| import OAI.Probability.InvariantIsing.Gaussian.GaussianGramExcessBound |
| import OAI.Probability.InvariantIsing.Gaussian.GaussianTransposeLowerEdge |
| import OAI.Probability.InvariantIsing.Gaussian.GaussianSingularUpperEdge |
| import OAI.Probability.InvariantIsing.Spectral.SpectralExcessScale |
| import OAI.Probability.InvariantIsing.Spectral.SpectralExcessContinuity |
|
|
| namespace OAI |
|
|
| /-! Both spectral edges of the physical Gaussian Gram matrix, with arbitrary real scaling. -/ |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology |
| namespace InvariantIsing |
|
|
| lemma marchenkoPasturLower_positive_part {α : ℝ} (hα : 0 ≤ α) : |
| marchenkoPasturLower α = (max (Real.sqrt α-1) 0)^2 := by |
| have hs := Real.sq_sqrt hα |
| have hn := Real.sqrt_nonneg α |
| by_cases ha : α ≤ 1 |
| · have hh : Real.sqrt α-1 ≤ 0 := by nlinarith |
| simp only [marchenkoPasturLower,ite_eq_left ha,max_eq_right hh,zero_pow (by decide : 2 ≠ 0)] |
| · have hh : 0 ≤ Real.sqrt α-1 := by nlinarith |
| rw [marchenkoPasturLower,ite_eq_right ha,marchenkoPasturA,max_eq_left hh] |
| ring |
|
|
| lemma positive_part_deficit (x b : ℝ) (hb : 0 ≤ b) : |
| max (max x 0-b) 0 = max (x-b) 0 := by |
| by_cases hx : 0 ≤ x |
| · rw [max_eq_left hx] |
| · have hx' : x ≤ 0 := le_of_not_ge hx |
| rw [max_eq_right hx',max_eq_right (by linarith : 0-b ≤ 0),max_eq_right (by linarith : x-b ≤ 0)] |
|
|
| theorem gaussianPattern_spectralExcess_tendsto {α : ℝ} (hα : 0 < α) |
| {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] |
| (Z : (N : ℕ) → Ω → EuclideanSpace ℝ (Fin N × Fin (gaussianPatternCount α N))) |
| (hZ : ∀ N, HasLaw (Z N) (stdGaussian _) P) (c : ℝ) : |
| TendstoInMeasure P (fun k ω => spectralExcess |
| (fun i => c*gaussianPatternEigenvalues (Z (k+1) ω) i) |
| (min (c*marchenkoPasturLower α) (c*marchenkoPasturB α)) |
| (max (c*marchenkoPasturLower α) (c*marchenkoPasturB α))) atTop (fun _ => 0) := by |
| let L := max (Real.sqrt α-1) 0 |
| let U := 1+Real.sqrt α |
| let du (k : ℕ) (ω : Ω) := max (gaussianPatternSingularMax (Z (k+1) ω)/Real.sqrt (k+1)-U) 0 |
| let dl (k : ℕ) (ω : Ω) := max (Real.sqrt α-1- |
| gaussianPatternSingularMin (gaussianPatternTranspose (k+1) (gaussianPatternCount α (k+1)) |
| (Z (k+1) ω))/Real.sqrt (k+1)) 0 |
| let E (k : ℕ) (ω : Ω) := spectralExcess |
| (fun i => c*gaussianPatternEigenvalues (Z (k+1) ω) i) |
| (min (c*marchenkoPasturLower α) (c*marchenkoPasturB α)) |
| (max (c*marchenkoPasturLower α) (c*marchenkoPasturB α)) |
| have hU : TendstoInMeasure P du atTop (fun _ => 0) := gaussianPatternSingularMax_upper_edge hα.le P Z hZ |
| have hL : TendstoInMeasure P dl atTop (fun _ => 0) := gaussianPatternTranspose_lower_edge hα P Z hZ |
| have hbound (k : ℕ) (ω : Ω) : E k ω ≤ |c| * ((U+du k ω)^2-U^2+2*L*dl k ω) := by |
| have hh := gaussianGram_spectralExcess_bound (Z (k+1) ω) L U (le_max_right _ _) |
| (by dsimp [U]; positivity) |
| have hb : 0 ≤ gaussianPatternSingularMin |
| (gaussianPatternTranspose (k+1) (gaussianPatternCount α (k+1)) (Z (k+1) ω))/Real.sqrt (k+1) := |
| div_nonneg (gaussianPatternSingularMin_nonneg (Nat.succ_pos k) _) (Real.sqrt_nonneg _) |
| rw [positive_part_deficit _ _ hb] at hh |
| have hlo : marchenkoPasturLower α = L^2 := marchenkoPasturLower_positive_part hα.le |
| have hup : marchenkoPasturB α = U^2 := rfl |
| have he := spectralExcess_scale_le (gaussianPatternEigenvalues (Z (k+1) ω)) (L^2) (U^2) c |
| dsimp only [E] |
| rw [hlo,hup] |
| exact he.trans (mul_le_mul_of_nonneg_left hh (abs_nonneg c)) |
| have hmeas (k : ℕ) : AEStronglyMeasurable (E k) P := by |
| have hh := continuous_spectralExcess |
| (fun z : EuclideanSpace ℝ (Fin (k+1) × Fin (gaussianPatternCount α (k+1))) => |
| fun i => c*gaussianPatternEigenvalues z i) |
| (fun i => continuous_const.mul ((continuous_apply i).comp (continuous_gaussianPatternEigenvalues _ _))) |
| (min (c*marchenkoPasturLower α) (c*marchenkoPasturB α)) |
| (max (c*marchenkoPasturLower α) (c*marchenkoPasturB α)) |
| exact (hh.measurable.aemeasurable.comp_aemeasurable (hZ (k+1)).aemeasurable).aestronglyMeasurable |
| apply (exists_seq_tendstoInMeasure_atTop_iff hmeas).mpr |
| intro ns hns |
| obtain ⟨u,hu,hua⟩ := (hU.comp hns.tendsto_atTop).exists_seq_tendsto_ae |
| obtain ⟨v,hv,hva⟩ := (hL.comp (hns.comp hu).tendsto_atTop).exists_seq_tendsto_ae |
| refine ⟨u ∘ v,hu.comp hv,?_⟩ |
| filter_upwards [hua,hva] with ω hωu hωv |
| have hau : Tendsto (fun k => du (ns (u (v k))) ω) atTop (𝓝 0) := hωu.comp hv.tendsto_atTop |
| have hav : Tendsto (fun k => dl (ns (u (v k))) ω) atTop (𝓝 0) := hωv |
| have ht := (((tendsto_const_nhds (x := U)).add hau).pow 2 |>.sub_const (U^2)).add |
| (hav.const_mul (2*L)) |>.const_mul |c| |
| have ht' : Tendsto (fun k => |c| *((U+du (ns (u (v k))) ω)^2-U^2+2*L*dl (ns (u (v k))) ω)) |
| atTop (𝓝 0) := by simpa only [add_zero,sub_self,mul_zero,zero_add] using ht |
| exact squeeze_zero (fun k => spectralExcess_nonneg _ _ _) |
| (fun k => hbound (ns (u (v k))) ω) ht' |
|
|
| end InvariantIsing |
|
|
| end |
|
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| end OAI |
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