| import OAI.Probability.InvariantIsing.Fields.FieldCompensatedTangent |
| import OAI.Probability.InvariantIsing.Fields.FieldScalarChain |
|
|
| namespace OAI |
|
|
| /-! Identification of the affine family with the already constructed |
| finite scalar values and physical conditional means. -/ |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory IsingPerceptron |
| open scoped NNReal |
|
|
| namespace InvariantIsing |
|
|
| def fieldAffineIncrements (L : List FieldAffineStep) (t : ℝ) : List (ℝ × ℝ≥0) := |
| L.map fun av => (av.exponent, Real.toNNReal (av.base + av.slope * t)) |
|
|
| theorem fieldAffineValue_scalar (L : List FieldAffineStep) (t : ℝ) : |
| (fun z => fieldAffineValue L (t, z)) = |
| fieldScalarValue (fieldAffineIncrements L t) (fun z => Real.log (Real.cosh z)) := by |
| induction L with |
| | nil => rfl |
| | cons av L ih => |
| change gaussianTransform (av.base + av.slope * t) av.exponent |
| (fun z => fieldAffineValue L (t, z)) = _ |
| rw [field_gaussianTransform_eq_operator, ih] |
| rfl |
|
|
| theorem fieldAffineFamily_mean (I : Set ℝ) (hI : IsOpen I) (L : List FieldAffineStep) |
| (hL : ∀ av ∈ L, ∀ t ∈ I, 0 < av.base + av.slope * t) (t : ℝ) : |
| (fun z => (fieldAffineFamily I hI L hL).X (t, z)) = |
| fieldScalarMean (fieldAffineIncrements L t) (fun z => Real.log (Real.cosh z)) Real.tanh := by |
| induction L with |
| | nil => rfl |
| | cons av L ih => |
| let htail := fun bv hb => hL bv (List.mem_cons_of_mem av hb) |
| let G := fieldAffineFamily I hI L htail |
| have hU : (fun z => G.U (t, z)) = |
| fieldScalarValue (fieldAffineIncrements L t) (fun z => Real.log (Real.cosh z)) := by |
| rw [show G.U = fieldAffineValue L from fieldAffineFamily_value I hI L htail] |
| exact fieldAffineValue_scalar L t |
| have hX : (fun z => G.X (t, z)) = |
| fieldScalarMean (fieldAffineIncrements L t) (fun z => Real.log (Real.cosh z)) Real.tanh := |
| ih htail |
| change gaussianTiltAverage (av.base + av.slope * t) av.exponent |
| (fun z => G.U (t, z)) (fun z => G.X (t, z)) = _ |
| have hmU : Measurable (fun z => G.U (t, z)) := G.mU.comp (by fun_prop) |
| have hmX : Measurable (fun z => G.X (t, z)) := G.mX.comp (by fun_prop) |
| rw [field_gaussianTiltAverage_eq_transition _ _ |
| hmU hmX, hU, hX] |
| rfl |
|
|
| theorem fieldAffineFamily_stationary (I : Set ℝ) (hI : IsOpen I) (L : List FieldAffineStep) |
| (hL : ∀ av ∈ L, ∀ t ∈ I, 0 < av.base + av.slope * t) |
| (hs : ∀ av ∈ L, av.slope = 0) (p : ℝ × ℝ) : |
| (fieldAffineFamily I hI L hL).T p = 0 := by |
| induction L generalizing p with |
| | nil => rfl |
| | cons av L ih => |
| let htail := fun bv hb => hL bv (List.mem_cons_of_mem av hb) |
| let G := fieldAffineFamily I hI L htail |
| have hT : (fun y => G.T (p.1, y)) = fun _ => 0 := by |
| funext y |
| exact ih htail (fun bv hb => hs bv (List.mem_cons_of_mem av hb)) (p.1, y) |
| change G.tangent av.base av.slope av.exponent p = 0 |
| unfold FieldSmoothFamily.tangent |
| rw [hs av List.mem_cons_self, hT] |
| simp only [zero_mul, add_zero, zero_div] |
| have hmU : Measurable (fun z => G.U (p.1, z)) := G.mU.comp (by fun_prop) |
| exact field_gaussianTiltAverage_const _ _ hmU (G.growth p.1) 0 p.2 |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
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|
