| import OAI.Probability.InvariantIsing.Pressure.RandomOrbitIntegration |
| import OAI.Probability.InvariantIsing.Pressure.UniformPhysicalTail |
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|
| namespace OAI |
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| /-! A localized tail bound under the exact joint conditional orbit hypothesis. -/ |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
| namespace InvariantIsing |
|
|
| lemma conditional_orbit_tail_bound {Ω : Type*} [MeasurableSpace Ω] {N : ℕ} |
| (hN : 0 < N) (P : Measure Ω) [IsProbabilityMeasure P] |
| (d : Ω → FieldSpectralData N) (Y : Ω → ℝ) (hd : Measurable d) (hY : Measurable Y) |
| (H : Measure (Orthogonal N)) [IsProbabilityMeasure H] |
| (hlaw : ConditionalFieldOrbitLaw P d Y H) (K ε B : ℝ) |
| (hbound : ∀ data : FieldSpectralData N, (∀ i, |data.1 i| ≤ K) → |
| H.real {U | ε ≤ |dataPhysicalPressure data U-(∫ V, dataPhysicalPressure data V ∂H)|} ≤ B) : |
| P {ω | (∀ i, |(d ω).1 i| ≤ K) ∧ |
| ε ≤ |Y ω-(∫ V, dataPhysicalPressure (d ω) V ∂H)|} ≤ ENNReal.ofReal B := by |
| let mean := fun data : FieldSpectralData N => ∫ V, dataPhysicalPressure data V ∂H |
| have hm : Measurable mean := measurable_dataPressureMean hN H |
| let E : Set (FieldSpectralData N × ℝ) := {z | (∀ i, |z.1.1 i| ≤ K) ∧ ε ≤ |z.2-mean z.1|} |
| have hK : MeasurableSet {z : FieldSpectralData N × ℝ | ∀ i, |z.1.1 i| ≤ K} := by |
| simp only [ofPred_forall] |
| apply MeasurableSet.iInter |
| intro i |
| exact measurableSet_le (((measurable_pi_apply i).comp |
| (measurable_fst.comp measurable_fst)).abs) measurable_const |
| have hE : MeasurableSet E := hK.inter |
| (measurableSet_le measurable_const (measurable_snd.sub (hm.comp measurable_fst)).abs) |
| have hp : Measurable (fun z : FieldSpectralData N × Orthogonal N => |
| (z.1,dataPhysicalPressure z.1 z.2)) := |
| measurable_fst.prodMk (measurable_dataPhysicalPressure hN) |
| have he := congrArg (fun Q : Measure (FieldSpectralData N × ℝ) => Q E) hlaw |
| rw [Measure.map_apply (hd.prodMk hY) hE,Measure.map_apply hp hE, |
| Measure.prod_apply (hp hE)] at he |
| let : IsProbabilityMeasure (P.map d) := |
| (Measure.isProbabilityMeasure_map_iff hd.aemeasurable).mpr inferInstance |
| change P ((fun ω => (d ω,Y ω)) ⁻¹' E) ≤ _ |
| rw [he] |
| calc |
| _ ≤ ∫⁻ _data : FieldSpectralData N, ENNReal.ofReal B ∂P.map d := by |
| apply lintegral_mono |
| intro data |
| dsimp only |
| by_cases hdata : ∀ i, |data.1 i| ≤ K |
| · have hevent : Prod.mk data ⁻¹' ((fun z : FieldSpectralData N × Orthogonal N => |
| (z.1,dataPhysicalPressure z.1 z.2)) ⁻¹' E)= |
| {U | ε ≤ |dataPhysicalPressure data U-mean data|} := by |
| ext U |
| change ((∀ i, |data.1 i| ≤ K) ∧ ε ≤ |dataPhysicalPressure data U-mean data|) ↔ _ |
| exact and_iff_right hdata |
| rw [hevent] |
| have hb := ENNReal.ofReal_le_ofReal (hbound data hdata) |
| rwa [measureReal_def,ENNReal.ofReal_toReal (measure_ne_top H _)] at hb |
| · have hevent : Prod.mk data ⁻¹' ((fun z : FieldSpectralData N × Orthogonal N => |
| (z.1,dataPhysicalPressure z.1 z.2)) ⁻¹' E)=∅ := by |
| ext U |
| change ((∀ i, |data.1 i| ≤ K) ∧ ε ≤ |dataPhysicalPressure data U-mean data|) ↔ False |
| simp only [hdata,false_and] |
| rw [hevent,measure_empty] |
| exact bot_le |
| _ = _ := by simp |
|
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| end InvariantIsing |
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| end |
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| end OAI |
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