openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Pressure /PhysicalPressureConcentration.lean
| import OAI.Probability.InvariantIsing.Cavity.CavityOrientedFamily | |
| import OAI.Probability.InvariantIsing.Pressure.PressureMean | |
| namespace OAI | |
| /-! Almost-sure pressure concentration for the actual orthogonal matrices. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped Topology | |
| namespace InvariantIsing | |
| theorem ae_physical_pressure_sub_mean_tendsto_zero (hpub : HaarConcentrationInput) | |
| {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] | |
| (U : (n : ℕ) → Ω → Orthogonal (n+3)) (hU : ∀ n, Measurable (U n)) | |
| (hHaar : ∀ n, (P.map (U n)).IsMulRightInvariant) | |
| (eig c : (n : ℕ) → Fin (n+3) → ℝ) | |
| (K : ℝ) (hK : 0 < K) (heig : ∀ n i, |eig n i| ≤ K) : | |
| ∀ᵐ ω ∂P, Tendsto (fun n => | |
| rotatedPressure (eig n) (matrixRotation (U n ω)⁻¹) (c n)- | |
| ∫ η, rotatedPressure (eig n) (matrixRotation (U n η)⁻¹) (c n) ∂P) | |
| atTop (𝓝 0) := by | |
| let V (n : ℕ) (ω : Ω) := cavityOrientationLift (by omega : 0 < n+3) (U n ω)⁻¹ | |
| have hm n : Measurable (V n) := | |
| (measurable_cavityOrientationLift _).comp (hU n).inv | |
| have hl n : P.map (V n)=cavityOrientedBaseLaw (by omega : 0 < n+3) (P.map (U n)) := by | |
| change P.map ((fun W => cavityOrientationLift (by omega : 0 < n+3) W⁻¹) ∘ U n)=_ | |
| exact (Measure.map_map ((measurable_cavityOrientationLift _).comp measurable_inv) (hU n)).symm | |
| have hv n : (P.map (V n)).IsMulLeftInvariant := by | |
| let : IsProbabilityMeasure (P.map (U n)) := | |
| (Measure.isProbabilityMeasure_map_iff (hU n).aemeasurable).mpr inferInstance | |
| let : (P.map (U n)).IsMulRightInvariant := hHaar n | |
| rw [hl n] | |
| exact cavityOrientedBaseLaw_leftInvariant _ _ | |
| have he n ω : rotatedPressure (eig n) (specialRotation (V n ω)) (c n)= | |
| rotatedPressure (eig n) (matrixRotation (U n ω)⁻¹) (c n) := by | |
| simp only [V,rotatedPressure,cavityOrientationLift_energy] | |
| have hh := ae_pressure_sub_mean_tendsto_zero hpub P V hm hv eig c K hK heig | |
| simpa only [he] using hh | |
| end InvariantIsing | |
| end | |
| end OAI | |