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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 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | import OAI.Probability.InvariantIsing.Arrays.TensorMinimumFluctuations
import OAI.Probability.InvariantIsing.Arrays.TensorGGMinimum
import OAI.Probability.InvariantIsing.Arrays.FullPerturbationVariance
namespace OAI
/-! For the complete perturbation, both coordinate minima force the
Ghirlanda–Guerra errors to vanish. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped BigOperators
namespace InvariantIsing
def tensorPerturbationPressureMean {N m : ℕ}
(μ : Measure (SpecialOrthogonal N)) (eig c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (u : Fin N → ℝ) (v : Fin m → ℝ) (t : ℝ)
(n : ℕ) (b h : ℕ → ℝ) : ℝ :=
let degree := fun j : Fin N => enumeratedSpectralDegree m j
let treeDegree := fun j : Fin N => enumeratedTreeDegree m j
∫ p, tensorDisorderPressure (diagonalPerturbedEigenvalues eig I v t) c I degree
(tensorPerturbationAmplitude N u) n p
∂(μ.prod (tensorRootTreeLaw I degree n b
(fun i => tensorPathProfile I degree n treeDegree h (i + 1))
(tensorPathProfile I degree n treeDegree h 0)))
def diagonalMinimumCenter (N : ℕ) (w : ℝ) : ℝ :=
let a := N * perturbationScale N
2 * (N / a ^ 2) * (a * w - a * (3 / 2))
/-- For the manuscript's full finite perturbation, genuine pressure-coordinate
minima give a GG residual bounded by two explicit rates tending to zero. -/
theorem fullPerturbationPressure_minimumGG (hhaar : HaarConcentrationInput)
(hgauss : GaussianLipschitzVarianceInput) :
∃ C : ℝ, 0 < C ∧
∀ N : ℕ, 3 ≤ N →
∀ μ : Measure (SpecialOrthogonal N), IsProbabilityMeasure μ → μ.IsMulLeftInvariant →
∀ m : ℕ, ∀ eig c : Fin N → ℝ, ∀ K : ℝ, 0 < K → (∀ i, |eig i| ≤ K) →
∀ I : Fin m → Finset (Fin N), ∀ u : Fin N → ℝ, (∀ j, u j ∈ Set.Icc (1 : ℝ) 2) →
∀ v : Fin m → ℝ, (∀ a, v a ∈ Set.Icc (1 : ℝ) 2) → ∀ t : ℝ, |t| ≤ 1 →
∀ n : ℕ, ∀ b : ℕ → ℝ, CascadeExponents n b →
∀ h : ℕ → ℝ, Monotone h → 0 ≤ h 0 → ∀ H : ℝ, h n ≤ H →
let M := tensorPerturbationPressureMean μ eig c I (n := n) (b := b) (h := h)
let degree := fun j : Fin N => enumeratedSpectralDegree m j
let treeDegree := fun j : Fin N => enumeratedTreeDegree m j
let L := 4 * (∫ T, (Real.log (rawTreeTotal n T).toReal) ^ 2
∂(rawCascadeLaw n b : Measure (RawTree n))) + H + 4 + C * (K + 4 * m + 8) ^ 2
perturbationScale N ≤ 1 / 32 → contactStep N ≤ 1 / 4 →
(∀ (j : Fin N) (w : ℝ), w ∈ Set.Icc (1 : ℝ) 2 →
-M u v t + perturbationWeight j * (u j - 3 / 2) ^ 2 ≤
-M (Function.update u j w) v t + perturbationWeight j * (w - 3 / 2) ^ 2) →
(∀ (a : Fin m) (w : ℝ), w ∈ Set.Icc (1 : ℝ) 2 →
-M u v t + (v a - 3 / 2) ^ 2 ≤ -M u (Function.update v a w) t + (w - 3 / 2) ^ 2) →
∀ j : Fin N, ∀ q : ℕ, ∀ B : ℝ, 0 ≤ B →
∀ D : SpecialOrthogonal N → (Fin (q + 1) → Spin N × LabeledLeaf n) → ℝ,
Measurable (Function.uncurry D) → (∀ U σ, |D U σ| ≤ B) →
|tensorSpectralGGResidual μ (diagonalPerturbedEigenvalues eig I v t) c I degree
(tensorPerturbationAmplitude N u) n b treeDegree h j D| ≤
contactGGRate N j L B +
2 * B * (∑ a, (degree j a : ℝ)) * diagonalContactRate N L := by
obtain ⟨C, hC, hvar⟩ := fullPerturbationPressure_variance hhaar hgauss
refine ⟨C, hC, ?_⟩
