| import Mathlib |
| import OAI.Probability.SKGap.Localization.ConditionalWellDefined |
| import OAI.Probability.SKGap.Localization.ConditionalHessianLaw |
| import OAI.Probability.SKGap.Gaussian.GaussianTiltMass |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Matrix Real Set MeasureTheory ProbabilityTheory |
| open scoped BigOperators Matrix.Norms.Frobenius ENNReal |
|
|
| theorem conditional_field_stability {j A R t0 T : ℝ} |
| (hj : 0 < j) (hj1 : j < 1) (hA : 1 ≤ A) (hsub : sqrt j*A < 1) |
| (hR : 0 ≤ R) (ht0 : 0 < t0) : |
| ∃ m ν σ0 : ℝ, 0 < m ∧ 0 < ν ∧ 0 < σ0 ∧ |
| ∃ U : Set (ScalarTimeDomain R t0 T), IsOpen U ∧ scalarEqualitySet j R t0 T ⊆ U ∧ |
| ∀ R1 : ℝ, 1 ≤ R1 → ∃ ε c : ℝ, ∃ N : ℕ, 0 < ε ∧ 0 < c ∧ 0 < N ∧ |
| ∀ (n : ℕ) [NeZero n], N ≤ n → ∀ x ∈ U, ∀ y : Fin n → ℝ, |
| empiricalLaw y=(x.1 : ProbabilityMeasure ℝ) → |
| (∑ i,if R1 < |y i| then y i^2 else 0) ≤ ν*(n:ℝ) → |
| ∀ σ : ℝ, |σ| ≤ σ0 → |
| (conditionalGoeLaw (j/(n:ℝ)) ((x.2:ℝ)+σ^2) (magnetization y) |
| (empiricalObservation j x.2 σ y)).real |
| {W | ((fun i k=>W (i,k)),y) ∈ badFieldSet n j A ε m} ≤ exp (-c*(n:ℝ)) := by |
| obtain ⟨m,ν,σ0,hm,hν,hσ0,U,hU,hKU,hbound⟩ := |
| conditional_empirical_simultaneous (T := T) hj hj1 hA hsub hR ht0 |
| let V : Set (ScalarTimeDomain R t0 T) := {x | 0 < scalarQMoment (x.1:ProbabilityMeasure ℝ)} |
| have hV : IsOpen V := isOpen_lt continuous_const |
| (continuous_scalarQMoment.comp (continuous_subtype_val.comp continuous_fst)) |
| have hKV : scalarEqualitySet j R t0 T ⊆ V := by |
| intro x hx |
| change 0 < scalarQMoment (x.1:ProbabilityMeasure ℝ) |
| rw [hx,scalarQMoment_curve hj hj1 (ht0.le.trans x.2.2.1)] |
| exact (scalar_fixed_pos hj hj1 (ht0.trans_le x.2.2.1)).2 |
| refine ⟨m/2,ν,σ0,half_pos hm,hν,hσ0,U∩V,hU.inter hV,fun x hx=>⟨hKU hx,hKV hx⟩,?_⟩ |
| intro R1 hR1 |
| obtain ⟨ε,c,N,hε,hc,hN,hb⟩ := hbound R1 hR1 |
| obtain ⟨M,hM⟩ := absorb_finite_exponential_factor 103 (half_pos hc) |
| refine ⟨ε,c/2,max N M,hε,half_pos hc,lt_of_lt_of_le hN (le_max_left _ _),?_⟩ |
| intro n _ hn x hx y hP hy σ hσ |
| have hq : 0 < scalarQMoment (empiricalLaw y) := by rw [hP]; exact hx.2 |
| apply (conditional_bad_le_freshbad hj.le (ht0.trans_le x.2.2.1) (by linarith : m/2 < m) y hq).trans |
| apply (hb n ((le_max_left N M).trans hn) x hx.1 y hP hy σ hσ).trans |
| simpa only [Nat.cast_ofNat,show (n:ℝ)*(-c)= -c*(n:ℝ) from by ring, |
| show (n:ℝ)*(-c+c/2)= -(c/2)*(n:ℝ) from by ring] using |
| hM n ((le_max_right N M).trans hn) (-c) |
| end SKGap |
| end |
| end |
|
|
| end OAI |
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|
