| import Mathlib |
| import OAI.Probability.SKGap.Stability.FieldIntegralReduction |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Matrix Real Set MeasureTheory ProbabilityTheory GaussianDensity |
| open scoped BigOperators Matrix.Norms.Frobenius ENNReal |
| |
| instance conditionalGoeLaw_probability {ι : Type*} [Fintype ι] [DecidableEq ι] [Nonempty ι] |
| (r v : ℝ) (m z : ι → ℝ) : IsProbabilityMeasure (conditionalGoeLaw r v m z) := by |
| unfold conditionalGoeLaw |
| infer_instance |
| |
| lemma conditionalFieldWeight_le_one {n : ℕ} [NeZero n] (j t σ : ℝ) |
| (E : Set (((Fin n×Fin n) → ℝ) × (Fin n → ℝ))) (y : Fin n → ℝ) : |
| conditionalFieldWeight j t σ E y ≤ 1 := prob_le_one |
| |
| lemma conditionalFieldWeight_of_real_bound {n : ℕ} [NeZero n] {j t σ c : ℝ} |
| (E : Set (((Fin n×Fin n) → ℝ) × (Fin n → ℝ))) (y : Fin n → ℝ) |
| (hb : (conditionalGoeLaw (j/(n:ℝ)) (t+σ^2) (magnetization y) (empiricalObservation j t σ y)).real |
| {W | (W,y)∈E} ≤ c) : conditionalFieldWeight j t σ E y ≤ ENNReal.ofReal c := by |
| rw [← ENNReal.ofReal_toReal (show conditionalFieldWeight j t σ E y ≠ ⊤ from measure_ne_top _ _)] |
| exact ENNReal.ofReal_le_ofReal hb |
| |
| lemma measurable_scalarIntegrand {n : ℕ} [NeZero n] (j t σ : ℝ) : |
| Measurable (scalarIntegrand (ι := Fin n) j t σ) := by |
| unfold scalarIntegrand productNormal scalarEmpirical scalarPsi scalarA scalarD scalarS scalarM scalarQMoment |
| simp only [integral_empiricalLaw] |
| unfold gaussianPDFReal |
| fun_prop |
| |
| lemma scalar_density_prefactor_le_one {n : ℕ} [NeZero n] {j t σ : ℝ} |
| (hj : 0 ≤ j) (ht : 0 ≤ t) (hσ : 0 < σ) (y : Fin n → ℝ) : |
| sqrt (scalarS j (scalarEmpirical y t σ)/ |
| (scalarS j (scalarEmpirical y t σ)+j*scalarQMoment (empiricalLaw y))) ≤ 1 := by |
| have hq : 0 ≤ j*scalarQMoment (empiricalLaw y) := mul_nonneg hj (scalarQMoment_bounds _).1 |
| have hs : 0 < scalarS j (scalarEmpirical y t σ) := add_pos_of_pos_of_nonneg |
| (add_pos_of_nonneg_of_pos ht (sq_pos_of_pos hσ)) hq |
| exact sqrt_le_one.mpr ((div_le_one (add_pos_of_pos_of_nonneg hs hq)).mpr (le_add_of_nonneg_right hq)) |
| |
| lemma scalarIntegrand_nonneg {n : ℕ} [NeZero n] (j t σ : ℝ) (y : Fin n → ℝ) : |
| 0 ≤ scalarIntegrand j t σ y := mul_nonneg (exp_pos _).le |
| (Finset.prod_nonneg (fun i _=>gaussianPDFReal_nonneg _ _ (y i))) |
| |
| lemma scalar_weighted_density_bound {n : ℕ} [NeZero n] {j t σ : ℝ} |
| (hj : 0 ≤ j) (ht : 0 ≤ t) (hσ : 0 < σ) (y : Fin n → ℝ) {w b : ℝ≥0∞} |
| (hw : w ≤ b) : |
| ENNReal.ofReal (sqrt (scalarS j (scalarEmpirical y t σ)/ |
| (scalarS j (scalarEmpirical y t σ)+j*scalarQMoment (empiricalLaw y)))*scalarIntegrand j t σ y)*w ≤ |
| ENNReal.ofReal (scalarIntegrand j t σ y)*b := by |
| apply mul_le_mul' _ hw |
| apply ENNReal.ofReal_le_ofReal |
| exact mul_le_of_le_one_left (scalarIntegrand_nonneg j t σ y) (scalar_density_prefactor_le_one hj ht hσ y) |
| |
| end SKGap |
| end |
| end |
| |
| end OAI |
| |
