| import OAI.Probability.FreeIsing.Brownian.EntropyEnergy |
| import OAI.Probability.FreeIsing.Brownian.PosteriorField |
| import OAI.Probability.FreeIsing.Gaussian.SiteInformation |
| import OAI.Probability.FreeIsing.Stochastic.FiniteIsingBrownianPosterior |
|
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped NNReal ENNReal |
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| namespace Problem367 |
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| local instance finiteEntropySmallTimeNonemptyIsingSpin : Nonempty IsingSpin := ⟨.neg⟩ |
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| theorem gaussian_channel_fourth_moment |
| {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] |
| {S G : Ω → ℝ} (hS : AEMeasurable S P) |
| (hSb : ∀ᵐ p ∂P, |S p| ≤ 1) {t : ℝ≥0} |
| (hG : HasLaw G (gaussianReal 0 t) P) (ht : (t : ℝ) ≤ 1) : |
| Integrable (fun p => |(t : ℝ) * S p + G p| ^ 4) P ∧ |
| (∫ p, |(t : ℝ) * S p + G p| ^ 4 ∂P) ≤ |
| (8 * (1 + Real.exp 32)) * (t : ℝ) := by |
| have ht0 := t.coe_nonneg |
| have hG4 := integrable_abs_pow_of_gaussianLaw hG 4 |
| have hdom : Integrable (fun p => (8 : ℝ) * ((t : ℝ) ^ 4 + |G p| ^ 4)) P := |
| ((integrable_const _).add hG4).const_mul 8 |
| have hbound : ∀ᵐ p ∂P, |(t : ℝ) * S p + G p| ^ 4 ≤ |
| 8 * ((t : ℝ) ^ 4 + |G p| ^ 4) := by |
| filter_upwards [hSb] with p hp |
| have hb : |(t : ℝ) * S p + G p| ≤ (t : ℝ) + |G p| := by |
| calc |
| _ ≤ |(t : ℝ) * S p| + |G p| := abs_add_le _ _ |
| _ ≤ (t : ℝ) + |G p| := by |
| rw [abs_mul, abs_of_nonneg ht0] |
| gcongr |
| simpa using mul_le_mul_of_nonneg_left hp ht0 |
| exact (pow_le_pow_left₀ (abs_nonneg _) hb 4).trans |
| (by convert add_pow_le ht0 (abs_nonneg (G p)) 4 using 1; norm_num) |
| have hi : Integrable (fun p => |(t : ℝ) * S p + G p| ^ 4) P := by |
| apply hdom.mono' (((hS.const_mul (t : ℝ)).add hG.aemeasurable).abs.pow_const 4).aestronglyMeasurable |
| filter_upwards [hbound] with p hp |
| change ‖|(t : ℝ) * S p + G p| ^ 4‖ ≤ _ |
| rw [Real.norm_eq_abs, abs_of_nonneg (by positivity)] |
| exact hp |
| have hstd : (∫ x : ℝ, |x| ^ 4 ∂gaussianReal 0 1) ≤ Real.exp 32 := by |
| calc |
| _ ≤ ∫ x : ℝ, (1 + |x|) ^ 4 ∂gaussianReal 0 1 := |
| integral_mono (integrable_standardGaussian_abs_pow 4) |
| (integrable_one_add_abs_pow_gaussian 4) |
| (fun x => pow_le_pow_left₀ (abs_nonneg x) (by linarith) 4) |
| _ ≤ _ := by |
| convert integral_one_add_abs_pow_gaussian_le (p := 4) (by decide) using 1; norm_num |
| have hnoise : (∫ p, |G p| ^ 4 ∂P) ≤ (t : ℝ) ^ 2 * Real.exp 32 := by |
| have heq := hG.integral_comp (f := fun x : ℝ => |x| ^ 4) (by fun_prop) |
| change (∫ p, |G p| ^ 4 ∂P) = _ at heq |
| rw [heq, integral_gaussianReal_abs_pow] |
| have hs : Real.sqrt (t : ℝ) ^ 4 = (t : ℝ) ^ 2 := by |
| rw [show (4 : ℕ) = 2 * 2 by decide, pow_mul, Real.sq_sqrt ht0] |
| rw [hs] |
| exact mul_le_mul_of_nonneg_left hstd (sq_nonneg _) |
| have ht2 : (t : ℝ) ^ 2 ≤ (t : ℝ) := by nlinarith |
| have ht4 : (t : ℝ) ^ 4 ≤ (t : ℝ) := by |
| have h := pow_le_pow_left₀ (sq_nonneg (t : ℝ)) ht2 2 |
| nlinarith |
| refine ⟨hi, ?_⟩ |
| calc |
| _ ≤ ∫ p, (8 : ℝ) * ((t : ℝ) ^ 4 + |G p| ^ 4) ∂P := |
| integral_mono_ae hi hdom hbound |
| _ = 8 * ((t : ℝ) ^ 4 + ∫ p, |G p| ^ 4 ∂P) := by |
