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c4cdbbd | 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 78 79 80 81 82 83 84 85 86 87 88 | import OAI.Probability.InvariantIsing.Cavity.CavityPhysicalBase
namespace OAI
/-! Exact independence, in bounded-test form, of the Haar base spectral
orbit and the small compression data in the concrete cavity coupling. -/
noncomputable section
open MeasureTheory
open scoped Matrix
namespace InvariantIsing
def cavityConjugate {N : ℕ} (V : Orthogonal N) (J : Matrix (Fin N) (Fin N) ℝ) :
Matrix (Fin N) (Fin N) ℝ :=
(V : Matrix (Fin N) (Fin N) ℝ) * J * (V : Matrix (Fin N) (Fin N) ℝ).transpose
lemma continuous_cavityConjugate (N : ℕ) :
Continuous (Function.uncurry (cavityConjugate (N := N))) :=
((continuous_subtype_val.comp continuous_fst).matrix_mul continuous_snd).matrix_mul
(continuous_subtype_val.comp continuous_fst).matrix_transpose
lemma cavityConjugate_mul {N : ℕ} (U V : Orthogonal N)
(J : Matrix (Fin N) (Fin N) ℝ) :
cavityConjugate (U * V) J = cavityConjugate U (cavityConjugate V J) := by
change ((U : Matrix (Fin N) (Fin N) ℝ) * (V : Matrix (Fin N) (Fin N) ℝ)) * J *
((U : Matrix (Fin N) (Fin N) ℝ) * (V : Matrix (Fin N) (Fin N) ℝ)).transpose = _
simp only [cavityConjugate, Matrix.transpose_mul, Matrix.mul_assoc]
lemma cavityPhysicalBase_reservoir_inv {N n m d : ℕ} (g : Fin (N + n) → Fin m)
(lam : Fin m → ℝ)
(B : (Fin m → Matrix (Fin n) (Fin n) ℝ) → Matrix (Fin (m * n)) (Fin d) ℝ)
(A₀ : Matrix (Fin d) (Fin d) ℝ)
(U : Orthogonal (N + n)) (V : Orthogonal N) :
cavityPhysicalBase g lam B A₀ (U * cavityReservoirRotation (n := n) V⁻¹) =
cavityConjugate V (cavityPhysicalBase g lam B A₀ U) := by
rw [cavityPhysicalBase_reservoir]
have hV : (V⁻¹ : Orthogonal N).val = (V : Matrix (Fin N) (Fin N) ℝ).transpose := by
ext i j
rfl
simp only [hV, Matrix.transpose_transpose, cavityConjugate]
theorem cavityBase_compression_factorization {N n m d : ℕ}
(g : Fin (N + n) → Fin m) (k : Fin m → ℕ)
(ek : ∀ a, {i : Fin (N + n) // g i = a} ≃ Fin (k a + n))
(e : ((a : Fin m) × Fin (k a)) ⊕ Fin d ≃ Fin N)
(lam : Fin m → ℝ) (lam₀ : Fin d → ℝ)
(B : (Fin m → Matrix (Fin n) (Fin n) ℝ) → Matrix (Fin (m * n)) (Fin d) ℝ)
(hBm : Measurable B)
(μ : Measure (Orthogonal (N + n))) [IsProbabilityMeasure μ] [μ.IsMulRightInvariant]
(ν : Measure (Orthogonal N)) [IsProbabilityMeasure ν] [ν.IsMulRightInvariant]
(hA : ∀ᵐ U ∂μ, ∀ a, (cavityCompressionGrams g U a).PosDef)
(hB : ∀ U, (∀ a, (cavityCompressionGrams g U a).PosDef) →
(B (cavityCompressionGrams g U)).transpose * B (cavityCompressionGrams g U) = 1)
(hBT : ∀ U, (∀ a, (cavityCompressionGrams g U a).PosDef) →
(B (cavityCompressionGrams g U)).transpose *
cavitySpectralStack (cavityCompressionGrams g U) = 0)
(f : Matrix (Fin N) (Fin N) ℝ → ℝ)
(q : (Fin m → Matrix (Fin n) (Fin n) ℝ) → ℝ)
(hf : Measurable f) (hq : Measurable q)
(Cf Cq : ℝ) (hCq : 0 ≤ Cq)
(hfb : ∀ J, ‖f J‖ ≤ Cf) (hqb : ∀ M, ‖q M‖ ≤ Cq) :
(∫ U, q (cavityCompressionGrams g U) *
f (cavityPhysicalBase g lam B (Matrix.diagonal lam₀) U) ∂μ) =
(∫ U, q (cavityCompressionGrams g U) ∂μ) *
∫ V, f (cavityConjugate V (Matrix.diagonal
(fun j => Sum.elim (fun a => lam a.1) lam₀ (e.symm j)))) ∂ν := by
let : OpensMeasurableSpace (Orthogonal N × Matrix (Fin N) (Fin N) ℝ) :=
inferInstanceAs (OpensMeasurableSpace (Orthogonal N × (Fin N → Fin N → ℝ)))
apply cavity_orbit_factorization_right_ae μ ν
(fun V => cavityReservoirRotation (n := n) V⁻¹)
(cavityPhysicalBase g lam B (Matrix.diagonal lam₀)) (cavityCompressionGrams g)
(measurable_cavityPhysicalBase g lam B hBm _) (measurable_cavityCompressionGrams g)
cavityConjugate (continuous_cavityConjugate N).measurable cavityConjugate_mul
_ _ _ _ f q hf hq Cf Cq hCq hfb hqb
· intro V
exact ae_of_all _ (fun U => cavityPhysicalBase_reservoir_inv g lam B _ U V)
· intro V
exact ae_of_all _ (fun U => cavityCompressionGrams_reservoir g U V⁻¹)
· filter_upwards [hA] with U hU
exact cavityPhysicalBase_mem_orbit g k ek e U lam lam₀ B (hB U hU) (hBT U hU) hU
end InvariantIsing
end
end OAI
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