| import OAI.Probability.FreeIsing.Gaussian.FiniteIsingGaussianEnergy |
| import OAI.Probability.FreeIsing.Stochastic.FiniteFieldEntropyIntegral |
| import OAI.Probability.FreeIsing.Trees.GibbsField |
|
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory |
| open scoped BigOperators NNReal |
|
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| namespace Problem367 |
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| local instance finiteIsingFieldEnergyNonemptySpin : Nonempty IsingSpin := ⟨.neg⟩ |
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| theorem finiteIsingEntropy_le_own_observation {d : ℕ} {beta : ℝ} |
| (hbeta : 0 ≤ beta) (D : Finset (TreeVertex d)) (v : D) (y : D → ℝ) : |
| FinitePosterior.fieldEntropy (finiteIsingEnergy beta D) |
| (fun (eta : D → IsingSpin) w => spinSign (eta w)) |
| (fun eta => spinSign (eta v)) y ≤ 2 * (|y v| + (d : ℝ) * beta) := by |
| classical |
| have hspin : ∀ eta : D → IsingSpin, spinSign (eta v) = -1 ∨ spinSign (eta v) = 1 := by |
| intro eta |
| cases eta v <;> simp [spinSign] |
| have hneg : ∃ eta : D → IsingSpin, spinSign (eta v) = -1 := ⟨fun _ => .neg, rfl⟩ |
| have hpos : ∃ eta : D → IsingSpin, spinSign (eta v) = 1 := ⟨fun _ => .pos, rfl⟩ |
| have hF := FinitePosterior.fieldEntropy_le_abs_field (finiteIsingEnergy beta D) |
| (fun (eta : D → IsingSpin) w => spinSign (eta w)) |
| (fun eta => spinSign (eta v)) y hspin hneg hpos |
| have hb := finite_tree_gibbs_field_uniform_bound D hbeta y v |
| change |FinitePosterior.fieldField (finiteIsingEnergy beta D) |
| (fun (eta : D → IsingSpin) w => spinSign (eta w)) |
| (fun eta => spinSign (eta v)) y - y v| ≤ (d : ℝ) * beta at hb |
| have ht := norm_le_norm_sub_add (FinitePosterior.fieldField (finiteIsingEnergy beta D) |
| (fun (eta : D → IsingSpin) w => spinSign (eta w)) |
| (fun eta => spinSign (eta v)) y) (y v) |
| simp only [Real.norm_eq_abs] at ht |
| linarith |
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| theorem HasFreeIsingLaw.integral_finiteIsingEntropy_le_own_observation |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| (D : Finset (TreeVertex d)) (v : D) (t : ℝ) : |
| let spin : (D → IsingSpin) → D → ℝ := fun eta w => spinSign (eta w) |
| let ρ := (μ.map D.restrict).prod (Measure.pi (fun _ : D => gaussianReal 0 1)) |
| (∫ x, FinitePosterior.fieldEntropy (finiteIsingEnergy beta D) spin |
| (fun eta => spinSign (eta v)) (FinitePosterior.finiteStandardObservation spin t x) ∂ρ) ≤ |
| 2 * ((∫ x, |FinitePosterior.finiteStandardObservation spin t x v| ∂ρ) + (d : ℝ) * beta) := by |
| classical |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure |
| let spin : (D → IsingSpin) → D → ℝ := fun eta w => spinSign (eta w) |
| let ρ := (μ.map D.restrict).prod (Measure.pi (fun _ : D => gaussianReal 0 1)) |
| have hcoords : ∀ eta w, spin eta w = -1 ∨ spin eta w = 1 := by |
| intro eta w |
| change spinSign (eta w) = -1 ∨ spinSign (eta w) = 1 |
| cases eta w <;> simp [spinSign] |
| have hobsInt : Integrable (FinitePosterior.finiteStandardObservation spin t) ρ := |
| FinitePosterior.integrable_finiteStandardObservation (μ.map D.restrict) spin hcoords t |
| have hcoordInt : Integrable (fun x => |FinitePosterior.finiteStandardObservation spin t x v|) ρ := |
| (hobsInt.eval v).abs |
| have hFInt : Integrable (fun x => FinitePosterior.fieldEntropy (finiteIsingEnergy beta D) spin |
| (fun eta => spinSign (eta v)) (FinitePosterior.finiteStandardObservation spin t x)) ρ := |
| FinitePosterior.integrable_finiteStandardEntropy (μ.map D.restrict) (finiteIsingEnergy beta D) |
| spin (fun eta => spinSign (eta v)) (fun eta => hcoords eta v) |
| ⟨fun _ => .neg, rfl⟩ ⟨fun _ => .pos, rfl⟩ hcoords t |
| have hb := integral_mono hFInt |
| ((hcoordInt.add (integrable_const ((d : ℝ) * beta))).const_mul 2) |
| (fun x => finiteIsingEntropy_le_own_observation hbeta D v |
| (FinitePosterior.finiteStandardObservation spin t x)) |
| change (∫ x, FinitePosterior.fieldEntropy (finiteIsingEnergy beta D) spin |
