| import OAI.Probability.InvariantIsing.Fields.FieldEndpointIntegrability |
| import OAI.Probability.InvariantIsing.Fields.FieldCanonicalVectorMoments |
|
|
| namespace OAI |
|
|
| /-! The biased Gaussian root followed by the full canonical recursion |
| has exactly the product endpoint law used in the terminal estimate. -/ |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory IsingPerceptron |
| open scoped BigOperators NNReal |
|
|
| namespace InvariantIsing |
|
|
| theorem fieldCanonicalVectorLaw_root (N : ℕ) (h : FieldStep) (b : Fin N → ℝ) : |
| fieldVectorTailEndpointKernel N (scalarFieldIncrements h) (scalarFieldIncrements_positive h) ∘ₘ |
| Measure.pi (fun i => gaussianReal (b i) (NNReal.mk (h.height 0) (h.nonneg 0))) = |
| fieldCanonicalVectorLaw N h b := by |
| rw [product_kernel_comp (fieldTailEndpointKernel (scalarFieldIncrements h) |
| (scalarFieldIncrements_positive h)) _ |
| (fieldVectorTailEndpointKernel_product N _ _)] |
| rfl |
|
|
| lemma fieldCanonicalEndpointLaw_integrable_id (h : FieldStep) (b : ℝ) : |
| Integrable (fun y : ℝ => y) (fieldCanonicalEndpointLaw h b) := by |
| have hp := (fieldCanonicalEndpointLaw_mgf h b 1).1 |
| have hn := (fieldCanonicalEndpointLaw_mgf h b (-1)).1 |
| apply (hp.add hn).mono' measurable_id.aestronglyMeasurable |
| apply ae_of_all |
| intro y |
| rw [Real.norm_eq_abs] |
| simp only [one_mul, neg_one_mul, Pi.add_apply, id_eq] |
| by_cases hy : 0 ≤ y |
| · rw [abs_of_nonneg hy] |
| linarith [Real.add_one_le_exp y, Real.exp_pos (-y)] |
| · rw [abs_of_nonpos (le_of_not_ge hy)] |
| linarith [Real.add_one_le_exp (-y), Real.exp_pos y] |
|
|
| lemma fieldCanonicalVectorLaw_linear_integrable (N : ℕ) (h : FieldStep) (b : Fin N → ℝ) |
| (F : (Fin N → ℝ) → ℝ) (hF : Measurable F) (hG : HasLinearGrowth F) : |
| Integrable F (fieldCanonicalVectorLaw N h b) := |
| integrable_linearGrowth_pi _ (fun i => fieldCanonicalEndpointLaw_integrable_id h (b i)) F hF hG |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
|
|
