweightedGradient (semigroup (coupling g) v f) y) r - |
| generator (coupling g) (unweightedGradient (semigroup (coupling g) r f)) y ≤ |
| (-κ + if y ∈ bad n g then A else 0)* |
| unweightedGradient (semigroup (coupling g) r f) y := by |
| rw [(unweightedGradient_hasDerivAt (coupling g) f y r).deriv] |
| exact hgradient n g hg (semigroup (coupling g) r f) y |
| obtain ⟨d,c,hd,hc,havoid⟩ := uniform_bad_avoidance G bad hκ hA hχ hK.le hnorm hmlsi hsmall hdrift |
| have hav := superpolynomial_bad_avoidance G bad hc havoid |
| have hgp : ∀ᶠ n : ℕ in atTop, ∀ g ∈ G n, ∀ f : Observables n, |
| γ*FiniteLaw.variance (mass g 0) f ≤ stationaryEnergy g f := Eventually.of_forall hgap |
| have hgd (D : ℝ) (_hD : 0 < D) : ∀ᶠ n : ℕ in atTop, ∀ g ∈ G n, |
| ∀ f : Observables n, ∀ r ∈ Set.Icc (0:ℝ) (D*Real.log n), ∀ y, |
| deriv (fun v => unweightedGradient (semigroup (coupling g) v f) y) r - |
| generator (coupling g) (unweightedGradient (semigroup (coupling g) r f)) y ≤ |
| (-κ + if y ∈ bad n g then A else 0)* |
| unweightedGradient (semigroup (coupling g) r f) y := |
| Eventually.of_forall (fun n g hg f r hr => hdrift n g hg f r hr.1) |
| have hrow : ∀ᶠ n : ℕ in atTop, ∀ g ∈ G n, ∀ i, ∑ j, coupling g i j ^ 2 ≤ K^2 := by |
| apply Eventually.of_forall |
| intro n g hg i |
| exact row_squares_le_opNorm (coupling g) (coupling_symm g) hK.le (hnorm n g hg) i |
| have hfields := uniform_field_tails_of_small_set G bad hκ hA hd.le hγ (sq_nonneg K) hrow hgp hgd hav |
| have hlag := uniform_positive_lag_of_small_set_gradient G bad hκ hA hd.le hγ hgp hgd hav hfields |
| have hlo := uniform_median_lower_of_small_set G bad hκ hA hd.le hγ hgp hgd hav |
| have hup := uniform_median_upper_of_mlsi G hχ hnorm hmlsi |
| apply ratio_cutoff_of_positive_lag β ε η hε hε1 hη G (c := 1/4) (C := 2/χ) |
| (by norm_num) hK hnorm hG _ hlag |
| filter_upwards [hlo,hup] with n hnlo hnup |
| intro g hg |
| exact ⟨by simpa only [one_div_mul_eq_div] using hnlo g hg,hnup g hg⟩ |
|
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| end SKRatio.Calculus |
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| end |
| end |
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| end OAI |
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