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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 | import OAI.Probability.InvariantIsing.Magnetic.MagneticCompactPartition
import OAI.Probability.InvariantIsing.Magnetic.MagneticPartitionContinuity
import OAI.Probability.InvariantIsing.Spectral.SpectralApproximationSqueeze
namespace OAI
/-! The finite external-field formula for an arbitrary compact spectral law. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Filter Set
open scoped Topology
namespace InvariantIsing
theorem compact_spectral_field_pressure_tendsto_of_mem
(hhaar : HaarConcentrationInput) (hgauss : GaussianLipschitzVarianceInput)
(hpub : PanchenkoTalagrandRestrictedFieldPairInput)
(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))
{A : Type*} [Fintype A] [DecidableEq A]
(group : (N : ℕ) → Fin N → A) (γ c : A → ℝ)
(hγ : ∀ a, 0 < γ a) (hγsum : ∑ a, γ a=1)
(hgroup : Tendsto (fun N a => (spinGroupSize (group N) a : ℝ)/N) atTop (𝓝 γ)) :
Tendsto (fun N => ∫ V, rotatedPressure (eig N) (matrixRotation V⁻¹) (fun i => c (group N i)) ∂μ N)
atTop (𝓝 (finiteMagneticFunctional (measureR (ν : Measure ℝ) b) γ c).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_magnetic_tendsto ν a b δ D hab ha hb hbound hδ γ c
(fun a => (hγ a).le) hγsum
apply spectral_approximation_squeeze _
(fun k N => ∫ V, rotatedPressure (fun i => (D k).lowerValue (eig N i))
(matrixRotation V⁻¹) (fun i => c (group N i)) ∂μ N)
(fun k N => ∫ V, rotatedPressure (fun i => (D k).upperValue (eig N i))
(matrixRotation V⁻¹) (fun i => c (group N i)) ∂μ N)
(fun k => (finiteMagneticFunctional (measureR ((D k).law : Measure ℝ) (D k).edge) γ c).toReal)
δ _ hL hδ
(fun k => ((D k).field_pressure_tendsto hhaar hgauss hpub μ eig hweak group γ c hγ hγsum hgroup).1)
(fun k => ((D k).field_pressure_tendsto hhaar hgauss hpub μ eig hweak group γ c hγ hγsum hgroup).2)
intro k
filter_upwards [eventually_ge_atTop 1] with N hN
exact (D k).mean_field_pressure_bounds (by omega) (μ N) (eig N) ha hb hab (heig N) (fun i => c (group N i))
end InvariantIsing
end
end OAI
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