| (fun eta => spinSign (eta v)) (FinitePosterior.finiteStandardObservation spin t x) ∂ρ) ≤ _ |
| calc |
| _ ≤ ∫ x, 2 * (|FinitePosterior.finiteStandardObservation spin t x v| + (d : ℝ) * beta) ∂ρ := hb |
| _ = _ := by rw [integral_const_mul, integral_add hcoordInt (integrable_const _)]; simp [ρ, spin] |
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| theorem finiteIsingEntropy_zero {d : ℕ} (beta : ℝ) (D : Finset (TreeVertex d)) (v : D) : |
| FinitePosterior.fieldEntropy (finiteIsingEnergy beta D) |
| (fun (eta : D → IsingSpin) w => spinSign (eta w)) |
| (fun eta => spinSign (eta v)) 0 = 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.fieldEntropy, hz] |
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| theorem HasFreeIsingLaw.integral_finiteIsingEntropyEnergy |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) |
| (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 ρ := (μ.map D.restrict).prod (Measure.pi (fun _ : D => gaussianReal 0 1)) |
| let C : ℝ → ℝ := fun t => ∫ x, |
| (1 + FinitePosterior.fieldMean (finiteIsingEnergy beta D) spin |
| (fun eta => spinSign (eta v)) (FinitePosterior.finiteStandardObservation spin t x) ^ 2) * |
| ∑ w, FinitePosterior.normalizedFieldCovariance (finiteIsingEnergy beta D) spin |
| (fun eta => spinSign (eta v)) (FinitePosterior.finiteStandardObservation spin t x) w ^ 2 ∂ρ |
| IntervalIntegrable C volume 0 T ∧ |
| (∫ t in 0..T, C t) = |
| (∫ x, FinitePosterior.fieldEntropy (finiteIsingEnergy beta D) spin |
| (fun eta => spinSign (eta v)) (FinitePosterior.finiteStandardObservation spin T x) ∂ρ) ∧ |
| 0 ≤ (∫ t in 0..T, C t) := by |
| classical |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure |
| have hcoords : ∀ (eta : D → IsingSpin) w, spinSign (eta w) = -1 ∨ spinSign (eta w) = 1 := by |
| intro eta w |
| cases eta w <;> simp [spinSign] |
| have h := FinitePosterior.integral_finiteStandardEntropyEnergy |
| (μ.map D.restrict) (finiteIsingEnergy beta D) |
| (hμ.restrict_real_eq_normalized_exp D hD) |
| (fun (eta : D → IsingSpin) w => spinSign (eta w)) (fun eta => spinSign (eta v)) |
| (fun eta => hcoords eta v) ⟨fun _ => .neg, rfl⟩ ⟨fun _ => .pos, rfl⟩ hcoords hT |
| simpa only [finiteIsingEntropy_zero, sub_zero] using h |
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| theorem HasFreeIsingLaw.integral_finiteIsingNormalizedEnergy_le |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| (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 ρ := (μ.map D.restrict).prod (Measure.pi (fun _ : D => gaussianReal 0 1)) |
| let Q : ℝ → ℝ := fun t => ∫ x, ∑ w, |
| FinitePosterior.normalizedFieldCovariance (finiteIsingEnergy beta D) spin |
| (fun eta => spinSign (eta v)) (FinitePosterior.finiteStandardObservation spin t x) w ^ 2 ∂ρ |
| IntervalIntegrable Q volume 0 T ∧ |
| (∫ t in 0..T, Q t) ≤ |
| 2 * ((∫ x, |FinitePosterior.finiteStandardObservation spin T x v| ∂ρ) + (d : ℝ) * beta) ∧ |
| 0 ≤ (∫ t in 0..T, Q t) := by |
| classical |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure |
| have hcoords : ∀ (eta : D → IsingSpin) w, spinSign (eta w) = -1 ∨ spinSign (eta w) = 1 := by |
| intro eta w |
| cases eta w <;> simp [spinSign] |
| have h := FinitePosterior.integral_finiteStandardNormalizedEnergy_le |
| (μ.map D.restrict) (finiteIsingEnergy beta D) |
| (hμ.restrict_real_eq_normalized_exp D hD) |
| (fun (eta : D → IsingSpin) w => spinSign (eta w)) (fun eta => spinSign (eta v)) |
| (fun eta => hcoords eta v) ⟨fun _ => .neg, rfl⟩ ⟨fun _ => .pos, rfl⟩ hcoords hT |
| have hbound := hμ.integral_finiteIsingEntropy_le_own_observation hbeta D v T |
| simp only [finiteIsingEntropy_zero, sub_zero] at h |
| exact ⟨h.1, h.2.1.trans hbound, h.2.2⟩ |
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| end Problem367 |
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| end |
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| end OAI |
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