File size: 3,647 Bytes
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 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 | import OAI.Probability.InvariantIsing.Arrays.NSpinComparison
namespace OAI
/-! Applying Haar concentration to the actual Gaussian cascade pressure. -/
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped BigOperators NNReal
namespace InvariantIsing
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
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)
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
/-- 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)
end InvariantIsing
end
end OAI
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