| import OAI.Probability.InvariantIsing.Pressure.Concentration |
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|
| namespace OAI |
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|
| /-! |
| # Almost-sure concentration without independence across dimensions |
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| A countable Borel–Cantelli argument upgrades Haar tails to almost-sure |
| concentration of the pressure about its medians. |
| -/ |
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|
| noncomputable section |
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|
| open MeasureTheory Filter |
| open scoped Topology NNReal ENNReal |
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| namespace InvariantIsing |
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|
| /-- Summable exponential tails imply almost-sure convergence to zero. No |
| independence between the random variables is required. -/ |
| lemma ae_tendsto_zero_of_exponential_tail {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (f : ℕ → Ω → ℝ) |
| (htail : ∀ ε : ℝ, 0 < ε → ∃ C a : ℝ, 0 ≤ C ∧ 0 < a ∧ |
| ∀ n, μ.real {ω | ε ≤ |f n ω|} ≤ C * Real.exp (-a * (n : ℝ))) : |
| ∀ᵐ ω ∂μ, Tendsto (fun n => f n ω) atTop (𝓝 0) := by |
| have hb (k : ℕ) : ∀ᵐ ω ∂μ, ∀ᶠ n in atTop, |
| ¬ (1 / (k + 1 : ℝ) ≤ |f n ω|) := by |
| have hk : 0 < 1 / (k + 1 : ℝ) := by positivity |
| obtain ⟨C, a, hC, ha, ht⟩ := htail _ hk |
| have hsum : Summable (fun n : ℕ => C * Real.exp (-a * (n : ℝ))) := by |
| simp_rw [mul_comm (-a), Real.exp_nat_mul] |
| exact (summable_geometric_of_lt_one (Real.exp_pos (-a)).le |
| (Real.exp_lt_one_iff.mpr (neg_neg_of_pos ha))).mul_left C |
| let q : ℕ → ℝ≥0 := fun n => NNReal.mk (C * Real.exp (-a * (n : ℝ))) |
| (mul_nonneg hC (Real.exp_pos _).le) |
| have hq : Summable q := (NNReal.summable_coe (f := q)).mp hsum |
| have hfinite : (∑' n, (q n : ENNReal)) ≠ ∞ := |
| ENNReal.tsum_coe_ne_top_iff_summable.mpr hq |
| have hbound (n : ℕ) : μ {ω | 1 / (k + 1 : ℝ) ≤ |f n ω|} ≤ (q n : ENNReal) := by |
| rw [← ENNReal.ofReal_toReal (measure_ne_top μ {ω | 1 / (k + 1 : ℝ) ≤ |f n ω|})] |
| change ENNReal.ofReal (μ.real {ω | 1 / (k + 1 : ℝ) ≤ |f n ω|}) ≤ _ |
| simpa only [q, ENNReal.ofReal_eq_coe_nnreal (mul_nonneg hC (Real.exp_pos _).le)] using |
| ENNReal.ofReal_le_ofReal (ht n) |
| exact ae_eventually_notMem (s := fun n => {ω | 1 / (k + 1 : ℝ) ≤ |f n ω|}) |
| (ne_top_of_le_ne_top hfinite (ENNReal.tsum_le_tsum hbound)) |
| filter_upwards [ae_all_iff.mpr hb] with ω hω |
| apply Metric.tendsto_atTop.mpr |
| intro ε hε |
| obtain ⟨k, hk⟩ := exists_nat_one_div_lt hε |
| obtain ⟨N, hN⟩ := eventually_atTop.mp (hω k) |
| refine ⟨N, ?_⟩ |
| intro n hn |
| simpa only [Real.dist_eq, sub_zero] using lt_trans (lt_of_not_ge (hN n hn)) hk |
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|
| /-- For an arbitrary joint realization of the Haar matrices, pressure is almost |
| surely close to a deterministic sequence of medians. -/ |
| theorem ae_pressure_sub_median_tendsto_zero (hpub : HaarConcentrationInput) |
| {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] |
| (U : (n : ℕ) → Ω → SpecialOrthogonal (n + 3)) |
| (hU : ∀ n, Measurable (U n)) |
| (hHaar : ∀ n, (P.map (U n)).IsMulLeftInvariant) |
| (eig c : (n : ℕ) → Fin (n + 3) → ℝ) |
| (K : ℝ) (hK : 0 < K) (heig : ∀ n i, |eig n i| ≤ K) : |
| ∃ m : ℕ → ℝ, ∀ᵐ ω ∂P, |
| Tendsto (fun n => rotatedPressure (eig n) (specialRotation (U n ω)) (c n) - m n) |
| atTop (𝓝 0) := by |
| obtain ⟨C, a, hC, ha, ht⟩ := haar_pressure_median_tail hpub |
| have hm (n : ℕ) : ∃ m : ℝ, |
| ∀ r : ℝ, 0 < r → |
| (P.map (U n)).real {V | r ≤ |rotatedPressure (eig n) (specialRotation V) (c n) - m|} ≤ |
| C * Real.exp (-a * (n + 3 : ℝ) * r ^ 2 / K ^ 2) := by |
| have hp : IsProbabilityMeasure (P.map (U n)) := |
| (Measure.isProbabilityMeasure_map_iff (hU n).aemeasurable).mpr inferInstance |
| obtain ⟨m, _, _, hm⟩ := ht (n + 3) (by omega) (P.map (U n)) hp (hHaar n) |
| (eig n) (c n) K hK (heig n) |
| exact ⟨m, by simpa only [Nat.cast_add, Nat.cast_ofNat] using hm⟩ |
| choose m hm using hm |
| refine ⟨m, ae_tendsto_zero_of_exponential_tail P _ ?_⟩ |
| intro ε hε |
| refine ⟨C, a * ε ^ 2 / K ^ 2, hC.le, by positivity, ?_⟩ |
| intro n |
| have hs : MeasurableSet {V : SpecialOrthogonal (n + 3) | |
| ε ≤ |rotatedPressure (eig n) (specialRotation V) (c n) - m n|} := |
| measurableSet_le measurable_const ((measurable_rotatedPressure _ _).sub_const _).abs |
| have hb := hm n ε hε |
| rw [measureReal_def, Measure.map_apply (hU n) hs] at hb |
| change P.real {ω | ε ≤ |rotatedPressure (eig n) (specialRotation (U n ω)) (c n) - m n|} ≤ _ at hb |
| refine hb.trans ?_ |
| apply mul_le_mul_of_nonneg_left _ hC.le |
| apply Real.exp_le_exp.mpr |
| have hnonneg : 0 ≤ a * ε ^ 2 / K ^ 2 := by positivity |
| calc |
| -a * ((n : ℝ) + 3) * ε ^ 2 / K ^ 2 = |
| -(a * ε ^ 2 / K ^ 2) * ((n : ℝ) + 3) := by ring |
| _ ≤ -(a * ε ^ 2 / K ^ 2) * (n : ℝ) := |
| mul_le_mul_of_nonpos_left (by linarith) (neg_nonpos.mpr hnonneg) |
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| end InvariantIsing |
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| end |
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| end OAI |
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