| rw [integral_const_mul, integral_add (integrable_const _) hG4] |
| simp |
| _ ≤ 8 * ((t : ℝ) + (t : ℝ) * Real.exp 32) := by |
| gcongr |
| exact hnoise.trans (mul_le_mul_of_nonneg_right ht2 (Real.exp_nonneg _)) |
| _ = _ := by ring |
|
|
| theorem finiteIsingBrownianEntropy_le_information |
| {d : ℕ} {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω] |
| (hbeta : 0 ≤ beta) (μ : Measure (TreeVertex d → IsingSpin)) |
| [IsProbabilityMeasure μ] (ν : TreeVertex d → Measure Ω) |
| {B : TreeVertex d → ℝ≥0 → Ω → ℝ} |
| (hB : ∀ i, IsPreBrownianReal (B i) (ν i)) |
| (hmB : ∀ i t, Measurable (B i t)) |
| (D : Finset (TreeVertex d)) (v : D) (t : ℝ≥0) (ht : (t : ℝ) ≤ 1) : |
| (∫ p, finiteIsingBrownianEntropy beta D v B t p |
| ∂μ.prod (Measure.infinitePi ν)) ≤ |
| 2 * Real.sqrt |
| (Real.cosh ((d : ℝ) * beta + 1) ^ 2 * |
| (∫ p, finiteIsingBrownianMean beta D v B t p ^ 2 |
| ∂μ.prod (Measure.infinitePi ν)) + |
| (16 * (1 + ((d : ℝ) * beta) ^ 2) * (1 + Real.exp 32)) * (t : ℝ)) := by |
| have (i : TreeVertex d) : IsProbabilityMeasure (ν i) := |
| (hB i).isGaussianProcess.isProbabilityMeasure |
| let P := μ.prod (Measure.infinitePi ν) |
| let M := finiteIsingBrownianMean beta D v B (t : ℝ) |
| let H := fun p => Real.artanh (M p) |
| let G := fun p : (TreeVertex d → IsingSpin) × (TreeVertex d → Ω) => B v t (p.2 v) |
| let S := fun p : (TreeVertex d → IsingSpin) × (TreeVertex d → Ω) => spinSign (p.1 v) |
| let X := fun p => (t : ℝ) * S p + G p |
| have hcoord := (measurePreserving_eval_infinitePi ν (v : TreeVertex d)).comp |
| (measurePreserving_snd (μ := μ) (ν := Measure.infinitePi ν)) |
| have hG : HasLaw G (gaussianReal 0 t) P := |
| (hB v).hasLaw_eval t |>.comp ⟨hcoord.measurable.aemeasurable, hcoord.map_eq⟩ |
| have hS : Measurable S := (measurable_of_finite spinSign).comp |
| ((measurable_pi_apply (v : TreeVertex d)).comp measurable_fst) |
| have hSb : ∀ᵐ p ∂P, |S p| ≤ 1 := Eventually.of_forall fun p => by |
| cases hp : p.1 v <;> simp [S, hp, spinSign] |
| have hX4 := gaussian_channel_fourth_moment hS.aemeasurable hSb hG ht |
| have hspin (eta : D → IsingSpin) : spinSign (eta v) = -1 ∨ spinSign (eta v) = 1 := by |
| 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 hm : Measurable M := |
| (FinitePosterior.continuous_fieldMean _ _ _).measurable.comp |
| (measurable_finiteIsingBrownianObservation hmB D t) |
| have hart : Measurable Real.artanh := by unfold Real.artanh; fun_prop |
| have hH : AEMeasurable H P := (hart.comp hm).aemeasurable |
| have hinterior (p) : M p ∈ Ioo (-1 : ℝ) 1 := |
| FinitePosterior.fieldMean_spin_mem_Ioo _ _ _ _ hspin hneg hpos |
| have htan (p) : Real.tanh (H p) = M p := Real.tanh_artanh (hinterior p) |
| have hfield : ∀ᵐ p ∂P, |H p - X p| ≤ (d : ℝ) * beta := by |
| apply Eventually.of_forall |
| intro p |
| have hb := finite_tree_gibbs_field_uniform_bound D hbeta |
| (finiteIsingBrownianObservation B D t p) v |
| change |H p - finiteIsingBrownianObservation B D t p v| ≤ |
| (d : ℝ) * beta at hb |
| simpa only [X, S, G, finiteIsingBrownianObservation, Real.toNNReal_coe] using hb |
| have hsecond := integral_sq_le_tanh_sq_add_fourth |
| (by positivity : 0 ≤ (d : ℝ) * beta) hH hX4.1 hfield |
| simp_rw [htan] at hsecond |
| have hH2 : MemLp H 2 P := |
