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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 45 46 47 48 49 50 51 52 53 54 55 | import OAI.Probability.InvariantIsing.Core.FiniteSpectrumStepUpper
import OAI.Probability.InvariantIsing.Core.TrialGridApproximation
namespace OAI
/-! Manuscript Proposition up:bound, in the equivalent eventual-epsilon
form against every trial of finite entropy. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Set Filter
open scoped BigOperators Topology
namespace InvariantIsing
theorem finiteSpectrum_meanPressure_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))
(p : OverlapPath) (S : ℝ) (hS : entropyFunctional p ≤ (S : EReal)) :
∀ ε : ℝ, 0 < ε → ∀ᶠ k in atTop,
(∫ U : SpecialOrthogonal (N k),
rotatedPressure (eig k) (specialRotation U) (fun _ => 0) ∂μ k) ≤
S + spectralFunctional (finiteR ρ lam hρpos hρsum) p + ε := by
intro ε hε
have hhalf : 0 < ε / 2 := half_pos hε
have hgrid := lowerGridPath_spectral_tendsto ρ lam hρpos hρsum p
obtain ⟨n, hn⟩ := (hgrid.eventually (gt_mem_nhds
(show spectralFunctional (finiteR ρ lam hρpos hρsum) p <
spectralFunctional (finiteR ρ lam hρpos hρsum) p + ε / 2 by linarith))).exists
have hup := finiteSpectrum_meanPressure_step_upper hhaar hgauss N hN hNlim m n
μ hμinv eig K hK heig I hdis hcover lam hlam ρ hρpos hρsum hρ
(uniformCut n) (uniformCut_strict n) (uniformCut_zero n) (uniformCut_last n)
(lowerGridPath p n)
(fun i => (1 - 1 / (n + 1 : ℕ)) * p ((i.val : ℝ) / (n + 1 : ℕ)))
(lowerGridPath_on_cell p n) (1 / (n + 1 : ℕ)) (by positivity)
(ae_of_all _ (lowerGridPath_below_one p n)) S
((lowerGridPath_entropy_le p n).trans hS) (ε / 2) hhalf
filter_upwards [hup] with k hk
linarith
end InvariantIsing
end
end OAI
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