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b317718 | 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 | import OAI.Probability.InvariantIsing.Arrays.TensorPerturbationVariance
import OAI.Probability.InvariantIsing.Arrays.MonomialEnumeration
import OAI.Probability.InvariantIsing.Spectral.SpectralDiagonalPerturbation
namespace OAI
/-! Uniform variance for both terms of the actual finite perturbation. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped BigOperators NNReal
namespace InvariantIsing
/-- The explicit complete monomial enumeration removes the degree-cap
assumption. The diagonal perturbation is the exact eigenvalue shift proved
in `rotatedEnergy_diagonalPerturbation`. -/
theorem fullPerturbationPressure_variance (hhaar : HaarConcentrationInput)
(hgauss : GaussianLipschitzVarianceInput) :
∃ C : ℝ, 0 < C ∧
∀ N : ℕ, 3 ≤ N →
∀ μ : Measure (SpecialOrthogonal N), IsProbabilityMeasure μ → μ.IsMulLeftInvariant →
∀ m : ℕ, ∀ eig c : Fin N → ℝ, ∀ K : ℝ, 0 < K → (∀ i, |eig i| ≤ K) →
∀ I : Fin m → Finset (Fin N), ∀ u : Fin N → ℝ, (∀ j, |u j| ≤ 2) →
∀ v : Fin m → ℝ, (∀ a, |v a| ≤ 2) → ∀ t : ℝ, |t| ≤ 1 →
∀ n : ℕ, ∀ b : ℕ → ℝ, CascadeExponents n b →
∀ h : ℕ → ℝ, Monotone h → 0 ≤ h 0 → ∀ H : ℝ, h n ≤ H →
let degree := fun j : Fin N => enumeratedSpectralDegree m j
let treeDegree := fun j : Fin N => enumeratedTreeDegree m j
let profile := tensorPathProfile I degree n treeDegree h
let P := tensorRootTreeLaw I degree n b (fun i => profile (i + 1)) (profile 0)
let F := tensorDisorderPressure (diagonalPerturbedEigenvalues eig I v t) c I degree
(tensorPerturbationAmplitude N u) n
MemLp F 2 (μ.prod P) ∧ variance F (μ.prod P) ≤
(4 * (∫ T, (Real.log (rawTreeTotal n T).toReal) ^ 2
∂(rawCascadeLaw n b : Measure (RawTree n))) + H + 4 +
C * (K + 4 * m + 8) ^ 2) / N := by
obtain ⟨C, hC, hv⟩ := tensorPerturbationPressure_variance hhaar hgauss
refine ⟨C, hC, ?_⟩
intro N hN μ hμ hμinv m eig c K hK heig I u hu v hvbound t ht n b hb h hh h0 H hH
degree treeDegree profile P F
have hNm : 0 < N := by omega
have hK' : 0 < K + 4 * m := by positivity
have heig' : ∀ i, |diagonalPerturbedEigenvalues eig I v t i| ≤ K + 4 * m :=
diagonalPerturbedEigenvalues_abs_le hNm eig I v hvbound t ht K heig
have hd : ∀ j : Fin N, (∑ a, (degree j a : ℝ)) ≤ (1 : ℝ) * ((j : ℝ) + 1) := by
intro j
simpa only [one_mul] using enumeratedSpectralDegree_real_sum_le m j
simpa only [mul_one] using hv N hN μ hμ hμinv m
(diagonalPerturbedEigenvalues eig I v t) c (K + 4 * m) hK' heig' I degree u hu
1 (by norm_num) hd n treeDegree b hb h hh h0 H hH
end InvariantIsing
end
end OAI
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