File size: 4,658 Bytes
7d9f027 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 | 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
|