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a89d282 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 | 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
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