openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Pressure /GeneralSpectralPressure.lean
| import OAI.Probability.InvariantIsing.Spectral.CompactSpectralPressure | |
| import OAI.Probability.InvariantIsing.Spectral.SpectralExcess | |
| namespace OAI | |
| /-! The general compact spectral law and no-outlier mean-pressure theorem. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory IsingPerceptron Filter Set | |
| open scoped Topology | |
| namespace InvariantIsing | |
| theorem general_spectral_mean_pressure_tendsto | |
| (hhaar : HaarConcentrationInput) (hgauss : GaussianLipschitzVarianceInput) | |
| (hpub : PanchenkoTalagrandFieldPairInput) | |
| (μ : (N : ℕ) → Measure (Orthogonal N)) [∀ N, IsProbabilityMeasure (μ N)] | |
| [∀ N, (μ N).IsMulRightInvariant] (eig : (N : ℕ) → Fin N → ℝ) | |
| (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) | |
| (hno : ∀ ε : ℝ, 0 < ε → ∀ᶠ N in atTop, ∀ i, a-ε ≤ eig N i ∧ eig N i ≤ b+ε) | |
| (hweak : Tendsto (fun k => empiricalSpectralLaw (Nat.succ_pos k) (eig (k+1))) | |
| atTop (𝓝 ν)) : | |
| Tendsto (fun N => ∫ V, rotatedPressure (eig N) (matrixRotation V⁻¹) (fun _ => 0) ∂μ N) | |
| atTop (𝓝 (variationalFunctional (measureR (ν : Measure ℝ) b)).toReal) := by | |
| have hab : a ≤ b := (hbound ha).2 | |
| have hc := compact_spectral_pressure_tendsto_of_mem hhaar hgauss hpub μ | |
| (fun N i => spectralClip a b (eig N i)) ν a b hcompact hbound ha hb | |
| (fun N i => spectralClip_mem hab (eig N i)) | |
| (spectral_clip_empirical_tendsto eig ν a b hbound hweak) | |
| have hη := spectralExcess_tendsto eig a b hno | |
| have hd : Tendsto (fun N => | |
| (∫ V, rotatedPressure (fun i => spectralClip a b (eig N i)) | |
| (matrixRotation V⁻¹) (fun _ => 0) ∂μ N)- | |
| ∫ V, rotatedPressure (eig N) (matrixRotation V⁻¹) (fun _ => 0) ∂μ N) | |
| atTop (𝓝 0) := by | |
| apply squeeze_zero_norm' _ (by simpa only [zero_div] using hη.div_const 2) | |
| filter_upwards [eventually_ge_atTop 1] with N hN | |
| simpa only [Real.norm_eq_abs] using spectral_clip_mean_pressure_close (by omega) | |
| (μ N) (eig N) a b (spectralExcess (eig N) a b) hab | |
| (spectralExcess_nonneg _ _ _) (spectralExcess_bounds _ _ _) | |
| have hh := hc.sub hd | |
| simpa only [sub_sub_cancel,sub_zero] using hh | |
| end InvariantIsing | |
| end | |
| end OAI | |