| import OAI.Probability.InvariantIsing.Fields.FieldTailMgf |
| import OAI.Probability.InvariantIsing.Fields.FieldFiniteDomain |
| import OAI.Probability.InvariantIsing.Magnetic.MagneticFieldInverseConvex |
|
|
| namespace OAI |
|
|
| /-! The canonical terminal field has a depth-independent second moment, |
| controlled by the initial bias and the final covariance height. -/ |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory IsingPerceptron |
| open scoped NNReal BigOperators |
|
|
| namespace InvariantIsing |
|
|
| lemma fieldTailVariance_total (h : FieldStep) : |
| h.height 0 + fieldTailVariance (scalarFieldIncrements h) = |
| h.height (Fin.last h.depth) := by |
| have hs := fieldIncrement_sum h |
| rw [Fin.sum_univ_succ] at hs |
| have htail : fieldTailVariance (scalarFieldIncrements h) = |
| ∑ i : Fin h.depth, (fieldIncrement h i.succ).2 := by |
| simp only [fieldTailVariance, scalarFieldIncrements, List.map_ofFn, |
| Function.comp_apply, NNReal.coe_mk, List.sum_ofFn] |
| rw [htail] |
| simpa [fieldIncrement] using hs |
|
|
| theorem fieldCanonicalEndpointLaw_mgf (h : FieldStep) (b t : ℝ) : |
| Integrable (fun y => Real.exp (t * y)) (fieldCanonicalEndpointLaw h b) ∧ |
| (∫ y, Real.exp (t * y) ∂fieldCanonicalEndpointLaw h b) ≤ |
| Real.exp (t * b + (t ^ 2 / 2 + |t|) * h.height (Fin.last h.depth)) := by |
| let L := scalarFieldIncrements h |
| have hL := scalarFieldIncrements_positive h |
| have hB := scalarFieldIncrements_exponent_le_one h |
| have hreg := fieldScalarValue_regular L hL measurable_logCosh logCosh_linearGrowth |
| let root : ℝ≥0 := ⟨h.height 0, h.nonneg 0⟩ |
| let κ := fieldTransitionKernel 0 root |
| (fieldScalarValue L (fun y => Real.log (Real.cosh y))) hreg.1 |
| have hκ (z : ℝ) := And.intro |
| (fieldTransitionKernel_integrable_exp root z 0 t _ hreg.1 hreg.2) |
| (fieldScalarTransition_mgf L hL root z 0 t le_rfl zero_le_one) |
| have he := kernel_exp_bound_comp κ (fieldTailEndpointKernel L hL) t |
| ((t ^ 2 / 2 + |t|) * root) ((t ^ 2 / 2 + |t|) * fieldTailVariance L) |
| hκ (fieldTailEndpointKernel_mgf L hL hB t) b |
| have hlaw : (fieldTailEndpointKernel L hL ∘ₖ κ) b = fieldCanonicalEndpointLaw h b := by |
| rw [Kernel.comp_apply] |
| dsimp only [κ] |
| rw [fieldTransitionKernel_zero _ _ hreg.1 hreg.2] |
| rfl |
| rw [hlaw] at he |
| refine ⟨he.1, he.2.trans_eq ?_⟩ |
| congr 1 |
| change t * b + (t ^ 2 / 2 + |t|) * h.height 0 + |
| (t ^ 2 / 2 + |t|) * fieldTailVariance (scalarFieldIncrements h) = _ |
| calc |
| _ = t * b + (t ^ 2 / 2 + |t|) * |
| (h.height 0 + fieldTailVariance (scalarFieldIncrements h)) := by ring |
| _ = _ := by rw [fieldTailVariance_total] |
|
|
| lemma sq_le_exp_two_add (x : ℝ) : x ^ 2 ≤ Real.exp (2 * x) + Real.exp (-2 * x) := by |
| have ha : |x| ≤ Real.exp |x| := by linarith [Real.add_one_le_exp |x|] |
| have hs := (sq_le_sq₀ (abs_nonneg x) (Real.exp_pos _).le).mpr ha |
| rw [sq_abs] at hs |
| rw [show (Real.exp |x|) ^ 2 = Real.exp (|x| + |x|) by |
| rw [pow_two, Real.exp_add]] at hs |
| by_cases hx : 0 ≤ x |
| · rw [abs_of_nonneg hx, show x + x = 2 * x by ring] at hs |
| linarith [Real.exp_pos (-2 * x)] |
| · rw [abs_of_nonpos (le_of_not_ge hx), show -x + -x = -2 * x by ring] at hs |
| linarith [Real.exp_pos (2 * x)] |
|
|
| theorem fieldCanonicalEndpointLaw_second_moment (h : FieldStep) (b : ℝ) : |
| Integrable (fun y : ℝ => y ^ 2) (fieldCanonicalEndpointLaw h b) ∧ |
| (∫ y, y ^ 2 ∂fieldCanonicalEndpointLaw h b) ≤ |
| Real.exp (2 * b + 4 * h.height (Fin.last h.depth)) + |
| Real.exp (-2 * b + 4 * h.height (Fin.last h.depth)) := by |
| have hp := fieldCanonicalEndpointLaw_mgf h b 2 |
| have hn := fieldCanonicalEndpointLaw_mgf h b (-2) |
| have hi : Integrable (fun y : ℝ => y ^ 2) (fieldCanonicalEndpointLaw h b) := by |
| apply (hp.1.add hn.1).mono' (by fun_prop) |
| exact ae_of_all _ (fun y => by |
| rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)] |
| exact sq_le_exp_two_add y) |
| refine ⟨hi, ?_⟩ |
| have hle := integral_mono hi (hp.1.add hn.1) sq_le_exp_two_add |
| simp only [Pi.add_apply] at hle |
| rw [integral_add hp.1 hn.1] at hle |
| norm_num at hp hn |
| have hn' : (∫ y, Real.exp (-2 * y) ∂fieldCanonicalEndpointLaw h b) ≤ |
| Real.exp (-2 * b + 4 * h.height (Fin.last h.depth)) := by |
| simpa only [neg_mul] using hn.2 |
| exact hle.trans (add_le_add hp.2 hn') |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
|
|
