| import Mathlib | |
| import OAI.Probability.SKGap.Localization.LocalVolumeFull | |
| import OAI.Probability.SKGap.Stability.FieldLogDet | |
| import OAI.Probability.SKGap.Stability.FieldIntegralMeasure | |
| import OAI.Probability.SKGap.Entropy.BadIntegralEndpoints | |
| namespace OAI | |
| section | |
| noncomputable section | |
| namespace SKGap | |
| open Matrix Real Set MeasureTheory ProbabilityTheory GaussianDensity | |
| open scoped BigOperators Matrix.Norms.Frobenius ENNReal | |
| theorem positive_time_root_stability {j A K t₀ T : ℝ} (hj : 0 < j) (hj1 : j < 1) | |
| (hA : 1 ≤ A) (hsub : sqrt j*A < 1) (hK : 0 ≤ K) | |
| (ht₀ : 0 < t₀) (hT : t₀ ≤ T) : | |
| ∃ ε c ρ a : ℝ,0 < ε ∧ 0 < c ∧ 0 < ρ ∧ 0 < a ∧ ∃ N : ℕ,0 < N ∧ | |
| ∀ (n : ℕ),N ≤ n → ∀ t ∈ Icc t₀ T, | |
| ((gaussianCoordinates (MatrixCoordinates (Fin n))).prod | |
| (Measure.pi (fun _ : Fin n=>gaussianReal 0 t.toNNReal))).real | |
| {p | (∀ z : Field n,vectorNorm (plantedInteraction j (goeMatrix (j/(n:ℝ)) p.1)*ᵥz) ≤ K*vectorNorm z) ∧ | |
| ∃ y : Field n, vectorNorm (tapField j (plantedInteraction j (goeMatrix (j/(n:ℝ)) p.1)) | |
| (fun i=>t+p.2 i) y) ≤ ρ*sqrt (n:ℝ) ∧ | |
| (goeMatrix (j/(n:ℝ)) p.1,y)∈badFieldSet n j A ε c} ≤ | |
| 4*exp (-min 1 (1/(π^2*j))*(n:ℝ))+exp (-a*(n:ℝ)) := by | |
| obtain ⟨m,ε,σb,a,hm,hε,hσb,ha,Nb,hbad⟩ := positive_time_bad_integral hj hj1 hA hsub ht₀ hT | |
| have hA0 : 0 < A := zero_lt_one.trans_le hA | |
| let r := min (ε/4) (m/(12*j*A)) | |
| have hr : 0 < r := lt_min (by positivity) (by positivity) | |
| have hrε : 2*r ≤ ε/2 := by have hh := min_le_left (ε/4) (m/(12*j*A));dsimp [r];linarith | |
| have hrm : 6*j*A*r ≤ m/2 := by | |
| have hh := (le_div_iff₀ (by positivity : 0 < 12*j*A)).mp (min_le_right (ε/4) (m/(12*j*A))) | |
| dsimp [r] | |
| nlinarith only [hh] | |
| obtain ⟨ε₁,hε₁,hvol⟩ := local_volume_uniform hj.le hK hr (half_pos ha) | |
| obtain ⟨γ,hγ,Nd,hNd,hdet⟩ := field_logdet_uniform hj hj1 hε₁ | |
| obtain ⟨σ₀,hσ₀,hσvol⟩ := hvol γ hγ | |
| let σ := min (σ₀/2) σb | |
| have hσ : 0 < σ := lt_min (by positivity) hσb | |
| have hσless : σ < σ₀ := (min_le_left _ _).trans_lt (by linarith) | |
| have hσle : σ ≤ σb := min_le_right _ _ | |
| obtain ⟨ρ,hρ,Nv,hNv,hv⟩ := hσvol σ hσ hσless | |
| refine ⟨ε/2,m/2,ρ,a/2,half_pos hε,half_pos hm,hρ,half_pos ha, | |
| max Nb (max Nd Nv),hNd.trans_le ((le_max_left _ _).trans (le_max_right _ _)),?_⟩ | |
| intro n hn t ht | |
| have hnb : Nb ≤ n := (le_max_left _ _).trans hn | |
| have hnd : Nd ≤ n := (le_max_left _ _).trans ((le_max_right _ _).trans hn) | |
| have hnv : Nv ≤ n := (le_max_right _ _).trans ((le_max_right _ _).trans hn) | |
| have hn0 : 0 < n := hNv.trans_le hnv | |
| let : NeZero n := ⟨Nat.ne_of_gt hn0⟩ | |
| let μ := gaussianCoordinates (MatrixCoordinates (Fin n)) | |
| let ν := Measure.pi (fun _ : Fin n=>gaussianReal 0 t.toNNReal) | |
| let D : Set (MatrixCoordinates (Fin n) → ℝ) := {g | ∃ y : Field n, | |
| (n:ℝ)*(j*varianceAverage y^2+ε₁) < | |
| log ((fieldJacobian j (plantedInteraction j (goeMatrix (j/(n:ℝ)) g)) y)ᵀ* | |
| fieldJacobian j (plantedInteraction j (goeMatrix (j/(n:ℝ)) g)) y+γ • 1).det} | |
| let I : ((MatrixCoordinates (Fin n) → ℝ) × (Fin n → ℝ)) → ℝ≥0∞ := fun p=> | |
| fieldIntegral j σ (plantedInteraction j (goeMatrix (j/(n:ℝ)) p.1)) (fun i=>t+p.2 i) | |
| {y | (goeMatrix (j/(n:ℝ)) p.1,y)∈badFieldSet n j A ε m} | |
| have hI : Measurable I := measurable_fieldIntegral j t σ | |
| (badFieldFlat n j A ε m) (measurableSet_badFieldFlat _ _ _ _ _) | |
| have hInt : ∫⁻ p,I p ∂μ.prod ν ≤ ENNReal.ofReal (exp (-a*(n:ℝ))) := by | |
| rw [lintegral_prod _ hI.aemeasurable] | |
| exact hbad n hnb t ht σ ⟨hσ,hσle⟩ | |
| have hMarkov := markov_exponential (μ.prod ν) hI (b := a/2) hInt | |
| have hrate : a-a/2=a/2 := by ring | |
| rw [hrate] at hMarkov | |
| have hDP : (μ.prod ν).real (D ×ˢ (Set.univ : Set (Fin n → ℝ))) ≤ | |
| 4*exp (-min 1 (1/(π^2*j))*(n:ℝ)) := by | |
| rw [measureReal_prod_prod] | |
| simp only [Measure.real,measure_univ,ENNReal.toReal_one,mul_one] | |
| exact hdet n hnd | |
| apply measureReal_union_detector (μ.prod ν) _ (D ×ˢ Set.univ) I | |
| (exp (-(a/2)*(n:ℝ))) _ _ hDP hMarkov | |
| rintro p ⟨hJ,y,hres,hy⟩ hp | |
| have hp' : p.1∉D := fun hh=>hp ⟨hh,mem_univ _⟩ | |
| dsimp [I] | |
| rw [fieldIntegral_eq_localVolume hσ] | |
| apply hv n hnv _ hJ _ y | |
| {z | (goeMatrix (j/(n:ℝ)) p.1,z)∈badFieldSet n j A ε m} | |
| · exact measurableSet_badField_section j A ε m _ | |
| · intro d hd | |
| exact badField_neighborhood hn0 hj.le hA0.le hε hr.le hrε hrm _ y d hy hd | |
| · exact not_lt.mp (fun hh=>hp' ⟨y,hh⟩) | |
| · exact hres | |
| end SKGap | |
| end | |
| end | |
| end OAI | |