| import OAI.Probability.FreeIsing.Gaussian.InwardGaussianCutChannels |
| import OAI.Probability.FreeIsing.Trees.SelectedInwardPathAttenuation |
|
|
| namespace OAI |
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|
| noncomputable section |
| open MeasureTheory |
| open scoped NNReal |
|
|
| namespace Problem367.InwardCutChannels |
| open PathSideBranches |
|
|
| theorem integral_inwardContinuationAverage_subset_le_exp_critical |
| {d k : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) (hd : 3 ≤ d) |
| (hcritical : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1) |
| {t : ℝ≥0} (ht : 0 < t) (p : TreeShell d (k + 1)) (L : ℕ) |
| (s : Finset ℕ) (hs : ∀ i ∈ s, k + 1 ≤ i ∧ i < k + 1 + L) : |
| (∫ z, inwardContinuationAverage μ t (Real.tanh beta) p L s z |
| ∂μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)) ≤ |
| Real.exp (-(ComponentSignalAttenuation.lossCoefficient (Real.tanh beta) * |
| min (Real.sqrt (t : ℝ)) 1) * (s.card : ℝ)) := by |
| let := hμ.isProbabilityMeasure |
| have hθ : Real.tanh beta ^ 2 ≤ 1 := by |
| have hn := tanh_nonneg_of_beta_nonneg hbeta |
| have h1 := Real.tanh_lt_one beta |
| nlinarith |
| apply integral_inwardContinuationAverage_le (by omega) μ t hθ p L s _ |
| intro w |
| exact integral_cutInwardGaussianField_subset_attenuation_le_exp_critical |
| hμ hbeta hd hcritical ht (appendCutContinuation p w).1 (k + 1) (by omega) s |
| (by intro i hi; simpa only [(appendCutContinuation p w).2] using hs i hi) |
|
|
| theorem integral_inwardContinuationAverage_le_exp_critical |
| {d k : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) (hd : 3 ≤ d) |
| (hcritical : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1) |
| {t : ℝ≥0} (ht : 0 < t) (p : TreeShell d (k + 1)) (L : ℕ) : |
| (∫ z, inwardContinuationAverage μ t (Real.tanh beta) p L |
| (Finset.Ico (k + 1) (k + 1 + L)) z |
| ∂μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)) ≤ |
| Real.exp (-(ComponentSignalAttenuation.lossCoefficient (Real.tanh beta) * |
| min (Real.sqrt (t : ℝ)) 1) * (L : ℝ)) := by |
| simpa only [Nat.card_Ico, Nat.add_sub_cancel_left] using |
| integral_inwardContinuationAverage_subset_le_exp_critical hμ hbeta hd hcritical ht p L |
| (Finset.Ico (k + 1) (k + 1 + L)) (fun _ hi => Finset.mem_Ico.mp hi) |
|
|
| end Problem367.InwardCutChannels |
|
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| end |
|
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| end OAI |
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|
