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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 | import OAI.Probability.InvariantIsing.Cavity.CavitySelectedDimensions
import OAI.Probability.InvariantIsing.Spectral.FiniteSpectralMeasure
namespace OAI
/-! The finite-alphabet pressure theorem along diverging dimensions. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Filter
open scoped Topology BigOperators
namespace InvariantIsing
theorem selected_finite_alphabet_pressure_tendsto
(hhaar : HaarConcentrationInput) (hgauss : GaussianLipschitzVarianceInput)
(hpub : PanchenkoTalagrandFieldPairInput) {m : ℕ}
(μ : (N : ℕ) → Measure (Orthogonal N)) [∀ N, IsProbabilityMeasure (μ N)]
[∀ N, (μ N).IsMulRightInvariant] (lam : Fin m → ℝ)
(D : ℕ → ℕ) (hD : ∀ k, 0 < D k) (hDlim : Tendsto D atTop atTop)
(g : (k : ℕ) → Fin (D k) → Fin m)
(ρ : Fin m → ℝ) (hρ : ∀ a, 0 < ρ a) (hρsum : ∑ a, ρ a=1)
(hρlim : Tendsto (fun k a => ((cavitySpectralGroup (g k) a).card : ℝ)/D k) atTop (𝓝 ρ)) :
Tendsto (fun k => ∫ V, rotatedPressure (fun i => lam (g k i))
(matrixRotation V⁻¹) (fun _ => 0) ∂μ (D k)) atTop
(𝓝 (variationalFunctional (finiteR ρ lam hρ hρsum)).toReal) := by
by_cases hm : 2 ≤ m
· exact cavity_selected_dimensions_pressure_tendsto hhaar hgauss hpub hm μ lam D hD hDlim g ρ hρ hρsum hρlim
have hm0 : 0 < m := by
obtain ⟨a,_⟩ := exists_pos_spectral_weight (fun a => (hρ a).le) hρsum
exact Nat.pos_of_ne_zero (by intro hz; subst m; exact Fin.elim0 a)
have hm1 : m=1 := by omega
subst m
have hlam : lam=fun _ => lam 0 := funext fun a => congrArg lam (Subsingleton.elim a 0)
have hR : finiteR ρ lam hρ hρsum=fun _ => lam 0 := by
funext x
rw [hlam]
exact finiteR_constant ρ hρ hρsum x (lam 0)
rw [hR, variationalFunctional_constant, EReal.toReal_coe]
have he : ∀ᶠ k in atTop, (∫ V, rotatedPressure (fun i => lam (g k i))
(matrixRotation V⁻¹) (fun _ => 0) ∂μ (D k))=lam 0/2 := by
apply Filter.Eventually.of_forall
intro k
have hpos : 0 < D k := hD k
have heig : (fun i => lam (g k i))=fun _ => lam 0 :=
funext fun i => congrArg lam (Subsingleton.elim (g k i) 0)
rw [heig]
simp only [rotatedPressure_constant hpos,integral_const,measureReal_def,measure_univ,
ENNReal.toReal_one,one_smul]
exact tendsto_const_nhds.congr' (he.mono fun _ h => h.symm)
end InvariantIsing
end
end OAI
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