| import OAI.Probability.InvariantIsing.Haar.MedianMean |
|
|
| namespace OAI |
|
|
| /-! Almost-sure concentration of the actual pressure about its expectation. -/ |
|
|
| noncomputable section |
|
|
| open MeasureTheory Filter |
| open scoped Topology |
|
|
| namespace InvariantIsing |
|
|
| /-- Deterministic reference obtained by removing the quadratic interaction. -/ |
| def noninteractingPressure {N : ℕ} (c : Fin N → ℝ) : ℝ := |
| (N : ℝ)⁻¹ * logPartition (fieldEnergy c) |
|
|
| lemma rotatedPressure_zero {N : ℕ} (U : Rotation N) (c : Fin N → ℝ) : |
| rotatedPressure (fun _ => 0) U c = noninteractingPressure c := by |
| simp [rotatedPressure, rotatedEnergy, noninteractingPressure] |
|
|
| lemma abs_pressure_sub_noninteracting_le {N : ℕ} (hN : 0 < N) |
| (eig c : Fin N → ℝ) (U : Rotation N) (K : ℝ) (heig : ∀ i, |eig i| ≤ K) : |
| |rotatedPressure eig U c - noninteractingPressure c| ≤ K / 2 := by |
| simpa only [sub_zero, rotatedPressure_zero] using |
| abs_rotatedPressure_sub_le hN eig (fun _ => 0) U c K (fun i => by simpa using heig i) |
|
|
| /-- Uniformly bounded spectra give almost-sure concentration about the mean, |
| for arbitrary deterministic fields and arbitrary dependence across dimensions. -/ |
| theorem ae_pressure_sub_mean_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 ω ∂P, Tendsto (fun n => |
| rotatedPressure (eig n) (specialRotation (U n ω)) (c n) - |
| ∫ η, rotatedPressure (eig n) (specialRotation (U n η)) (c n) ∂P) |
| atTop (𝓝 0) := by |
| let F : (n : ℕ) → SpecialOrthogonal (n + 3) → ℝ := fun n V => |
| rotatedPressure (eig n) (specialRotation V) (c n) - noninteractingPressure (c n) |
| let f : ℕ → Ω → ℝ := fun n ω => F n (U n ω) |
| have hFm (n : ℕ) : Measurable (F n) := |
| (measurable_rotatedPressure (eig n) (c n)).sub_const _ |
| have hfm (n : ℕ) : Measurable (f n) := (hFm n).comp (hU n) |
| have hfb (n : ℕ) (ω : Ω) : |f n ω| ≤ K / 2 := |
| abs_pressure_sub_noninteracting_le (by omega) _ _ _ K (heig n) |
| obtain ⟨C, a, hC, ha, hp⟩ := hpub |
| have hm (n : ℕ) : ∃ m : ℝ, |
| (1 / 2 : ℝ) ≤ P.real {ω | f n ω ≤ m} ∧ |
| (1 / 2 : ℝ) ≤ P.real {ω | m ≤ f n ω} ∧ |
| ∀ r : ℝ, 0 < r → P.real {ω | r ≤ |f n ω - m|} ≤ |
| C * Real.exp (-a * (n + 3 : ℝ) * r ^ 2 / K ^ 2) := by |
| have hprob : IsProbabilityMeasure (P.map (U n)) := |
| (Measure.isProbabilityMeasure_map_iff (hU n).aemeasurable).mpr inferInstance |
| have hLip : ∀ V W, |F n V - F n W| ≤ K * frobeniusDistance V W := by |
| intro V W |
| dsimp only [F] |
| rw [sub_sub_sub_cancel_right] |
| exact abs_rotatedPressure_sub_rotation_le (by omega) _ _ _ _ K hK.le (heig n) |
| obtain ⟨m, hl, hu, ht⟩ := hp (n + 3) (by omega) (P.map (U n)) hprob |
| (hHaar n) (F n) (hFm n) K hK hLip |
| refine ⟨m, ?_, ?_, ?_⟩ |
| · simpa only [f, Set.preimage_ofPred_eq, measureReal_def, Measure.map_apply (hU n) |
| (measurableSet_le (hFm n) measurable_const)] using hl |
| · simpa only [f, Set.preimage_ofPred_eq, measureReal_def, Measure.map_apply (hU n) |
| (measurableSet_le measurable_const (hFm n))] using hu |
| · intro r hr |
| simpa only [f, Set.preimage_ofPred_eq, measureReal_def, Measure.map_apply (hU n) |
| (measurableSet_le measurable_const ((hFm n).sub_const m).abs), |
| Nat.cast_add, Nat.cast_ofNat] using ht r hr |
| choose m hl hu ht using hm |
| have htail : ∀ r : ℝ, 0 < r → ∃ C a : ℝ, 0 ≤ C ∧ 0 < a ∧ |
| ∀ n, P.real {ω | r ≤ |f n ω - m n|} ≤ C * Real.exp (-a * (n : ℝ)) := by |
| intro r hr |
| refine ⟨C, a * r ^ 2 / K ^ 2, hC.le, by positivity, ?_⟩ |
| intro n |
| refine (ht n r hr).trans (mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr ?_) hC.le) |
| have hnonneg : 0 ≤ a * r ^ 2 / K ^ 2 := by positivity |
| calc |
| -a * ((n : ℝ) + 3) * r ^ 2 / K ^ 2 = |
| -(a * r ^ 2 / K ^ 2) * ((n : ℝ) + 3) := by ring |
| _ ≤ -(a * r ^ 2 / K ^ 2) * (n : ℝ) := |
| mul_le_mul_of_nonpos_left (by linarith) (neg_nonpos.mpr hnonneg) |
| have hmean := tendsto_mean_sub_median_zero P f hfm m (K / 2) (by positivity) |
| hfb hl hu htail |
| have hae := ae_tendsto_zero_of_exponential_tail P (fun n ω => f n ω - m n) htail |
| have hfi (n : ℕ) : Integrable (f n) P := |
| (integrable_const (K / 2)).mono' (hfm n).aestronglyMeasurable |
| (ae_of_all _ fun ω => by simpa only [Real.norm_eq_abs] using hfb n ω) |
| filter_upwards [hae] with ω hω |
| have hresult := hω.sub hmean |
| have he (n : ℕ) : |
| f n ω - m n - ((∫ η, f n η ∂P) - m n) = |
| rotatedPressure (eig n) (specialRotation (U n ω)) (c n) - |
| ∫ η, rotatedPressure (eig n) (specialRotation (U n η)) (c n) ∂P := by |
| have hadd := integral_add (hfi n) (integrable_const (noninteractingPressure (c n))) |
| have heq : (fun η => f n η + noninteractingPressure (c n)) = |
| (fun η => rotatedPressure (eig n) (specialRotation (U n η)) (c n)) := by |
| funext η |
| simp [f, F] |
| rw [heq] at hadd |
| simp only [integral_const, probReal_univ, one_smul] at hadd |
| dsimp only [f, F] |
| linarith |
| simpa only [he, sub_self] using hresult |
| |
| end InvariantIsing |
| |
| end |
| |
| end OAI |
| |
