File size: 2,852 Bytes
c4cdbbd | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 | 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
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