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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 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 | import OAI.Probability.InvariantIsing.Haar.Rotation
import OAI.Probability.IsingPerceptron.ArraySynchronization
namespace OAI
/-!
# Spectral overlap geometry
The finite projected overlaps are Gram matrices. Panchenko's ultrametricity
theorem applies to their limit arrays satisfying the Ghirlanda–Guerra identities.
-/
noncomputable section
open MeasureTheory
open scoped BigOperators Matrix
namespace InvariantIsing
/-- Overlap restricted to a deterministic spectral coordinate group. -/
def projectedOverlap {N : ℕ} (U : Rotation N) (I : Finset (Fin N))
(σ τ : Spin N) : ℝ :=
(N : ℝ)⁻¹ * ∑ i ∈ I, U (spinVector σ) i * U (spinVector τ) i
lemma projectedOverlap_symm {N : ℕ} (U : Rotation N) (I : Finset (Fin N))
(σ τ : Spin N) : projectedOverlap U I σ τ = projectedOverlap U I τ σ := by
simp only [projectedOverlap, mul_comm]
/-- Each spectral overlap component is a positive semidefinite Gram kernel. -/
lemma projectedOverlap_posSemidef {N m : ℕ} (U : Rotation N) (I : Finset (Fin N))
(σ : Fin m → Spin N) :
Matrix.PosSemidef (fun i j => projectedOverlap U I (σ i) (σ j)) := by
classical
let A : Matrix (Fin m) (Fin N) ℝ := fun i k =>
if k ∈ I then U (spinVector (σ i)) k else 0
have hp := (Matrix.posSemidef_self_mul_conjTranspose A).smul
(show (0 : ℝ) ≤ (N : ℝ)⁻¹ by positivity)
have he : (fun i j : Fin m => projectedOverlap U I (σ i) (σ j)) =
(N : ℝ)⁻¹ • (A * Aᴴ) := by
ext i j
change (N : ℝ)⁻¹ * (∑ k ∈ I, U (spinVector (σ i)) k * U (spinVector (σ j)) k) =
(N : ℝ)⁻¹ * ∑ k, (if k ∈ I then U (spinVector (σ i)) k else 0) *
(if k ∈ I then U (spinVector (σ j)) k else 0)
congr 1
simp only [ite_mul, mul_ite, zero_mul, mul_zero,
Finset.sum_ite_mem, Finset.univ_inter, Finset.inter_self]
rw [he]
exact hp
lemma projectedOverlap_self_nonneg {N : ℕ} (U : Rotation N) (I : Finset (Fin N))
(σ : Spin N) : 0 ≤ projectedOverlap U I σ σ := by
apply mul_nonneg (by positivity)
exact Finset.sum_nonneg fun i _ => mul_self_nonneg _
/-- Full spectral overlap agrees with the ordinary spin overlap, because an
orthogonal rotation preserves the inner product. -/
lemma projectedOverlap_univ {N : ℕ} (U : Rotation N) (σ τ : Spin N) :
projectedOverlap U Finset.univ σ τ =
(N : ℝ)⁻¹ * ∑ i, spinValue (σ i) * spinValue (τ i) := by
simp only [projectedOverlap]
congr 1
have h := U.inner_map_map (spinVector σ) (spinVector τ)
simpa only [EuclideanSpace.inner_eq_star_dotProduct, dotProduct, star_trivial,
spinVector_apply, RCLike.inner_apply, starRingEnd_apply, star_trivial, mul_comm] using h
/-- Panchenko (2013), Theorem 1, in the exact bounded-Gram-array specialization
needed after proving GG for spectral overlap limits. The proof is already in
`IsingPerceptron.gg_ultrametric`; GG remains an explicit hypothesis here. -/
theorem ultrametric_of_gram_gg {Ω : Type*} [MeasurableSpace Ω]
{B : Ω → IsingPerceptron.RealArray} {μ : Measure Ω} [IsProbabilityMeasure μ]
(hB : Measurable B) (hsym : ∀ ω i j, B ω i j = B ω j i)
(hwe : IsingPerceptron.WeaklyExchangeable B μ)
(hgg : IsingPerceptron.HasGhirlandaGuerra B μ) {D : ℝ}
(hgram : ∀ ω n, Matrix.PosSemidef (IsingPerceptron.arrayBlock B n ω))
(hdiag : ∀ ω i, B ω i i = D) :
∀ᵐ ω ∂μ, IsingPerceptron.IsUltrametricArray (B ω) :=
IsingPerceptron.gg_ultrametric hB hsym hwe hgg hgram hdiag
end InvariantIsing
end
end OAI
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