| import OAI.Probability.FreeIsing.Trees.InfiniteCanonicalPathResponse |
| import OAI.Probability.FreeIsing.Trees.InfiniteConeCavity |
| import OAI.Probability.FreeIsing.Trees.SelectedPathAttenuation |
|
|
| namespace OAI |
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| noncomputable section |
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|
| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal BigOperators |
|
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| namespace Problem367 |
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| open PlantedSymmetry CavityFieldLimit PathSideBranches |
|
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| theorem HasFreeIsingLaw.rootResponseProduct_eq_forward_product |
| {d : ℕ} {β : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh β) μ) (hβ : 0 ≤ β) |
| (t : ℝ≥0) (v : TreeVertex d) : |
| ∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t), |
| rootResponseProduct hβ v (observation t p) = |
| Real.tanh β ^ v.1.length * ∏ k ∈ Finset.range v.1.length, |
| responseAttenuation (Real.tanh β) (forwardGaussianField μ t v k p) := by |
| have hfield (i : Fin v.1.length) := |
| hμ.infiniteCavityField_path_eq_forward_posterior hβ t v i i.2 |
| filter_upwards [ae_all_iff.mpr hfield] with p hp |
| unfold rootResponseProduct |
| congr 1 |
| apply Finset.prod_congr rfl |
| intro k hk |
| exact congrArg (responseAttenuation (Real.tanh β)) (hp ⟨k, Finset.mem_range.mp hk⟩) |
|
|
| theorem HasFreeIsingLaw.integral_rootResponseProduct_sq_le_critical |
| {d : ℕ} {β : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh β) μ) (hβ : 0 ≤ β) (hd : 3 ≤ d) |
| (hcritical : ((d - 1 : ℕ) : ℝ) * Real.tanh β ^ 2 = 1) |
| {t : ℝ≥0} (ht : 0 < t) (v : TreeVertex d) : |
| (∫ p, rootResponseProduct hβ v (observation t p) ^ 2 |
| ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ≤ |
| (Real.tanh β ^ v.1.length) ^ 2 * |
| Real.exp (-(ComponentSignalAttenuation.lossCoefficient (Real.tanh β) * |
| min (Real.sqrt (t : ℝ)) 1) * (v.1.length : ℝ)) := by |
| calc |
| _ = (Real.tanh β ^ v.1.length) ^ 2 * |
| ∫ p, ∏ k ∈ Finset.range v.1.length, |
| responseAttenuation (Real.tanh β) (forwardGaussianField μ t v k p) ^ 2 |
| ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t) := by |
| rw [← integral_const_mul] |
| apply integral_congr_ae |
| filter_upwards [hμ.rootResponseProduct_eq_forward_product hβ t v] with p hp |
| rw [hp, mul_pow, Finset.prod_pow] |
| _ ≤ _ := mul_le_mul_of_nonneg_left |
| (integral_forwardGaussianField_attenuation_le_exp_critical hμ hβ hd hcritical ht v) |
| (sq_nonneg _) |
|
|
| theorem HasFreeIsingLaw.integral_gaussianRootResponse_sq_le_critical |
| {d : ℕ} {β : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh β) μ) (hβ : 0 ≤ β) (hd : 3 ≤ d) |
| (hcritical : ((d - 1 : ℕ) : ℝ) * Real.tanh β ^ 2 = 1) |
| {t : ℝ≥0} (ht : 0 < t) (v : TreeVertex d) : |
| (∫ p, gaussianPosteriorResponse μ t (ConnectedHull.root d) v p ^ 2 |
| ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ≤ |
| (Real.tanh β ^ v.1.length) ^ 2 * |
| Real.exp (-(ComponentSignalAttenuation.lossCoefficient (Real.tanh β) * |
| min (Real.sqrt (t : ℝ)) 1) * (v.1.length : ℝ)) := by |
| have he := integral_congr_ae ((hμ.gaussian_rootResponse_eq_canonical_product hβ t v).fun_comp |
| (fun z : ℝ => z ^ 2)) |
| simp only [Function.comp_apply] at he |
| rw [he] |
| exact hμ.integral_rootResponseProduct_sq_le_critical hβ hd hcritical ht v |
|
|
| theorem HasFreeIsingLaw.integral_infiniteGibbsRootResponse_sq_le_critical |
| {d : ℕ} {β : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh β) μ) (hβ : 0 ≤ β) (hd : 3 ≤ d) |
| (hcritical : ((d - 1 : ℕ) : ℝ) * Real.tanh β ^ 2 = 1) |
| {t : ℝ≥0} (ht : 0 < t) (v : TreeVertex d) : |
| (∫ p, infiniteGibbsResponse β (ConnectedHull.root d) v (observation t p) ^ 2 |
| ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ≤ |
| (Real.tanh β ^ v.1.length) ^ 2 * |
| Real.exp (-(ComponentSignalAttenuation.lossCoefficient (Real.tanh β) * |
| min (Real.sqrt (t : ℝ)) 1) * (v.1.length : ℝ)) := by |
| have he := integral_congr_ae ((hμ.infiniteGibbsRootResponse_eq_canonical_product hβ t v).fun_comp |
| (fun z : ℝ => z ^ 2)) |
| simp only [Function.comp_apply] at he |
| rw [he] |
| exact hμ.integral_rootResponseProduct_sq_le_critical hβ hd hcritical ht v |
|
|
| end Problem367 |
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|
| end |
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| end OAI |
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|
