| import OAI.Probability.FreeIsing.Estimates.FiniteIsingZeroField |
| import OAI.Probability.FreeIsing.Gaussian.FiniteIsingGaussianTesting |
| import OAI.Probability.FreeIsing.Posterior.GaussianEnergy |
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|
| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal NNReal BigOperators |
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| namespace Problem367 |
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| local instance finiteIsingGaussianEnergyNonemptySpin : Nonempty IsingSpin := ⟨.neg⟩ |
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| theorem HasFreeIsingLaw.hasDerivAt_finiteIsingGaussianEnergy |
| {d : ℕ} {beta : ℝ} {mu : Measure (TreeVertex d → IsingSpin)} |
| (hmu : HasFreeIsingLaw (Real.tanh beta) mu) |
| (D : Finset (TreeVertex d)) (hD : ConnectedFinset D) (v : D) |
| {t : ℝ} (ht : 0 < t) : |
| let spin : (D → IsingSpin) → D → ℝ := fun eta w => spinSign (eta w) |
| let rho := (mu.map D.restrict).prod (Measure.pi (fun _ : D => gaussianReal 0 1)) |
| HasDerivAt |
| (fun r => ∫ x, FinitePosterior.fieldMean (finiteIsingEnergy beta D) spin |
| (fun eta => spinSign (eta v)) |
| (FinitePosterior.finiteStandardObservation spin r x) ^ 2 ∂rho) |
| (∫ x, ∑ w, FinitePosterior.fieldCovariance (finiteIsingEnergy beta D) spin |
| (fun eta => spinSign (eta v)) |
| (FinitePosterior.finiteStandardObservation spin t x) w ^ 2 ∂rho) t := by |
| classical |
| let : IsProbabilityMeasure mu := hmu.isProbabilityMeasure |
| apply FinitePosterior.hasDerivAt_finiteStandardEnergy |
| (mu.map D.restrict) (finiteIsingEnergy beta D) |
| (hmu.restrict_real_eq_normalized_exp D hD) |
| (fun eta w => spinSign (eta w)) (fun eta => spinSign (eta v)) 1 |
| · intro eta |
| cases eta v <;> norm_num [spinSign] |
| · intro eta w |
| cases eta w <;> simp [spinSign] |
| · exact ht |
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| theorem finiteIsingGaussianEnergy_zero |
| {d : ℕ} (beta : ℝ) (D : Finset (TreeVertex d)) (v : D) |
| (rho : Measure ((D → IsingSpin) × (D → ℝ))) : |
| (∫ x, FinitePosterior.fieldMean (finiteIsingEnergy beta D) |
| (fun (eta : D → IsingSpin) w => spinSign (eta w)) |
| (fun eta => spinSign (eta v)) |
| (FinitePosterior.finiteStandardObservation |
| (fun (eta : D → IsingSpin) w => spinSign (eta w)) 0 x) ^ 2 ∂rho) = 0 := by |
| have hz := finiteIsingPosterior_spin_zero beta D v |
| change FinitePosterior.fieldMean (finiteIsingEnergy beta D) |
| (fun (eta : D → IsingSpin) w => spinSign (eta w)) |
| (fun eta => spinSign (eta v)) 0 = 0 at hz |
| simp [FinitePosterior.finiteStandardObservation, FinitePosterior.gaussianObservationCurve, hz] |
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| end Problem367 |
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| end |
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| end OAI |
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