File size: 2,353 Bytes
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 56 | import OAI.Probability.InvariantIsing.Cavity.CavityFinitePressureLimit
import OAI.Probability.InvariantIsing.Spectral.FiniteSpectralMeasure
namespace OAI
/-! The finite-alphabet conditional pressure theorem, including the
one-point spectral law. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Filter
open scoped Topology BigOperators
namespace InvariantIsing
theorem 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)
(g : (N : ℕ) → Fin N → Fin m)
(ρ : Fin m → ℝ) (hρ : ∀ a, 0 < ρ a) (hρsum : ∑ a, ρ a=1)
(hρlim : Tendsto (fun N a => ((cavitySpectralGroup (g N) a).card : ℝ)/N) atTop (𝓝 ρ)) :
Tendsto (fun N => ∫ V, rotatedPressure (fun i => lam (g N i))
(matrixRotation V⁻¹) (fun _ => 0) ∂μ N) atTop
(𝓝 (variationalFunctional (finiteR ρ lam hρ hρsum)).toReal) := by
by_cases hm : 2 ≤ m
· exact cavity_finite_pressure_tendsto hhaar hgauss hpub hm μ lam 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 N in atTop, (∫ V, rotatedPressure (fun i => lam (g N i))
(matrixRotation V⁻¹) (fun _ => 0) ∂μ N)=lam 0/2 := by
apply eventually_atTop.mpr
refine ⟨1,fun N hN => ?_⟩
have hpos : 0 < N := by omega
have heig : (fun i => lam (g N i))=fun _ => lam 0 :=
funext fun i => congrArg lam (Subsingleton.elim (g N 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
|