| import OAI.Probability.InvariantIsing.Pressure.RandomOrbitTail |
| import OAI.Probability.InvariantIsing.Pressure.RandomSpectralBound |
| import OAI.Probability.InvariantIsing.Pressure.LocalizedAlmostSure |
|
|
| namespace OAI |
|
|
| /-! Almost-sure conditional Haar concentration for random spectra and fields. -/ |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology ENNReal |
| namespace InvariantIsing |
|
|
| theorem ae_random_orbit_pressure_sub_mean (hhaar : HaarConcentrationInput) |
| {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] |
| (d : (N : ℕ) → Ω → FieldSpectralData N) (Y : ℕ → Ω → ℝ) |
| (hd : ∀ N, Measurable (d N)) (hY : ∀ N, Measurable (Y N)) |
| (H : (N : ℕ) → Measure (Orthogonal N)) [∀ N, IsProbabilityMeasure (H N)] |
| [∀ N, (H N).IsMulRightInvariant] |
| (hlaw : ∀ N, ConditionalFieldOrbitLaw P (d N) (Y N) (H N)) |
| (hbound : ∀ᵐ ω ∂P, ∃ K : ℝ, ∀ᶠ k in atTop, spectralRadius ((d (k+1) ω).1) ≤ K) : |
| ∀ᵐ ω ∂P, Tendsto (fun k => Y (k+1) ω- |
| ∫ U, dataPhysicalPressure (d (k+1) ω) U ∂H (k+1)) atTop (𝓝 0) := by |
| apply ae_tendsto_zero_of_localized_exponential_tail P |
| (fun k ω => Y (k+1) ω-∫ U, dataPhysicalPressure (d (k+1) ω) U ∂H (k+1)) |
| (fun k ω => spectralRadius ((d (k+1) ω).1)) _ hbound |
| intro K ε hK hε |
| obtain ⟨C,a,hC,ha,ht⟩ := uniform_physical_pressure_mean_tail hhaar |
| obtain ⟨N₀,hN₀⟩ := ht K ε hK hε |
| refine ⟨C,a*(ε/2)^2/K^2,hC.le,by positivity,max N₀ 3,?_⟩ |
| intro n hn |
| have hb := conditional_orbit_tail_bound (Nat.succ_pos n) P (d (n+1)) (Y (n+1)) |
| (hd (n+1)) (hY (n+1)) (H (n+1)) (hlaw (n+1)) K ε |
| (C*Real.exp (-a*(n+1 : ℝ)*(ε/2)^2/K^2)) (fun data hdata => by |
| simpa only [dataPhysicalPressure,Nat.cast_add,Nat.cast_one] using |
| hN₀ (n+1) (by omega) (by omega) (H (n+1)) inferInstance inferInstance |
| data.1 data.2 hdata) |
| have he : {ω | (∀ i, |(d (n+1) ω).1 i| ≤ K) ∧ |
| ε ≤ |Y (n+1) ω-∫ U, dataPhysicalPressure (d (n+1) ω) U ∂H (n+1)|}= |
| {ω | ε ≤ |Y (n+1) ω-∫ U, dataPhysicalPressure (d (n+1) ω) U ∂H (n+1)| ∧ |
| spectralRadius ((d (n+1) ω).1) ≤ K} := by |
| ext ω |
| simp only [mem_ofPred_eq,spectralRadius_le_iff,and_comm] |
| rw [he] at hb |
| have hreal := ENNReal.toReal_mono (ENNReal.ofReal_ne_top) hb |
| rw [ENNReal.toReal_ofReal (by positivity)] at hreal |
| refine hreal.trans ?_ |
| apply mul_le_mul_of_nonneg_left _ hC.le |
| apply Real.exp_le_exp.mpr |
| have hc : 0 ≤ a*(ε/2)^2/K^2 := by positivity |
| calc |
| -a*(n+1 : ℝ)*(ε/2)^2/K^2= -(a*(ε/2)^2/K^2)*((n : ℝ)+1) := by ring |
| _ ≤ -(a*(ε/2)^2/K^2)*(n : ℝ) := by nlinarith |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
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