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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 | import OAI.Probability.InvariantIsing.Pressure.RandomOrbitAlmostSure
import OAI.Probability.InvariantIsing.Pressure.GeneralSpectralPressure
namespace OAI
/-! The almost-sure random-spectrum pressure theorem, under the conditional
orbit law in Proposition gen:random. No global integrability is assumed. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Filter Set
open scoped Topology
namespace InvariantIsing
theorem random_pressure_tendsto_ae
(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)
(hno : ∀ᵐ ω ∂P, ∀ ε : ℝ, 0 < ε → ∀ᶠ N in atTop,
∀ i, a-ε ≤ eig N ω i ∧ eig N ω i ≤ b+ε)
(hweak : ∀ᵐ ω ∂P, Tendsto (fun k =>
empiricalSpectralLaw (Nat.succ_pos k) (eig (k+1) ω)) atTop (𝓝 ν)) :
∀ᵐ ω ∂P, Tendsto (fun N => Y N ω) atTop
(𝓝 (variationalFunctional (measureR (ν : Measure ℝ) b)).toReal) := by
have hbounded : ∀ᵐ ω ∂P, ∃ K : ℝ, ∀ᶠ k in atTop,
spectralRadius (eig (k+1) ω) ≤ K := hno.mono fun ω hω =>
spectralRadius_eventually_bounded_of_no_outliers (fun N => eig N ω) a b hω
have ht := ae_random_orbit_pressure_sub_mean hhaar P
(fun N ω => (eig N ω,fun _ => 0)) Y
(fun N => (heig N).prodMk measurable_const) hY H hlaw hbounded
filter_upwards [ht,hno,hweak] with ω hω hn hw
have hm := general_spectral_mean_pressure_tendsto hhaar hgauss hpub H
(fun N => eig N ω) ν a b hcompact hbound ha hb hn hw
have hadd := hω.add (hm.comp (tendsto_add_atTop_nat 1))
simp only [dataPhysicalPressure,Function.comp_apply,sub_add_cancel,zero_add] at hadd
exact (tendsto_add_atTop_iff_nat 1).mp hadd
end InvariantIsing
end
end OAI
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