| import Mathlib |
| import OAI.Probability.SKGap.Stability.FieldTonelli |
| import OAI.Probability.SKGap.Localization.ConditionalIntegralRate |
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| namespace OAI |
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| section |
| noncomputable section |
| namespace SKGap |
| open Matrix Real Set MeasureTheory ProbabilityTheory GaussianDensity |
| open scoped BigOperators Matrix.Norms.Frobenius ENNReal |
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| lemma conditionalFieldWeight_cylinder {n : ℕ} [NeZero n] (j t σ : ℝ) |
| (E : Set (Fin n → ℝ)) (y : Fin n → ℝ) : |
| conditionalFieldWeight j t σ (Set.univ ×ˢ E) y=E.indicator (fun _=>1) y := by |
| by_cases hy : y∈E <;> simp [conditionalFieldWeight,hy] |
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| lemma field_integral_scalar_bound {n : ℕ} [NeZero n] {j t σ : ℝ} |
| (hj : 0 ≤ j) (ht : 0 ≤ t) (hσ : 0 < σ) (E : Set (Fin n → ℝ)) (hE : MeasurableSet E) : |
| (∫⁻ g, ∫⁻ x, fieldIntegral j σ (plantedInteraction j (goeMatrix (j/(n:ℝ)) g)) |
| (fun i=>t+x i) E |
| ∂Measure.pi (fun _ : Fin n=>gaussianReal 0 t.toNNReal) |
| ∂gaussianCoordinates (MatrixCoordinates (Fin n))) ≤ |
| ∫⁻ y in E, ENNReal.ofReal (scalarIntegrand j t σ y) := by |
| have hh := field_integral_gaussian_reduction hj ht hσ (Set.univ ×ˢ E) (MeasurableSet.univ.prod hE) |
| simp only [mem_prod,mem_univ,true_and,ofPred_mem_eq,conditionalFieldWeight_cylinder] at hh |
| rw [hh,← lintegral_indicator hE] |
| apply lintegral_mono |
| intro y |
| by_cases hy : y∈E |
| · simp only [indicator_of_mem hy,mul_one] |
| simpa only [mul_one] using (scalar_weighted_density_bound hj ht hσ y (le_rfl (a := (1:ℝ≥0∞)))) |
| · simp [hy] |
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| theorem smalltime_bad_integral {j q₀ : ℝ} (hj : 0 < j) (hj1 : j < 1) (hq₀ : 0 < q₀) : |
| ∃ t₀ σ₀ a : ℝ, 0 < t₀ ∧ 0 < σ₀ ∧ 0 < a ∧ ∃ N : ℕ, |
| ∀ (n : ℕ) [NeZero n],N ≤ n → ∀ t ∈ Icc 0 t₀,∀ σ ∈ Ioc 0 σ₀, |
| (∫⁻ g, ∫⁻ x, fieldIntegral j σ (plantedInteraction j (goeMatrix (j/(n:ℝ)) g)) |
| (fun i=>t+x i) {y | q₀ ≤ overlap y} |
| ∂Measure.pi (fun _ : Fin n=>gaussianReal 0 t.toNNReal) |
| ∂gaussianCoordinates (MatrixCoordinates (Fin n))) ≤ ENNReal.ofReal (exp (-a*(n:ℝ))) := by |
| obtain ⟨t₀,ht₀,δ,hδ,a,ha,N,hN⟩ := scalar_rate_iii hj hj1 hq₀ |
| refine ⟨t₀,min δ 1,a,ht₀,lt_min hδ zero_lt_one,ha,N,?_⟩ |
| intro n _ hn t ht σ hσ |
| have hE : MeasurableSet {y : Fin n → ℝ | q₀ ≤ overlap y} := |
| (isClosed_le continuous_const (by unfold overlap magnetization;fun_prop)).measurableSet |
| apply (field_integral_scalar_bound hj.le ht.1 hσ.1 _ hE).trans |
| exact hN n hn (NeZero.pos n) t ht σ ⟨hσ.1.le,hσ.2⟩ _ hE |
| (fun y hy=>by rw [scalarQMoment_empirical_overlap];exact hy) |
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| theorem positive_time_bad_integral {j A t₀ T : ℝ} (hj : 0 < j) (hj1 : j < 1) |
| (hA : 1 ≤ A) (hsub : sqrt j*A < 1) (ht₀ : 0 < t₀) (hT : t₀ ≤ T) : |
| ∃ m ε σ₀ a : ℝ,0 < m ∧ 0 < ε ∧ 0 < σ₀ ∧ 0 < a ∧ ∃ N : ℕ, |
| ∀ (n : ℕ) [NeZero n],N ≤ n → ∀ t ∈ Icc t₀ T,∀ σ ∈ Ioc 0 σ₀, |
| (∫⁻ g, ∫⁻ x, fieldIntegral j σ (plantedInteraction j (goeMatrix (j/(n:ℝ)) g)) |
| (fun i=>t+x i) {y | (goeMatrix (j/(n:ℝ)) g,y)∈badFieldSet n j A ε m} |
| ∂Measure.pi (fun _ : Fin n=>gaussianReal 0 t.toNNReal) |
| ∂gaussianCoordinates (MatrixCoordinates (Fin n))) ≤ ENNReal.ofReal (exp (-a*(n:ℝ))) := by |
| obtain ⟨m,ε,σ₀,a,hm,hε,hσ₀,ha,N,hN⟩ := conditional_bad_integral_rate hj hj1 hA hsub ht₀ hT |
| refine ⟨m,ε,σ₀,a,hm,hε,hσ₀,ha,N,?_⟩ |
| intro n _ hn t ht σ hσ |
| have hh := field_integral_gaussian_reduction hj.le (ht₀.le.trans ht.1) hσ.1 |
| (badFieldFlat n j A ε m) (measurableSet_badFieldFlat _ _ _ _ _) |
| change _=∫⁻ y,reducedBadDensity j t σ A ε m y at hh |
| erw [hh] |
| exact hN n hn t ht σ hσ |
| end SKGap |
| end |
| end |
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| end OAI |
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