| import OAI.Probability.InvariantIsing.Arrays.NSpinComparison |
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| namespace OAI |
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| /-! Applying Haar concentration to the actual Gaussian cascade pressure. -/ |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory |
| open scoped BigOperators NNReal |
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| namespace InvariantIsing |
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| lemma measurable_rotatedFieldTerminal_joint {N : ℕ} (eig c : Fin N → ℝ) : |
| Measurable (fun p : SpecialOrthogonal N × (Fin N → ℝ) => |
| rotatedFieldTerminal eig (specialRotation p.1) c p.2) := by |
| have hE (σ : Spin N) : Measurable (fun p : SpecialOrthogonal N × (Fin N → ℝ) => |
| rotatedEnergy eig (specialRotation p.1) σ + fieldEnergy (c + p.2) σ) := by |
| apply Measurable.add |
| · exact (Finset.measurable_sum _ fun i _ => |
| (((measurable_specialRotation_eval (spinVector σ) i).comp measurable_fst).pow_const 2).const_mul |
| (eig i)).const_mul (1 / 2 : ℝ) |
| · unfold fieldEnergy |
| fun_prop |
| unfold rotatedFieldTerminal logPartition |
| exact ((Finset.measurable_sum _ fun σ _ => (hE σ).exp).const_mul |
| (Fintype.card (Spin N) : ℝ)⁻¹).log |
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| lemma measurable_rotatedCascadeValue_joint {N : ℕ} (n : ℕ) |
| (b : ℕ → ℝ) (v : ℕ → ℝ≥0) (eig c : Fin N → ℝ) : |
| Measurable (fun p : SpecialOrthogonal N × (Fin N → ℝ) => |
| rotatedCascadeValue n b v eig (specialRotation p.1) c p.2) := by |
| induction n generalizing b v with |
| | zero => exact measurable_rotatedFieldTerminal_joint eig c |
| | succ n ih => |
| have hp : Measurable (fun p : (SpecialOrthogonal N × (Fin N → ℝ)) × (Fin N → ℝ) => |
| (p.1.1, p.1.2 + p.2)) := |
| measurable_fst.fst.prodMk (measurable_fst.snd.add measurable_snd) |
| have hm := (((ih (fun i => b (i + 1)) (fun i => v (i + 1))).comp hp).const_mul (b 0)).exp |
| exact hm.stronglyMeasurable.integral_prod_right.measurable.log.div_const (b 0) |
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| lemma measurable_rotatedEnrichedPressure {N : ℕ} (n : ℕ) (b : ℕ → ℝ) |
| (v : ℕ → ℝ≥0) (root : ℝ≥0) (eig c : Fin N → ℝ) : |
| Measurable (fun U : SpecialOrthogonal N => |
| rotatedEnrichedPressure n b v root eig (specialRotation U) c) := by |
| exact (measurable_rotatedCascadeValue_joint n b v eig c).stronglyMeasurable.integral_prod_right.measurable.const_mul (N : ℝ)⁻¹ |
|
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| /-- Uniform Haar tails for the finite enriched Ising pressure, conditional |
| only on the precise published concentration input. -/ |
| theorem haar_enrichedPressure_median_tail (hpub : HaarConcentrationInput) : |
| ∃ C c₀ : ℝ, 0 < C ∧ 0 < c₀ ∧ |
| ∀ N : ℕ, 3 ≤ N → |
| ∀ μ : Measure (SpecialOrthogonal N), |
| IsProbabilityMeasure μ → μ.IsMulLeftInvariant → |
| ∀ n : ℕ, ∀ b : ℕ → ℝ, ∀ v : ℕ → ℝ≥0, ∀ root : ℝ≥0, |
| IsingPerceptron.CascadeExponents n b → |
| ∀ eig c : Fin N → ℝ, ∀ K : ℝ, 0 < K → (∀ i, |eig i| ≤ K) → |
| ∃ m : ℝ, |
| (1 / 2 : ℝ) ≤ μ.real {U | rotatedEnrichedPressure n b v root eig (specialRotation U) c ≤ m} ∧ |
| (1 / 2 : ℝ) ≤ μ.real {U | m ≤ rotatedEnrichedPressure n b v root eig (specialRotation U) c} ∧ |
| ∀ r : ℝ, 0 < r → |
| μ.real {U | r ≤ |rotatedEnrichedPressure n b v root eig (specialRotation U) c - m|} ≤ |
| C * Real.exp (-c₀ * (N : ℝ) * r ^ 2 / K ^ 2) := by |
| obtain ⟨C, c₀, hC, hc, hp⟩ := hpub |
| refine ⟨C, c₀, hC, hc, ?_⟩ |
| intro N hN μ hμ hμinv n b v root hb eig c K hK heig |
| exact hp N hN μ hμ hμinv _ |
| (measurable_rotatedEnrichedPressure n b v root eig c) K hK |
| (fun U V => abs_rotatedEnrichedPressure_sub_rotation_le (by omega) |
| n b v root hb eig U V c K hK.le heig) |
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| end InvariantIsing |
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| end |
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| end OAI |
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