| import OAI.Probability.InvariantIsing.Pressure.RandomPressureBound |
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| namespace OAI |
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| /-! The complete in-probability, L1 and expectation clauses of Proposition gen:random. -/ |
| noncomputable section |
| open MeasureTheory ProbabilityTheory IsingPerceptron Filter Set |
| open scoped Topology |
| namespace InvariantIsing |
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| theorem random_pressure_limit |
| (hhaar : HaarConcentrationInput) (hgauss : GaussianLipschitzVarianceInput) |
| (hpub : PanchenkoTalagrandFieldPairInput) |
| {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] |
| (eig : (N : ℕ) → Ω → Fin N → ℝ) (Y : ℕ → Ω → ℝ) |
| (heig : ∀ N, Measurable (eig N)) (hY : ∀ N, Measurable (Y N)) |
| (H : (N : ℕ) → Measure (Orthogonal N)) [∀ N, IsProbabilityMeasure (H N)] |
| [∀ N, (H N).IsMulRightInvariant] |
| (hlaw : ∀ N, ConditionalFieldOrbitLaw P (fun ω => (eig N ω,fun _ => 0)) (Y N) (H N)) |
| (ν : ProbabilityMeasure ℝ) (a b : ℝ) |
| (hcompact : IsCompact (ν : Measure ℝ).support) |
| (hbound : (ν : Measure ℝ).support ⊆ Icc a b) |
| (ha : a∈(ν : Measure ℝ).support) (hb : b∈(ν : Measure ℝ).support) |
| (hweak : TendstoInMeasure P (fun k ω => LevyProkhorov.ofMeasure |
| (empiricalSpectralLaw (Nat.succ_pos k) (eig (k+1) ω))) atTop |
| (fun _ => LevyProkhorov.ofMeasure ν)) |
| (hexcess : TendstoInMeasure P (fun k ω => spectralExcess (eig (k+1) ω) a b) |
| atTop (fun _ => 0)) |
| (hUI : UniformIntegrable (fun k => Y (k+1)) 1 P) : |
| TendstoInMeasure P (fun k => Y (k+1)) atTop |
| (fun _ => (variationalFunctional (measureR (ν : Measure ℝ) b)).toReal) ∧ |
| Tendsto (fun k => eLpNorm (fun ω => Y (k+1) ω- |
| (variationalFunctional (measureR (ν : Measure ℝ) b)).toReal) 1 P) atTop (𝓝 0) ∧ |
| Tendsto (fun k => ∫ ω, Y (k+1) ω ∂P) atTop |
| (𝓝 (variationalFunctional (measureR (ν : Measure ℝ) b)).toReal) := by |
| have hp := random_pressure_tendsto_in_measure hhaar hgauss hpub P eig Y heig hY H hlaw |
| ν a b hcompact hbound ha hb hweak hexcess |
| exact ⟨hp,pressure_L1_of_probability_and_uniformIntegrability P _ _ hUI hp⟩ |
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| end InvariantIsing |
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| end |
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| end OAI |
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