| import OAI.Probability.InvariantIsing.Pressure.UniformHaarMeanTail | |
| import OAI.Probability.InvariantIsing.Cavity.CavityOrientedFamily | |
| import OAI.Probability.InvariantIsing.Spectral.SpectralCountComparison | |
| namespace OAI | |
| /-! A dimension-uniform centered tail for the physical Haar pressure, independent | |
| of the external field. This can be conditioned on random spectral and field data. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Filter Set | |
| open scoped Topology | |
| namespace InvariantIsing | |
| theorem uniform_physical_pressure_mean_tail (hpub : HaarConcentrationInput) : | |
| ∃ C a : ℝ, 0 < C ∧ 0 < a ∧ | |
| ∀ K ε : ℝ, 0 < K → 0 < ε → ∃ N₀ : ℕ, | |
| ∀ N : ℕ, N₀ ≤ N → 3 ≤ N → | |
| ∀ μ : Measure (Orthogonal N), IsProbabilityMeasure μ → μ.IsMulRightInvariant → | |
| ∀ eig c : Fin N → ℝ, (∀ i, |eig i| ≤ K) → | |
| μ.real {U | ε ≤ |rotatedPressure eig (matrixRotation U⁻¹) c- | |
| ∫ V, rotatedPressure eig (matrixRotation V⁻¹) c ∂μ|} ≤ | |
| C*Real.exp (-a*(N : ℝ)*(ε/2)^2/K^2) := by | |
| obtain ⟨C,a,hC,ha,ht⟩ := uniform_bounded_haar_mean_tail hpub | |
| refine ⟨C,a,hC,ha,?_⟩ | |
| intro K ε hK hε | |
| obtain ⟨N₀,hN₀⟩ := ht K ε hK hε | |
| refine ⟨N₀,?_⟩ | |
| intro N hN hN3 μ hμ hHaar eig c heig | |
| let : IsProbabilityMeasure μ := hμ | |
| let : μ.IsMulRightInvariant := hHaar | |
| have hpos : 0 < N := by omega | |
| let η := cavityOrientedBaseLaw hpos μ | |
| let f := fun U : SpecialOrthogonal N => rotatedPressure eig (specialRotation U) c-noninteractingPressure c | |
| have hf : Measurable f := (measurable_rotatedPressure eig c).sub_const _ | |
| have hfb U : |f U| ≤ K := | |
| (abs_pressure_sub_noninteracting_le hpos eig c (specialRotation U) K heig).trans (by linarith) | |
| have hlip U V : |f U-f V| ≤ K*frobeniusDistance U V := by | |
| dsimp only [f] | |
| rw [sub_sub_sub_cancel_right] | |
| exact abs_rotatedPressure_sub_rotation_le hpos eig U V c K hK.le heig | |
| have hh := hN₀ N hN hN3 η inferInstance (cavityOrientedBaseLaw_leftInvariant hpos μ) | |
| f hf hfb hlip | |
| have hmean : (∫ U, f U ∂η)= | |
| (∫ V, rotatedPressure eig (matrixRotation V⁻¹) c ∂μ)-noninteractingPressure c := by | |
| rw [integral_sub (integrable_countPressure hpos η eig c K heig) (integrable_const _)] | |
| simp only [integral_const,probReal_univ,one_smul] | |
| rw [cavity_oriented_pressure] | |
| have hset : MeasurableSet {U | ε ≤ |f U-(∫ V, f V ∂η)|} := | |
| measurableSet_le measurable_const (hf.sub_const _).abs | |
| change (cavityOrientedBaseLaw hpos μ).real _ ≤ _ at hh | |
| have hT : Measurable (fun V : Orthogonal N => cavityOrientationLift hpos V⁻¹) := | |
| (measurable_cavityOrientationLift hpos).comp measurable_inv | |
| rw [measureReal_def,cavityOrientedBaseLaw,Measure.map_apply hT hset] at hh | |
| convert hh using 1 | |
| congr 1 | |
| ext U | |
| change (ε ≤ |_ - _|) ↔ (ε ≤ |_ - _|) | |
| rw [hmean] | |
| simp only [f,rotatedPressure,cavityOrientationLift_energy] | |
| ring_nf | |
| end InvariantIsing | |
| end | |
| end OAI | |