openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Magnetic /MagneticConvergingFieldUpper.lean
| import OAI.Probability.InvariantIsing.Magnetic.MagneticConvergingVariationalUpper | |
| import OAI.Probability.InvariantIsing.Core.FiniteVariationalBounds | |
| import OAI.Probability.InvariantIsing.Fields.SpinGroupFieldPressure | |
| import OAI.Probability.InvariantIsing.Magnetic.RestrictedPressureFluctuation | |
| namespace OAI | |
| /-! The finite-spectrum pressure upper bound after taking the path infimum, | |
| under Haar and Gaussian concentration hypotheses. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory IsingPerceptron Set Filter | |
| open scoped BigOperators Topology | |
| namespace InvariantIsing | |
| theorem finiteSpectrum_magneticFieldPressure_converging_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)) | |
| {A : Type*} [Fintype A] [DecidableEq A] | |
| (group : ∀ k, Fin (N k) → A) (count : ℕ → A → ℕ) | |
| (hcount : ∀ k a, count k a ≤ spinGroupSize (group k) a) | |
| (γs mags : ℕ → A → ℝ) (γ mag : A → ℝ) | |
| (hγs : Tendsto γs atTop (𝓝 γ)) (hmags : Tendsto mags atTop (𝓝 mag)) | |
| (hγ : ∀ a, 0 ≤ γ a) (hγsum : ∑ a, γ a=1) | |
| (hmag : ∀ a, |mag a|<1) | |
| (hc : ∀ k a, (count k a : ℝ)=spinGroupSize (group k) a*((1+mags k a)/2)) | |
| (hgroup : ∀ k a, (spinGroupSize (group k) a : ℝ)=N k*γs k a) (b : A → ℝ) : | |
| ∀ ε : ℝ, 0 < ε → ∀ᶠ k in atTop, | |
| (∫ U : SpecialOrthogonal (N k), | |
| restrictedRotatedPressure (spinGroupSlice (group k) (count k)) (eig k) (specialRotation U) (fun i => b (group k i)) ∂μ k) ≤ | |
| (magneticVariationalFunctional (finiteR ρ lam hρpos hρsum) γ mag).toReal + | |
| (∑ a, γ a*b a*mag a) + ε := by | |
| intro ε hε | |
| obtain ⟨B,hB,hfluc⟩ := haar_restrictedPressure_mean_fluctuation hhaar | |
| have hf : Tendsto (fun k => ∑ a, γs k a*b a*mags k a) atTop | |
| (𝓝 (∑ a, γ a*b a*mag a)) := by | |
| apply tendsto_finsetSum | |
| intro a _ | |
| exact (((tendsto_pi_nhds.mp hγs) a).mul_const (b a)).mul ((tendsto_pi_nhds.mp hmags) a) | |
| have hup := finiteSpectrum_magneticPressure_converging_variational_upper hhaar hgauss | |
| N hN hNlim m μ hμinv eig K hK heig I hdis hcover lam hlam ρ hρpos hρsum hρ | |
| group count hcount γs mags γ mag hγs hmags hγ hγsum hmag hc hgroup (ε/2) (half_pos hε) | |
| filter_upwards [hup, hf.eventually (Iio_mem_nhds (show (∑ a, γ a*b a*mag a)< | |
| (∑ a, γ a*b a*mag a)+ε/2 by linarith))] with k hk hfk | |
| rw [mean_restrictedRotatedPressure_group_field (by have := hN k; omega) (μ k) | |
| (group k) (count k) (hcount k) (γs k) (mags k) b (hgroup k) (hc k) (eig k) | |
| (hfluc (N k) (hN k) (μ k) inferInstance (hμinv k) _ | |
| (spinGroupSlice_nonempty (group k) (count k) (hcount k)) (eig k) (fun _ => 0) K hK (heig k)).1] | |
| linarith | |
| end InvariantIsing | |
| end | |
| end OAI | |