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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 | import OAI.Probability.InvariantIsing.Spectral.CompactPartitionPressure
import OAI.Probability.InvariantIsing.Spectral.CompactPartitionContinuity
import OAI.Probability.InvariantIsing.Spectral.SpectralApproximationSqueeze
namespace OAI
/-! The zero-field pressure limit for arbitrary compact spectral laws,
when the finite spectra lie between the limiting support edges. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Filter Set
open scoped Topology
namespace InvariantIsing
theorem compact_spectral_pressure_tendsto_of_mem
(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)
(ν : ProbabilityMeasure ℝ) (a b : ℝ)
(hcompact : IsCompact (ν : Measure ℝ).support)
(hbound : (ν : Measure ℝ).support ⊆ Icc a b)
(ha : a∈(ν : Measure ℝ).support) (hb : b∈(ν : Measure ℝ).support)
(heig : ∀ N i, eig N i∈Icc a b)
(hweak : Tendsto (fun k => empiricalSpectralLaw (Nat.succ_pos k) (eig (k+1)))
atTop (𝓝 ν)) :
Tendsto (fun N => ∫ V, rotatedPressure (eig N) (matrixRotation V⁻¹) (fun _ => 0) ∂μ N)
atTop (𝓝 (variationalFunctional (measureR (ν : Measure ℝ) b)).toReal) := by
let δ := fun k : ℕ => (1 : ℝ)/(k+1)
let D (k : ℕ) : CompactSpectralPartition ν a b (δ k) := Classical.choice
(exists_compact_spectral_partition ν a b (δ k) hcompact hbound (by dsimp [δ]; positivity))
have hab : a ≤ b := (hbound ha).2
have hδ : Tendsto δ atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)
have hL := compact_partition_variational_tendsto ν a b δ D hab ha hb hbound hδ
apply spectral_approximation_squeeze _
(fun k N => ∫ V, rotatedPressure (fun i => (D k).lowerValue (eig N i))
(matrixRotation V⁻¹) (fun _ => 0) ∂μ N)
(fun k N => ∫ V, rotatedPressure (fun i => (D k).upperValue (eig N i))
(matrixRotation V⁻¹) (fun _ => 0) ∂μ N)
(fun k => (variationalFunctional (measureR ((D k).law : Measure ℝ) (D k).edge)).toReal)
δ _ hL hδ
(fun k => ((D k).pressure_tendsto hhaar hgauss hpub μ eig hweak).1)
(fun k => ((D k).pressure_tendsto hhaar hgauss hpub μ eig hweak).2)
intro k
filter_upwards [eventually_ge_atTop 1] with N hN
exact (D k).mean_pressure_bounds (by omega) (μ N) (eig N) ha hb hab (heig N)
end InvariantIsing
end
end OAI
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