| import OAI.Probability.InvariantIsing.Magnetic.MagneticCanonicalSpin |
|
|
| namespace OAI |
|
|
| /-! The one-site canonical endpoint law and its final spin draw. The |
| ordinary root Gaussian is retained explicitly. -/ |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory IsingPerceptron |
| open scoped NNReal |
|
|
| namespace InvariantIsing |
|
|
| def fieldTailEndpointKernel : (L : List (ℝ × ℝ≥0)) → |
| (∀ av ∈ L, 0 < av.1) → Kernel ℝ ℝ |
| | [], _ => Kernel.id |
| | av :: L, hL => |
| let ht := fun bv hb => hL bv (List.mem_cons_of_mem av hb) |
| let hreg := fieldScalarValue_regular L ht measurable_logCosh logCosh_linearGrowth |
| fieldTailEndpointKernel L ht ∘ₖ |
| fieldTransitionKernel av.1 av.2 |
| (fieldScalarValue L (fun z => Real.log (Real.cosh z))) hreg.1 |
| |
| instance fieldTailEndpointKernel_markov (L : List (ℝ × ℝ≥0)) |
| (hL : ∀ av ∈ L, 0 < av.1) : IsMarkovKernel (fieldTailEndpointKernel L hL) := by |
| induction L with |
| | nil => change IsMarkovKernel Kernel.id; infer_instance |
| | cons av L ih => |
| have ht := fun bv hb => hL bv (List.mem_cons_of_mem av hb) |
| let _ := ih ht |
| change IsMarkovKernel (_ ∘ₖ _) |
| infer_instance |
| |
| lemma fieldTailSpinKernel_endpoint (L : List (ℝ × ℝ≥0)) |
| (hL : ∀ av ∈ L, 0 < av.1) : |
| fieldTailSpinKernel L hL = scalarSpinKernel ∘ₖ fieldTailEndpointKernel L hL := by |
| induction L with |
| | nil => simp [fieldTailSpinKernel, fieldTailEndpointKernel] |
| | cons av L ih => |
| have ht := fun bv hb => hL bv (List.mem_cons_of_mem av hb) |
| change fieldTailSpinKernel L ht ∘ₖ _ = scalarSpinKernel ∘ₖ (fieldTailEndpointKernel L ht ∘ₖ _) |
| rw [ih ht, Kernel.comp_assoc] |
| |
| def fieldCanonicalEndpointLaw (h : FieldStep) (b : ℝ) : Measure ℝ := |
| fieldTailEndpointKernel (scalarFieldIncrements h) (scalarFieldIncrements_positive h) ∘ₘ |
| gaussianReal b (NNReal.mk (h.height 0) (h.nonneg 0)) |
| |
| instance fieldCanonicalEndpointLaw_probability (h : FieldStep) (b : ℝ) : |
| IsProbabilityMeasure (fieldCanonicalEndpointLaw h b) := by |
| unfold fieldCanonicalEndpointLaw |
| infer_instance |
| |
| lemma fieldTransitionKernel_zero (v : ℝ≥0) (F : ℝ → ℝ) (hF : Measurable F) |
| (hG : HasLinearGrowth F) (z : ℝ) : |
| fieldTransitionKernel 0 v F hF z = gaussianReal z v := by |
| rw [fieldTransitionKernel_eq_tilted 0 v F hF hG] |
| simp |
| |
| theorem fieldCanonicalSpinLaw_endpoint (h : FieldStep) (b : ℝ) : |
| fieldCanonicalSpinLaw h b = scalarSpinKernel ∘ₘ fieldCanonicalEndpointLaw h b := by |
| have hreg := fieldScalarValue_regular (scalarFieldIncrements h) |
| (scalarFieldIncrements_positive h) measurable_logCosh logCosh_linearGrowth |
| rw [fieldCanonicalEndpointLaw, Measure.comp_assoc, ← fieldTailSpinKernel_endpoint] |
| rw [fieldCanonicalSpinLaw, Kernel.comp_apply, fieldTransitionKernel_zero _ _ hreg.1 hreg.2] |
| |
| end InvariantIsing |
| |
| end |
| |
| end OAI |
| |