| (memLp_two_iff_integrable_sq hH.aestronglyMeasurable).mpr hsecond.1 |
| have hinfo : (∫ p, H p ^ 2 ∂P) ≤ |
| Real.cosh ((d : ℝ) * beta + 1) ^ 2 * (∫ p, M p ^ 2 ∂P) + |
| (16 * (1 + ((d : ℝ) * beta) ^ 2) * (1 + Real.exp 32)) * (t : ℝ) := by |
| apply hsecond.2.trans |
| have hb := mul_le_mul_of_nonneg_left hX4.2 |
| (by positivity : 0 ≤ 2 * (1 + ((d : ℝ) * beta) ^ 2)) |
| nlinarith |
| have hE := (finiteIsingBrownianEntropy_moment hbeta μ ν hB hmB D v t).1 |
| calc |
| _ ≤ ∫ p, 2 * |H p| ∂P := by |
| apply integral_mono hE ((hH2.integrable (by norm_num)).abs.const_mul 2) |
| intro p |
| exact FinitePosterior.fieldEntropy_le_abs_field _ _ _ _ hspin hneg hpos |
| _ = 2 * ∫ p, |H p| ∂P := integral_const_mul _ _ |
| _ ≤ 2 * Real.sqrt (∫ p, H p ^ 2 ∂P) := |
| mul_le_mul_of_nonneg_left (integral_abs_le_sqrt_integral_sq hH hsecond.1) (by norm_num) |
| _ ≤ _ := mul_le_mul_of_nonneg_left (Real.sqrt_le_sqrt hinfo) (by norm_num) |
|
|
| theorem brownianTerminal_information_eq_gaussian |
| {d : ℕ} {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure (TreeVertex d → IsingSpin)) [IsProbabilityMeasure μ] |
| (ν : TreeVertex d → Measure Ω) |
| {B : TreeVertex d → ℝ≥0 → Ω → ℝ} |
| (hB : ∀ i, IsPreBrownianReal (B i) (ν i)) |
| (hmB : ∀ i t, Measurable (B i t)) (t : ℝ≥0) (v : TreeVertex d) : |
| (∫ p, ((μ.prod (Measure.infinitePi ν))[fun z => spinSign (z.1 v) | |
| brownianTerminalSigma B t] p) ^ 2 ∂μ.prod (Measure.infinitePi ν)) = |
| ∫ z, (PlantedSymmetry.posteriorMagnetization |
| (μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)) t v z) ^ 2 |
| ∂μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t) := by |
| have (i : TreeVertex d) : IsProbabilityMeasure (ν i) := |
| (hB i).isGaussianProcess.isProbabilityMeasure |
| let T : ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) → |
| (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) := |
| fun p => (p.1, fun i => B i t (p.2 i)) |
| have hT : MeasurePreserving T (μ.prod (Measure.infinitePi ν)) |
| (μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)) := |
| (MeasurePreserving.id μ).prod (brownian_array_eval_measurePreserving ν hB hmB t) |
| have hspin : Measurable (fun z : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) => |
| spinSign (z.1 v)) := (measurable_of_finite spinSign).comp |
| ((measurable_pi_apply v).comp measurable_fst) |
| have hpull := condExp_comp_measurePreserving_comap hT hspin.stronglyMeasurable |
| (PlantedSymmetry.integrable_plantedSpin _ v) |
| (PlantedSymmetry.observationSigma_le (TreeVertex d) t) |
| have hsame : (PlantedSymmetry.observationSigma (TreeVertex d) (t : ℝ)).comap T = |
| brownianTerminalSigma B t := by |
| rw [PlantedSymmetry.observationSigma, MeasurableSpace.comap_comp] |
| change MeasurableSpace.comap |
| (fun p : (TreeVertex d → IsingSpin) × (TreeVertex d → Ω) => |
| fun i => (t : ℝ) * spinSign (p.1 i) + B i t (p.2 i)) inferInstance = _ |
| simp only [brownianTerminalSigma, Real.toNNReal_coe] |
| rw [hsame] at hpull |
| calc |
| _ = ∫ p, (PlantedSymmetry.posteriorMagnetization |
| (μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)) t v (T p)) ^ 2 |
| ∂μ.prod (Measure.infinitePi ν) := by |
| apply integral_congr_ae |
| filter_upwards [hpull] with p hp |
