File size: 3,572 Bytes
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 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 | import OAI.Probability.InvariantIsing.Spectral.SelectedSpectralPressure
import OAI.Probability.InvariantIsing.Spectral.SelectedOrbitAlmostSure
import OAI.Probability.InvariantIsing.Spectral.SelectedSpectralExcess
import Mathlib.MeasureTheory.Measure.LevyProkhorovMetric
import Mathlib.MeasureTheory.Function.ConvergenceInMeasure
namespace OAI
/-! Proposition gen:random: convergence in probability from weak empirical
convergence and vanishing edge excess, with no independence across dimensions. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Filter Set
open scoped Topology
namespace InvariantIsing
theorem random_pressure_tendsto_in_measure
(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)
(hweak : TendstoInMeasure P (fun k ω => LevyProkhorov.ofMeasure
(empiricalSpectralLaw (Nat.succ_pos k) (eig (k+1) ω))) atTop
(fun _ => LevyProkhorov.ofMeasure ν))
(hexcess : TendstoInMeasure P (fun k ω => spectralExcess (eig (k+1) ω) a b)
atTop (fun _ => 0)) :
TendstoInMeasure P (fun k => Y (k+1)) atTop
(fun _ => (variationalFunctional (measureR (ν : Measure ℝ) b)).toReal) := by
apply (exists_seq_tendstoInMeasure_atTop_iff
(fun k => (hY (k+1)).aestronglyMeasurable)).mpr
intro ns hns
obtain ⟨u,hu,hwu⟩ := (hweak.comp hns.tendsto_atTop).exists_seq_tendsto_ae
obtain ⟨v,hv,hev⟩ := (hexcess.comp (hns.comp hu).tendsto_atTop).exists_seq_tendsto_ae
let s := ns ∘ u ∘ v
have hs : StrictMono s := hns.comp (hu.comp hv)
have hws : ∀ᵐ ω ∂P, Tendsto (fun k => empiricalSpectralLaw (Nat.succ_pos (s k))
(eig (s k+1) ω)) atTop (𝓝 ν) := by
filter_upwards [hwu] with ω hω
exact (LevyProkhorov.continuous_toMeasure_probabilityMeasure.tendsto _).comp
(hω.comp hv.tendsto_atTop)
have hbs : ∀ᵐ ω ∂P, ∃ K : ℝ, ∀ᶠ k in atTop, spectralRadius (eig (s k+1) ω) ≤ K := by
filter_upwards [hev] with ω hω
exact selected_spectralRadius_bounded s (fun k => eig (s k+1) ω) a b hω
have ht := ae_selected_orbit_pressure_sub_mean hhaar P s hs.id_le
(fun k ω => (eig (s k+1) ω,fun _ => 0)) (fun k => Y (s k+1))
(fun k => (heig (s k+1)).prodMk measurable_const) (fun k => hY (s k+1)) H
(fun k => hlaw (s k+1)) hbs
refine ⟨u ∘ v,hu.comp hv,?_⟩
filter_upwards [ht,hws,hev] with ω htω hwω heω
have hm := selected_spectral_pressure_tendsto hhaar hgauss hpub H
(fun k => s k+1) (fun k => Nat.succ_pos _) ((tendsto_add_atTop_nat 1).comp hs.tendsto_atTop)
(fun k => eig (s k+1) ω) ν a b hcompact hbound ha hb
(selected_no_outliers_of_excess (fun k => s k+1) (fun k => eig (s k+1) ω) a b heω) hwω
have hh := htω.add hm
simpa only [s,Function.comp_apply,dataPhysicalPressure,sub_add_cancel,zero_add] using hh
end InvariantIsing
end
end OAI
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