intro N hN μ hμ hμinv m eig c K hK heig I u hu v hv t ht n b hb h hh h0 H hH
M degree treeDegree L he hs hminu hminv j q B hB D hD hDb
let : IsProbabilityMeasure μ := hμ
have hNp : 0 < N := by omega
have hua : ∀ j, |u j| ≤ 2 := fun j => abs_le.mpr ⟨by linarith [(hu j).1], (hu j).2⟩
have hva : ∀ a, |v a| ≤ 2 := fun a => abs_le.mpr ⟨by linarith [(hv a).1], (hv a).2⟩
let amplitude := tensorPerturbationAmplitude N u
let R := μ.prod (tensorRootTreeLaw I degree n b
(fun i => tensorPathProfile I degree n treeDegree h (i + 1))
(tensorPathProfile I degree n treeDegree h 0))
have hgu (w : ℝ) (hw : |w| ≤ 2) :
MemLp (tensorDisorderPressure (diagonalPerturbedEigenvalues eig I v t) c I degree
(Function.update amplitude j (perturbationAmplitude N j * w)) n) 2 R ∧
variance (tensorDisorderPressure (diagonalPerturbedEigenvalues eig I v t) c I degree
(Function.update amplitude j (perturbationAmplitude N j * w)) n) R ≤ L / N := by
simpa only [tensorPerturbationAmplitude_update] using
hvar N hN μ hμ hμinv m eig c K hK heig I (Function.update u j w)
(update_abs_le_two hua j hw) v hva t ht n b hb h hh h0 H hH
have hgv (a : Fin m) (w : ℝ) (hw : |w| ≤ 2) :
MemLp (tensorDisorderPressure (diagonalPerturbedEigenvalues eig I (Function.update v a w) t)
c I degree amplitude n) 2 R ∧
variance (tensorDisorderPressure (diagonalPerturbedEigenvalues eig I (Function.update v a w) t)
c I degree amplitude n) R ≤ L / N := by
exact hvar N hN μ hμ hμinv m eig c K hK heig I u hua (Function.update v a w)
(update_abs_le_two hva a hw) t ht n b hb h hh h0 H hH
have hdiag (a : Fin m) : tensorNamespacedObservableAverage μ (diagonalPerturbedEigenvalues eig I v t)
c I degree amplitude n b treeDegree h
(fun U x => |projectedOverlap (specialRotation U) (I a) x.1 x.1 - diagonalMinimumCenter N (v a)|) ≤
diagonalContactRate N L := by
have hd := tensorDiagonal_fluctuation_at_minimum hNp μ eig c I degree amplitude n b treeDegree h
hh h0 hb v t a (v a) L (hv a) he hs (fun w hw => (hgv a w hw).1)
(fun w hw => (hgv a w hw).2) (by
dsimp only
intro w hw
simpa only [M, tensorPerturbationPressureMean, Function.update_eq_self] using hminv a w hw)
simpa only [Function.update_eq_self, diagonalMinimumCenter] using hd
have hgm : ∀ w ∈ Set.Icc (1 : ℝ) 2,
-tensorGaussianCoordinatePressure μ (diagonalPerturbedEigenvalues eig I v t) c I degree amplitude
n b treeDegree h j (u j) + perturbationWeight j * (u j - 3 / 2) ^ 2 ≤
-tensorGaussianCoordinatePressure μ (diagonalPerturbedEigenvalues eig I v t) c I degree amplitude
n b treeDegree h j w + perturbationWeight j * (w - 3 / 2) ^ 2 := by
intro w hw
simpa only [amplitude, M, tensorPerturbationPressureMean, tensorGaussianCoordinatePressure, ← tensorPerturbationAmplitude_update,
Function.update_eq_self] using hminu j w hw
have hg := tensorGaussianGG_at_minimum hNp μ (diagonalPerturbedEigenvalues eig I v t) c I degree
amplitude n b treeDegree h hh h0 hb j (u j) L (hu j) (by linarith) hs rfl
(fun w hw => (hgu w hw).1) (fun w hw => (hgu w hw).2) hgm B hB D hD hDb
(fun a => diagonalMinimumCenter N (v a)) (fun _ => diagonalContactRate N L) hdiag
simpa only [← Finset.sum_mul, mul_assoc] using hg
end InvariantIsing
end
end OAI
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