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a89d282 | 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.Spectral.CompactSpectralPartition
import OAI.Probability.InvariantIsing.Spectral.SpectralPartitionPressure
import OAI.Probability.InvariantIsing.Cavity.CavitySpectralSqueeze
namespace OAI
/-! The finite positive spectral partitions bracket physical mean pressure. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Filter Set
open scoped Topology Classical Function
namespace InvariantIsing.CompactSpectralPartition
variable {ν : ProbabilityMeasure ℝ} {a b δ : ℝ} (D : CompactSpectralPartition ν a b δ)
def lowerValue (x : ℝ) : ℝ :=
D.value (positiveSpectralLabel D.weight D.lower
(spectralPartitionIndex D.cells (spectralBallCell_cover _) x))
def upperValue (x : ℝ) : ℝ :=
D.value (positiveSpectralLabel D.weight D.upper
(spectralPartitionIndex D.cells (spectralBallCell_cover _) x))
lemma rounding_bounds (ha : a∈(ν : Measure ℝ).support) (hb : b∈(ν : Measure ℝ).support)
(x : ℝ) (hx : x∈Icc a b) : D.lowerValue x-δ ≤ x ∧ x ≤ D.upperValue x+δ := by
have he := D.endpoint_bounds ha hb
exact positive_spectral_rounding_bounds (ν : Measure ℝ) D.cells (spectralBallCell_cover _)
D.value a b δ D.lower D.upper he.1 he.2
(positive_spectral_ball_cell_close (ν : Measure ℝ) D.center D.radius a δ
D.radius_le D.residual_zero) x hx
lemma rounded_mem (hab : a ≤ b) (x : ℝ) :
D.lowerValue x∈Icc a b ∧ D.upperValue x∈Icc a b :=
⟨D.value_mem hab _,D.value_mem hab _⟩
lemma pressure_tendsto
(hhaar : HaarConcentrationInput) (hgauss : GaussianLipschitzVarianceInput)
(hpub : PanchenkoTalagrandFieldPairInput)
(μ : (N : ℕ) → Measure (Orthogonal N)) [∀ N, IsProbabilityMeasure (μ N)]
[∀ N, (μ N).IsMulRightInvariant] (eig : (N : ℕ) → Fin N → ℝ)
(hweak : Tendsto (fun k => empiricalSpectralLaw (Nat.succ_pos k) (eig (k+1)))
atTop (𝓝 ν)) :
Tendsto (fun N => ∫ V, rotatedPressure (fun i => D.lowerValue (eig N i))
(matrixRotation V⁻¹) (fun _ => 0) ∂μ N) atTop
(𝓝 (variationalFunctional (measureR (D.law : Measure ℝ) D.edge)).toReal) ∧
Tendsto (fun N => ∫ V, rotatedPressure (fun i => D.upperValue (eig N i))
(matrixRotation V⁻¹) (fun _ => 0) ∂μ N) atTop
(𝓝 (variationalFunctional (measureR (D.law : Measure ℝ) D.edge)).toReal) := by
have hh (fallback : D.Positive) := spectral_partition_pressure_tendsto hhaar hgauss hpub μ eig ν hweak
D.cells (spectralBallCell_cover _) (spectralBallCell_disjoint _)
(spectralBallCell_measurable _ (fun _ => measurableSet_ball))
(spectralBallCell_null_boundary (ν : Measure ℝ) _ D.null_boundary)
D.value fallback D.upper D.le_upper
exact ⟨hh D.lower,hh D.upper⟩
lemma mean_pressure_bounds {N : ℕ} (hN : 0 < N)
(μ : Measure (Orthogonal N)) [IsProbabilityMeasure μ] (eig : Fin N → ℝ)
(ha : a∈(ν : Measure ℝ).support) (hb : b∈(ν : Measure ℝ).support)
(hab : a ≤ b) (heig : ∀ i, eig i∈Icc a b) :
(∫ V, rotatedPressure (fun i => D.lowerValue (eig i))
(matrixRotation V⁻¹) (fun _ => 0) ∂μ)-δ/2 ≤
∫ V, rotatedPressure eig (matrixRotation V⁻¹) (fun _ => 0) ∂μ ∧
(∫ V, rotatedPressure eig (matrixRotation V⁻¹) (fun _ => 0) ∂μ) ≤
(∫ V, rotatedPressure (fun i => D.upperValue (eig i))
(matrixRotation V⁻¹) (fun _ => 0) ∂μ)+δ/2 := by
let K := max |a| |b|
have hbound (x : ℝ) (hx : x∈Icc a b) : |x| ≤ K := by
apply abs_le.mpr
constructor
· have hh := neg_abs_le a
have hk : |a| ≤ K := le_max_left _ _
linarith [hx.1]
· exact hx.2.trans ((le_abs_self b).trans (le_max_right _ _))
exact cavity_meanPressure_squeeze hN μ eig (fun i => D.lowerValue (eig i))
(fun i => D.upperValue (eig i)) (fun _ => 0) δ K
(fun i => hbound _ (heig i))
(fun i => hbound _ (D.rounded_mem hab (eig i)).1)
(fun i => hbound _ (D.rounded_mem hab (eig i)).2)
(fun i => (D.rounding_bounds ha hb _ (heig i)).1)
(fun i => (D.rounding_bounds ha hb _ (heig i)).2)
end InvariantIsing.CompactSpectralPartition
end
end OAI
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