| import OAI.Probability.InvariantIsing.Cavity.CavityPhysicalLogLimit |
| import OAI.Probability.InvariantIsing.Cavity.CavityCanonicalHaarLog |
|
|
| namespace OAI |
|
|
| /-! The spectral/GG logarithmic limit in the canonical coordinates of the |
| physical pressure increment. -/ |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory IsingPerceptron Filter |
| open scoped Topology BigOperators |
|
|
| namespace InvariantIsing |
|
|
| theorem cavity_canonical_capped_log_limit {m d n : ℕ} |
| (N depth : ℕ → ℕ) (hN : ∀ r, 0<N r) (k : ℕ → Fin m → ℕ) |
| (e : (r : ℕ) → (((a : Fin m) × Fin (k r a)) ⊕ Fin d) ≃ Fin (N r)) |
| (a₀ : Fin d → Fin m) (hk : ∀ r a, d ≤ k r a) |
| (hgroups : ∀ r a, 0<cavityBaseGroupDimension (k r) a₀ a) |
| (hdims : ∀ a, Tendsto (fun r => cavityBaseGroupDimension (k r) a₀ a) atTop atTop) |
| (μ : (r : ℕ) → Measure (Orthogonal (N r))) [∀ r, IsProbabilityMeasure (μ r)] |
| (θ : (r : ℕ) → Measure (LabeledTree (depth r))) [∀ r, IsProbabilityMeasure (θ r)] |
| (μG : (r : ℕ) → (a : Fin m) → Measure (Orthogonal (cavityBaseGroupDimension (k r) a₀ a))) |
| [∀ r a, IsProbabilityMeasure (μG r a)] [∀ r a, (μG r a).IsMulRightInvariant] |
| (lam : Fin m → ℝ) (v : ℕ → Fin m → ℝ) (u : ℕ → ℕ → ℝ) |
| {c : ℝ} (hc : 0<c) |
| (hfrac : ∀ r a, c ≤ (cavityBaseGroupDimension (k r) a₀ a : ℝ)/N r) |
| (ρ : Fin m → ℝ) (hρ : ∀ a, 0<ρ a) (hρsum : ∑ a, ρ a=1) |
| (hρlim : Tendsto (fun r a => (cavityBaseGroupDimension (k r) a₀ a : ℝ)/N r) atTop (𝓝 ρ)) |
| (Q₀ : ProbabilityMeasure (SpectralArray (m+1))) |
| (hlim : Tendsto (fun r => cavityRotationArrayLaw (μ r) (θ r) |
| (diagonalPerturbedEigenvalues |
| (fun i => lam ((cavityBaseGroupEquiv (k r) (e r) a₀).symm i).1) |
| (cavitySpectralGroup (fun i => ((cavityBaseGroupEquiv (k r) (e r) a₀).symm i).1)) (v r) 1) |
| (cavitySpectralGroup (fun i => ((cavityBaseGroupEquiv (k r) (e r) a₀).symm i).1)) |
| (u r)) atTop (𝓝 Q₀)) |
| (hgg : HasEntryGhirlandaGuerra (fun x i j => x (i,j)) (Q₀ : Measure (SpectralArray (m+1)))) |
| (hG : ∀ᵐ x ∂(Q₀ : Measure (SpectralArray (m+1))), SpectralGram x) |
| (d₀ : Fin (m+1) → ℝ) (hd0 : ∀ j, 0≤d₀ j) |
| (hd : ∀ᵐ x ∂(Q₀ : Measure (SpectralArray (m+1))), ∀ i j, (x (i,i) j : ℝ)=d₀ j) |
| (hE : ∀ e : ℕ → ℕ, Function.Injective e → |
| (Q₀ : Measure (SpectralArray (m+1))).map (permuteSpectralArray e)=Q₀) |
| (hP : ∀ᵐ x ∂(Q₀ : Measure (SpectralArray (m+1))), SpectralPartitionGeometry m x) |
| (hn : ∀ᵐ x ∂(Q₀ : Measure (SpectralArray (m+1))), ∀ j, 0≤(x (0,1) j : ℝ)) |
| (hoff : ∀ j l, ∀ Φ : ℝ → ℝ, Continuous Φ → ∀ B : ℝ, 0≤B → (∀ t, |Φ t|≤B) → |
| spectralOffWardResidual Q₀ ρ lam j l Φ=0) |
| (hdiag : ∀ j l, spectralDiagonalWardResidual Q₀ ρ lam j l=0) |
| (A : CavityFactorBlocks d n) (cap : ℝ) (hcap : 0≤cap) : |
| let p := spectralSpinQuantilePath Q₀ hP hn |
| let Qf := fun r => cavityLabeledDisorderLaw r (chainExponent (uniformCut r)) |
| (cavityFiniteRootCovariance ρ lam hρ hρsum a₀ (cavityStrictUniformPath p r) |
| (cavityStrictUniformLevels p r)) |
| (cavityFiniteNoiseCovariance ρ lam hρ hρsum a₀ (cavityStrictUniformPath p r) |
| (uniformCut r) (cavityStrictUniformLevels p r)) |
| let η := fun r => cavityLabeledPriorKernel r |
| (cavityFiniteCovariancePath ρ lam hρ hρsum a₀ (cavityStrictUniformPath p r) |
| (cavityStrictUniformLevels p r) r) (uniformSpinPrior n) |
| Tendsto (fun r => |
| (∫ ω, cavityCanonicalHaarLog (k r) (e r) a₀ (hk r) lam (v r) (u r) 1 cap 0 A ω |
| ∂(((μ r).prod (θ r)).prod gaussianCoordinates).prod (Measure.pi (μG r))) - |
| ∫ ω, Real.log (∫ x, Real.exp (min (cavityLabeledPotential r A.1 A.2.1 A.2.2 (ω,x)) cap) |
| ∂η r ω) ∂Qf r) atTop (𝓝 0) := by |
| intro p Qf η |
| let dims := fun r => cavityBaseGroupDimension (k r) a₀ |
| let E := fun r => cavityBaseGroupEquiv (k r) (e r) a₀ |
| let frames := fun r => cavityCanonicalGroupFrame (k r) (e r) a₀ (hk r) |
| let eig := fun r => diagonalPerturbedEigenvalues (fun i => lam ((E r).symm i).1) |
| (cavitySpectralGroup (fun i => ((E r).symm i).1)) (v r) 1 |
| have hinj : Function.Injective (fun j : Fin d => (a₀ j,j)) := by |
| intro i j hij |
| exact congrArg Prod.snd hij |
| have h := cavity_physical_capped_log_limit N depth hN dims hgroups hdims E μ θ μG frames |
| (fun r a => cavityCanonicalGroupFrame_gram (k r) (e r) a₀ (hk r) a) eig u hc hfrac |
| ρ lam hρ hρsum hρlim Q₀ hlim hgg hG d₀ hd0 hd hE hP hn hoff hdiag |
| a₀ (fun j => (a₀ j,j)) hinj (fun _ => rfl) A.1 A.2.1 A.2.2 cap hcap |
| simpa only [cavityCanonicalHaarLog, cavityHaarLogObservable, cavityHaarSpinPotential, |
| one_mul, zero_mul, sub_zero, dims, E, frames, eig, p, Qf, η] using h |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
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|
