| 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 |
| |
