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c4cdbbd | 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 | import OAI.Probability.InvariantIsing.Core.FiniteSpectrumUpper
import OAI.Probability.InvariantIsing.Core.FiniteVariationalBounds
namespace OAI
/-! The finite-spectrum pressure upper bound after taking the path infimum. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Set Filter
open scoped BigOperators Topology
namespace InvariantIsing
theorem finiteSpectrum_meanPressure_variational_upper
(hhaar : HaarConcentrationInput) (hgauss : GaussianLipschitzVarianceInput)
(N : ℕ → ℕ) (hN : ∀ k, 3 ≤ N k) (hNlim : Tendsto N atTop atTop) (m : ℕ)
(μ : (k : ℕ) → Measure (SpecialOrthogonal (N k))) [∀ k, IsProbabilityMeasure (μ k)]
(hμinv : ∀ k, (μ k).IsMulLeftInvariant)
(eig : (k : ℕ) → Fin (N k) → ℝ)
(K : ℝ) (hK : 0 < K) (heig : ∀ k i, |eig k i| ≤ K)
(I : (k : ℕ) → Fin m → Finset (Fin (N k)))
(hdis : ∀ k, Set.PairwiseDisjoint (Set.univ : Set (Fin m)) (I k))
(hcover : ∀ k, Finset.univ.biUnion (I k) = Finset.univ)
(lam : Fin m → ℝ) (hlam : ∀ k a i, i ∈ I k a → eig k i = lam a)
(ρ : Fin m → ℝ) (hρpos : ∀ a, 0 < ρ a) (hρsum : ∑ a, ρ a = 1)
(hρ : Tendsto (fun k a => ((I k a).card : ℝ) / N k) atTop (𝓝 ρ)) :
∀ ε : ℝ, 0 < ε → ∀ᶠ k in atTop,
(∫ U : SpecialOrthogonal (N k),
rotatedPressure (eig k) (specialRotation U) (fun _ => 0) ∂μ k) ≤
(variationalFunctional (finiteR ρ lam hρpos hρsum)).toReal + ε := by
intro ε hε
obtain ⟨p, S, hS, hp⟩ := exists_finiteEntropy_trial_near_finiteVariational
ρ lam hρpos hρsum (half_pos hε)
have hup := finiteSpectrum_meanPressure_upper hhaar hgauss N hN hNlim m μ hμinv
eig K hK heig I hdis hcover lam hlam ρ hρpos hρsum hρ p S hS.le (ε / 2) (half_pos hε)
filter_upwards [hup] with k hk
linarith
end InvariantIsing
end
end OAI
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