| import OAI.Probability.InvariantIsing.Cavity.CavityMagneticBlockModulus |
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| namespace OAI |
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| /-! L1 replacement transports the physical constrained overlap tests |
| to the magnetic block paths used in the variational problem. -/ |
| noncomputable section |
| open MeasureTheory ProbabilityTheory IsingPerceptron Filter |
| open scoped Topology BoundedContinuousFunction |
| namespace InvariantIsing |
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|
| lemma integrable_overlap_weighted_test (q a : OverlapPath) (Φ : ℝ →ᵇ ℝ) : |
| Integrable (fun s => Φ (q s)*(a s-q s)) pathMeasure := by |
| apply integrable_of_measurable_abs_le |
| ((Φ.continuous.measurable.comp q.measurable).mul (a.measurable.sub q.measurable)) |
| intro s |
| change |Φ (q s)*(a s-q s)| ≤ ‖Φ‖ |
| rw [abs_mul] |
| have hd : |a s-q s| ≤ 1 := abs_le.mpr |
| ⟨by linarith [a.nonneg s,q.le_one s], by linarith [a.le_one s,q.nonneg s]⟩ |
| exact (mul_le_mul (Φ.norm_coe_le_norm _) hd (abs_nonneg _) (norm_nonneg Φ)).trans_eq (mul_one _) |
|
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| lemma overlap_weighted_test_sub_le (q a b : OverlapPath) (Φ : ℝ →ᵇ ℝ) : |
| |(∫ s, Φ (q s)*(a s-q s) ∂pathMeasure) - |
| ∫ s, Φ (q s)*(b s-q s) ∂pathMeasure| ≤ |
| ‖Φ‖ * ∫ s, |a s-b s| ∂pathMeasure := by |
| rw [← integral_sub (integrable_overlap_weighted_test q a Φ) |
| (integrable_overlap_weighted_test q b Φ)] |
| have he : (fun s => Φ (q s)*(a s-q s)-Φ (q s)*(b s-q s)) = |
| (fun s => Φ (q s)*(a s-b s)) := by funext s; ring |
| rw [he] |
| calc |
| _ ≤ ∫ s, |Φ (q s)*(a s-b s)| ∂pathMeasure := abs_integral_le_integral_abs |
| _ ≤ ∫ s, ‖Φ‖ * |a s-b s| ∂pathMeasure := by |
| apply integral_mono |
| · have hi := (integrable_overlap_weighted_test q a Φ).sub |
| (integrable_overlap_weighted_test q b Φ) |
| change Integrable (fun s => Φ (q s)*(a s-q s)-Φ (q s)*(b s-q s)) pathMeasure at hi |
| rw [he] at hi |
| exact hi.abs |
| · exact (a.integrable.sub b.integrable).abs.const_mul _ |
| · intro s |
| change |Φ (q s)*(a s-b s)| ≤ ‖Φ‖ * |a s-b s| |
| rw [abs_mul] |
| exact mul_le_mul_of_nonneg_right (Φ.norm_coe_le_norm _) (abs_nonneg _) |
| _ = _ := integral_const_mul _ _ |
|
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| theorem overlap_weighted_test_l1_transport (q a b : ℕ → OverlapPath) |
| (hab : Tendsto (fun n => ∫ s, |a n s-b n s| ∂pathMeasure) atTop (𝓝 0)) |
| (Φ : ℝ →ᵇ ℝ) : |
| Tendsto (fun n => (∫ s, Φ (q n s)*(a n s-q n s) ∂pathMeasure) - |
| ∫ s, Φ (q n s)*(b n s-q n s) ∂pathMeasure) atTop (𝓝 0) := by |
| apply squeeze_zero_norm (fun n => ?_) (by simpa using hab.const_mul ‖Φ‖) |
| simpa only [Real.norm_eq_abs] using overlap_weighted_test_sub_le (q n) (a n) (b n) Φ |
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| end InvariantIsing |
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| end |
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| end OAI |
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