| import OAI.Probability.InvariantIsing.Arrays.NSpinCascade |
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| namespace OAI |
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| /-! Conditional cascade fluctuations for the actual Ising terminal. -/ |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory |
| open scoped NNReal |
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| namespace InvariantIsing |
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| /-- The conditional cascade fluctuation is bounded uniformly in the Ising |
| terminal, rotation, external field, Gaussian root and number of spins. -/ |
| theorem rotatedCascadeLog_centered_square {N : ℕ} (hN : 0 < N) |
| (n : ℕ) (b : ℕ → ℝ) (v : ℕ → ℝ≥0) (hb : IsingPerceptron.CascadeExponents n b) |
| (eig : Fin N → ℝ) (U : Rotation N) (c z : Fin N → ℝ) : |
| Integrable (fun T => (rotatedCascadeLog n eig U c z T - |
| rotatedCascadeValue n b v eig U c z) ^ 2) (rotatedCascadeLaw N n b v) ∧ |
| (∫ T, (rotatedCascadeLog n eig U c z T - rotatedCascadeValue n b v eig U c z) ^ 2 |
| ∂rotatedCascadeLaw N n b v) ≤ |
| 4 * ∫ T, (Real.log (IsingPerceptron.rawTreeTotal n T).toReal) ^ 2 |
| ∂(IsingPerceptron.rawCascadeLaw n b : Measure (IsingPerceptron.RawTree n)) := by |
| have h := IsingPerceptron.linearGrowth_noiseCascade_recursion n b hb |
| (fun i => vectorGaussianLaw N (v i)) (fun i _ => vectorGaussianLaw_moments hN (v i)) |
| (measurable_rotatedFieldTerminal eig U c) (rotatedFieldTerminal_linearGrowth eig U c) z |
| have he := IsingPerceptron.noise_leaf_log_eq_ratio n b hb |
| (fun i => vectorGaussianLaw N (v i)) |
| (fun _ => measurable_fst.add measurable_snd) (measurable_rotatedFieldTerminal eig U c) z |
| let L := fun T => Real.log |
| ((IsingPerceptron.terminalTreeFactor n b (fun i => vectorGaussianLaw N (v i)) |
| (fun _ p => p.1 + p.2) (rotatedFieldTerminal eig U c) z T).toReal / |
| (IsingPerceptron.noiseTreeTotal (Fin N → ℝ) n T).toReal) |
| have heq : (fun T => (rotatedCascadeLog n eig U c z T - rotatedCascadeValue n b v eig U c z) ^ 2) |
| =ᵐ[rotatedCascadeLaw N n b v] |
| (fun T => (L T - rotatedCascadeValue n b v eig U c z) ^ 2) := by |
| filter_upwards [he] with T hT |
| exact congrArg (fun x => (x - rotatedCascadeValue n b v eig U c z) ^ 2) hT |
| exact ⟨h.2.2.2.1.congr (Filter.EventuallyEq.symm heq), |
| (integral_congr_ae heq).trans_le h.2.2.2.2⟩ |
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| end InvariantIsing |
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| end |
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| end OAI |
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