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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 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 | 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
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