| import OAI.Probability.FreeIsing.Brownian.PosteriorBareResponse |
| import OAI.Probability.FreeIsing.Critical.CanonicalResponse |
| import OAI.Probability.FreeIsing.Gaussian.TreeResponseObservedPosterior |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal |
|
|
| namespace Problem367 |
|
|
| open PlantedSymmetry |
|
|
| namespace CriticalCanonicalRowEnergy |
|
|
| def rate (beta : ℝ) (t : ℝ≥0) : ℝ := |
| ComponentSignalAttenuation.lossCoefficient (Real.tanh beta) * |
| min (Real.sqrt (t : ℝ)) 1 |
|
|
| theorem rate_pos {d : ℕ} {beta : ℝ} (hb : 0 ≤ beta) |
| (hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1) |
| {t : ℝ≥0} (ht : 0 < t) : 0 < rate beta t := by |
| have htheta : 0 < Real.tanh beta := by |
| have hn := tanh_nonneg_of_beta_nonneg hb |
| by_contra h |
| have hz : Real.tanh beta = 0 := le_antisymm (not_lt.mp h) hn |
| simp [hz] at hc |
| exact mul_pos (ComponentSignalAttenuation.lossCoefficient_pos htheta |
| (Real.tanh_lt_one beta)) |
| (lt_min (Real.sqrt_pos.mpr (show (0 : ℝ) < t from ht)) zero_lt_one) |
|
|
| theorem branching_ratio {d : ℕ} {beta : ℝ} |
| (hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1) (t : ℝ≥0) : |
| ((d - 1 : ℕ) : ℝ) * (Real.tanh beta ^ 2 * Real.exp (-rate beta t)) = |
| Real.exp (-rate beta t) := by rw [← mul_assoc, hc, one_mul] |
|
|
| theorem ratio_lt_one {d : ℕ} {beta : ℝ} (hb : 0 ≤ beta) |
| (hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1) |
| {t : ℝ≥0} (ht : 0 < t) : |
| ((d - 1 : ℕ) : ℝ) * (Real.tanh beta ^ 2 * Real.exp (-rate beta t)) < 1 := by |
| rw [branching_ratio hc] |
| exact Real.exp_lt_one_iff.mpr (neg_neg_of_pos (rate_pos hb hc ht)) |
|
|
| theorem exp_div_one_sub_exp_le_inv {x : ℝ} (hx : 0 < x) : |
| Real.exp (-x) / (1 - Real.exp (-x)) ≤ x⁻¹ := by |
| have hq : Real.exp (-x) < 1 := Real.exp_lt_one_iff.mpr (neg_neg_of_pos hx) |
| have he := mul_le_mul_of_nonneg_right (Real.add_one_le_exp x) (Real.exp_pos (-x)).le |
| have hcancel : Real.exp x * Real.exp (-x) = 1 := by |
| rw [← Real.exp_add, add_neg_cancel, Real.exp_zero] |
| rw [hcancel] at he |
| apply (div_le_iff₀ (sub_pos.mpr hq)).mpr |
| rw [inv_mul_eq_div] |
| exact (le_div_iff₀ hx).mpr (by nlinarith) |
|
|
| end CriticalCanonicalRowEnergy |
|
|
| open CriticalCanonicalRowEnergy |
|
|
| theorem HasFreeIsingLaw.gaussian_rootResponse_critical_radial_bound |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d) |
| (hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 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 beta ^ 2 * Real.exp (-rate beta t)) ^ v.1.length := by |
| have h := hμ.integral_gaussianRootResponse_sq_le_critical hb hd hc ht v |
| simpa only [rate, mul_pow, ← pow_mul, Nat.mul_comm, |
| ← Real.exp_nat_mul, mul_comm] using h |
|
|
| theorem HasFreeIsingLaw.gaussian_rootResponse_critical_row_energy |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d) |
| (hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1) |
| {t : ℝ≥0} (ht : 0 < t) : |
| (∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t), |
| Summable (fun v => gaussianPosteriorResponse μ t (ConnectedHull.root d) v p ^ 2)) ∧ |
| Integrable (fun p => ∑' v, gaussianPosteriorResponse μ t (ConnectedHull.root d) v p ^ 2) |
| (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ∧ |
| (∫ p, ∑' v, gaussianPosteriorResponse μ t (ConnectedHull.root d) v p ^ 2 |
| ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ≤ |
| 1 + (d : ℝ) * Real.tanh beta ^ 2 * Real.exp (-rate beta t) / |
| (1 - Real.exp (-rate beta t)) := by |
| have hold := hμ.gaussian_posteriorResponse_row_energy hb ht hc.le |
| (ConnectedHull.root d) rfl |
| refine ⟨hold.1, hold.2.1, ?_⟩ |
| have h := integral_tsum_sq_le_of_tree_radial_bound (C := 1) |
| (fun v => (hμ.gaussian_posteriorResponse_sq_decay hb ht (ConnectedHull.root d) v rfl).1) |
| (fun v => by simpa only [one_mul] using |
| hμ.gaussian_rootResponse_critical_radial_bound hb hd hc ht v) |
| (mul_nonneg (sq_nonneg _) (Real.exp_pos _).le) (ratio_lt_one hb hc ht) |
| simpa only [one_mul, branching_ratio hc, mul_assoc] using h |
|
|
| theorem HasFreeIsingLaw.integral_gaussian_rootResponse_row_le_inv_rate |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d) |
| (hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1) |
| {t : ℝ≥0} (ht : 0 < t) : |
| (∫ p, ∑' v, gaussianPosteriorResponse μ t (ConnectedHull.root d) v p ^ 2 |
| ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ≤ |
| 1 + (d : ℝ) * Real.tanh beta ^ 2 / rate beta t := by |
| apply (hμ.gaussian_rootResponse_critical_row_energy hb hd hc ht).2.2.trans |
| have h := mul_le_mul_of_nonneg_left |
| (exp_div_one_sub_exp_le_inv (rate_pos hb hc ht)) |
| (mul_nonneg (Nat.cast_nonneg d) (sq_nonneg (Real.tanh beta))) |
| simpa only [mul_div_assoc, div_eq_mul_inv, mul_assoc] using |
| add_le_add (show (1 : ℝ) ≤ 1 from le_rfl) h |
| |
| theorem HasFreeIsingLaw.gaussian_rootResponse_critical_row_tail |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d) |
| (hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1) |
| {t : ℝ≥0} (ht : 0 < t) (L : ℕ) : |
| (∫ p, ∑' v : TreeVertex d, if L < v.1.length then |
| gaussianPosteriorResponse μ t (ConnectedHull.root d) v p ^ 2 else 0 |
| ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ≤ |
| (d : ℝ) * Real.tanh beta ^ 2 * Real.exp (-rate beta t) * |
| Real.exp (-rate beta t) ^ L / (1 - Real.exp (-rate beta t)) := by |
| have h := integral_tsum_sq_cutoff_le_of_tree_radial_bound (C := 1) |
| (fun v => (hμ.gaussian_posteriorResponse_sq_decay hb ht (ConnectedHull.root d) v rfl).1) |
| (fun v => by simpa only [one_mul] using |
| hμ.gaussian_rootResponse_critical_radial_bound hb hd hc ht v) |
| (mul_nonneg (sq_nonneg _) (Real.exp_pos _).le) (ratio_lt_one hb hc ht) L |
| simpa only [one_mul, branching_ratio hc, mul_assoc] using h |
|
|
| theorem HasFreeIsingLaw.lintegral_gaussian_rootResponse_row_le_inv_rate |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d) |
| (hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1) |
| {t : ℝ≥0} (ht : 0 < t) : |
| (∫⁻ p, ∑' v, ENNReal.ofReal |
| (gaussianPosteriorResponse μ t (ConnectedHull.root d) v p ^ 2) |
| ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ≤ |
| ENNReal.ofReal (1 + (d : ℝ) * Real.tanh beta ^ 2 / rate beta t) := by |
| have hrow := hμ.gaussian_rootResponse_critical_row_energy hb hd hc ht |
| have he := hrow.1.mono fun p hp => |
| ENNReal.ofReal_tsum_of_nonneg (fun v => sq_nonneg |
| (gaussianPosteriorResponse μ t (ConnectedHull.root d) v p)) hp |
| rw [← lintegral_congr_ae he, |
| ← ofReal_integral_eq_lintegral_ofReal hrow.2.1 |
| (Eventually.of_forall fun p => tsum_nonneg (fun v => sq_nonneg _))] |
| exact ENNReal.ofReal_le_ofReal |
| (hμ.integral_gaussian_rootResponse_row_le_inv_rate hb hd hc ht) |
| |
| theorem HasFreeIsingLaw.brownian_rootResponse_critical_row_energy |
| {d : ℕ} {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω] |
| {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d) |
| (hc : ((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)) {t : ℝ≥0} (ht : 0 < t) : |
| (∀ᵐ p ∂μ.prod (Measure.infinitePi ν), |
| Summable (fun v => brownianPosteriorResponse μ ν B t (ConnectedHull.root d) v p ^ 2)) ∧ |
| Integrable (fun p => ∑' v, |
| brownianPosteriorResponse μ ν B t (ConnectedHull.root d) v p ^ 2) |
| (μ.prod (Measure.infinitePi ν)) ∧ |
| (∫ p, ∑' v, brownianPosteriorResponse μ ν B t (ConnectedHull.root d) v p ^ 2 |
| ∂μ.prod (Measure.infinitePi ν)) ≤ |
| 1 + (d : ℝ) * Real.tanh beta ^ 2 / rate beta t := by |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure |
| have (i : TreeVertex d) : IsProbabilityMeasure (ν i) := |
| (hB i).isGaussianProcess.isProbabilityMeasure |
| have hT := (MeasurePreserving.id μ).prod |
| (brownian_array_eval_measurePreserving ν hB hmB t) |
| have hg := hμ.gaussian_rootResponse_critical_row_energy hb hd hc ht |
| have heq := ae_all_iff.mpr (fun v => brownian_past_posteriorResponse_eq_gaussian |
| μ ν hB hmB t (ConnectedHull.root d) v) |
| have he : (fun p => ∑' v, |
| brownianPosteriorResponse μ ν B t (ConnectedHull.root d) v p ^ 2) =ᵐ[ |
| μ.prod (Measure.infinitePi ν)] |
| (fun q => ∑' v, gaussianPosteriorResponse μ t (ConnectedHull.root d) v q ^ 2) ∘ |
| Prod.map id (fun p i => B i t (p i)) := by |
| filter_upwards [heq] with p hp |
| exact tsum_congr fun v => congrArg (fun x : ℝ => x ^ 2) (hp v) |
| refine ⟨?_, (hT.integrable_comp_of_integrable hg.2.1).congr he.symm, ?_⟩ |
| · filter_upwards [hT.quasiMeasurePreserving.ae hg.1, heq] with p hp hpeq |
| exact hp.congr fun v => congrArg (fun x : ℝ => x ^ 2) (hpeq v).symm |
| · rw [integral_congr_ae he, hT.hasLaw.integral_comp hg.2.1.aestronglyMeasurable] |
| exact hμ.integral_gaussian_rootResponse_row_le_inv_rate hb hd hc ht |
| |
| end Problem367 |
| |
| end |
| |
| end OAI |
| |