| exact congrArg (fun x : ℝ => x ^ 2) hp |
| _ = _ := hT.hasLaw.integral_comp |
| (((stronglyMeasurable_condExp).mono |
| (PlantedSymmetry.observationSigma_le (TreeVertex d) t)).pow 2).aestronglyMeasurable |
|
|
| theorem HasFreeIsingLaw.finiteIsingBrownian_critical_information_upper |
| {d : ℕ} (hd : 3 ≤ d) {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω] |
| {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (hcritical : ((d - 1 : ℕ) : ℝ) * (Real.tanh beta) ^ 2 = 1) |
| (ν : TreeVertex d → Measure Ω) |
| {B : TreeVertex d → ℝ≥0 → Ω → ℝ} |
| (hB : ∀ i, IsPreBrownianReal (B i) (ν i)) |
| (hmB : ∀ i t, Measurable (B i t)) |
| (D : Finset (TreeVertex d)) (hD : ConnectedFinset D) (v : D) |
| (t : ℝ≥0) (ht : (t : ℝ) ≤ 1) : |
| (∫ p, finiteIsingBrownianMean beta D v B t p ^ 2 |
| ∂μ.prod (Measure.infinitePi ν)) ≤ |
| ((1 + (Real.tanh beta) ^ 2) * |
| Real.sqrt (1 / ((1 - (Real.tanh beta) ^ 2) * (Real.tanh beta) ^ 4))) * |
| Real.sqrt (t : ℝ) := by |
| let := hμ.isProbabilityMeasure |
| have h := hμ.finiteIsingBrownian_information_le ν hB hmB D hD v t |
| dsimp only at h |
| rw [brownianTerminal_information_eq_gaussian μ ν hB hmB t v] at h |
| exact h.trans (hμ.gaussian_site_critical_information_upper hd hb hcritical t ht v) |
|
|
| def criticalEntropyEnvelope (d : ℕ) (beta t : ℝ) : ℝ := |
| 2 * Real.sqrt |
| (Real.cosh ((d : ℝ) * beta + 1) ^ 2 * |
| (((1 + (Real.tanh beta) ^ 2) * |
| Real.sqrt (1 / ((1 - (Real.tanh beta) ^ 2) * (Real.tanh beta) ^ 4))) * |
| Real.sqrt t) + |
| (16 * (1 + ((d : ℝ) * beta) ^ 2) * (1 + Real.exp 32)) * t) |
|
|
| theorem HasFreeIsingLaw.finiteIsingBrownianEntropy_le_criticalEnvelope |
| {d : ℕ} (hd : 3 ≤ d) {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω] |
| {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (hcritical : ((d - 1 : ℕ) : ℝ) * (Real.tanh beta) ^ 2 = 1) |
| (ν : TreeVertex d → Measure Ω) |
| {B : TreeVertex d → ℝ≥0 → Ω → ℝ} |
| (hB : ∀ i, IsPreBrownianReal (B i) (ν i)) |
| (hmB : ∀ i t, Measurable (B i t)) |
| (D : Finset (TreeVertex d)) (hD : ConnectedFinset D) (v : D) |
| (t : ℝ≥0) (ht : (t : ℝ) ≤ 1) : |
| (∫ p, finiteIsingBrownianEntropy beta D v B t p |
| ∂μ.prod (Measure.infinitePi ν)) ≤ criticalEntropyEnvelope d beta t := by |
| let := hμ.isProbabilityMeasure |
| apply (finiteIsingBrownianEntropy_le_information hb μ ν hB hmB D v t ht).trans |
| apply mul_le_mul_of_nonneg_left _ (by norm_num : (0 : ℝ) ≤ 2) |
| apply Real.sqrt_le_sqrt |
| apply add_le_add_left |
| exact mul_le_mul_of_nonneg_left |
| (hμ.finiteIsingBrownian_critical_information_upper hd hb hcritical ν hB hmB D hD v t ht) |
| (sq_nonneg _) |
|
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| @[simp] theorem criticalEntropyEnvelope_zero (d : ℕ) (beta : ℝ) : |
| criticalEntropyEnvelope d beta 0 = 0 := by simp [criticalEntropyEnvelope] |
|
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| theorem continuous_criticalEntropyEnvelope (d : ℕ) (beta : ℝ) : |
| Continuous (criticalEntropyEnvelope d beta) := by |
| unfold criticalEntropyEnvelope |
| fun_prop |
|
|
| end Problem367 |
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| end |
|
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| end OAI |
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