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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
namespace OAI
/-! Almost-sure extension of Theorem7.1 to a joint conditional orbit law. -/
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped Topology
namespace InvariantIsing
theorem random_field_pressure_tendsto_ae
(hhaar : HaarConcentrationInput) (hgauss : GaussianLipschitzVarianceInput)
(hpub : PanchenkoTalagrandRestrictedFieldPairInput)
{Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
(d : (N : ℕ) → Ω → FieldSpectralData N) (Y : ℕ → Ω → ℝ)
(hd : ∀ N, Measurable (d N)) (hY : ∀ N, Measurable (Y N))
(H : (N : ℕ) → Measure (Orthogonal N)) [∀ N, IsProbabilityMeasure (H N)]
[∀ N, (H N).IsMulRightInvariant]
(hlaw : ∀ N, ConditionalFieldOrbitLaw P (d N) (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 ≤ (d N ω).1 i ∧ (d N ω).1 i ≤ b+ε)
(hweak : ∀ᵐ ω ∂P, Tendsto (fun k =>
empiricalSpectralLaw (Nat.succ_pos k) (d (k+1) ω).1) atTop (𝓝 ν))
(ξ : ProbabilityMeasure ℝ) (hξ : Integrable (fun x : ℝ => x) (ξ : Measure ℝ))
(hw : ∀ᵐ ω ∂P, Tendsto (fun k => fieldWassersteinOne
(empiricalSpectralLaw (Nat.succ_pos k) (d (k+1) ω).2) ξ) atTop (𝓝 0)) :
∀ᵐ ω ∂P, Tendsto (fun N => Y N ω) atTop
(𝓝 (magneticFieldFunctional (ν : Measure ℝ) b ξ)) := by
have hbounded : ∀ᵐ ω ∂P, ∃ K : ℝ, ∀ᶠ k in atTop,
spectralRadius ((d (k+1) ω).1) ≤ K := hno.mono fun ω hω =>
spectralRadius_eventually_bounded_of_no_outliers (fun N => (d N ω).1) a b hω
have ht := ae_random_orbit_pressure_sub_mean hhaar P d Y hd hY H hlaw hbounded
filter_upwards [ht,hno,hweak,hw] with ω hω hn hweakω hwω
have hm := general_mean_wasserstein_field_pressure_tendsto hhaar hgauss hpub H
(fun N => (d N ω).1) ν a b hcompact hbound ha hb hn hweakω
(fun N => (d N ω).2) ξ hξ hwω
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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