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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 | import OAI.Probability.InvariantIsing.Spectral.CompactSpectralPartition
import OAI.Probability.InvariantIsing.Spectral.MeasureVariationalContinuity
namespace OAI
/-! The finite positive-mass partition values converge to the compact spectral value. -/
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped Topology
namespace InvariantIsing
lemma compact_partition_laws_tendsto (ν : ProbabilityMeasure ℝ) (a b : ℝ)
(δ : ℕ → ℝ) (D : (k : ℕ) → CompactSpectralPartition ν a b (δ k))
(hδ : Tendsto δ atTop (𝓝 0)) :
Tendsto (fun k => (D k).law) atTop (𝓝 ν) := by
have hh := spectral_quantization_weak ν
(fun k => spectralBallQuantizer (D k).center (D k).radius a)
(fun k => measurable_spectralBallQuantizer _ _ _) δ hδ (fun k => (D k).quantizer_close)
apply hh.congr
intro k
apply Subtype.ext
exact (D k).law_eq_map.symm
lemma compact_partition_edges_tendsto (ν : ProbabilityMeasure ℝ) (a b : ℝ)
(δ : ℕ → ℝ) (D : (k : ℕ) → CompactSpectralPartition ν a b (δ k))
(hab : a ≤ b) (ha : a∈(ν : Measure ℝ).support) (hb : b∈(ν : Measure ℝ).support)
(hδ : Tendsto δ atTop (𝓝 0)) :
Tendsto (fun k => (D k).edge) atTop (𝓝 b) := by
have hlo k : b-δ k ≤ (D k).edge := by linarith [(D k).endpoint_bounds ha hb]
have hhi k : (D k).edge ≤ b := ((D k).value_mem hab (D k).upper).2
exact tendsto_of_tendsto_of_tendsto_of_le_of_le
(by simpa only [sub_zero] using tendsto_const_nhds.sub hδ)
tendsto_const_nhds hlo hhi
lemma compact_partition_variational_tendsto (ν : ProbabilityMeasure ℝ) (a b : ℝ)
(δ : ℕ → ℝ) (D : (k : ℕ) → CompactSpectralPartition ν a b (δ k))
(hab : a ≤ b) (ha : a∈(ν : Measure ℝ).support) (hb : b∈(ν : Measure ℝ).support)
(hbound : (ν : Measure ℝ).support ⊆ Icc a b) (hδ : Tendsto δ atTop (𝓝 0)) :
Tendsto (fun k => (variationalFunctional (measureR ((D k).law : Measure ℝ) (D k).edge)).toReal)
atTop (𝓝 (variationalFunctional (measureR (ν : Measure ℝ) b)).toReal) := by
let K := |a|+|b|+1
have hK : 0 < K := by dsimp [K]; positivity
have haK : -K ≤ a := by dsimp [K]; linarith [neg_abs_le a,abs_nonneg b]
have hbK : b ≤ K := by dsimp [K]; linarith [le_abs_self b,abs_nonneg a]
exact measureVariational_tendsto_weak (fun k => (D k).law) ν
(fun k => (D k).edge) b (compact_partition_laws_tendsto ν a b δ D hδ)
(compact_partition_edges_tendsto ν a b δ D hab ha hb hδ) hK hbK
(fun k => (((D k).value_mem hab (D k).upper).2).trans hbK)
(fun k => ((D k).law_support hab).mono (fun _ hx => ⟨haK.trans hx.1,hx.2⟩))
(by
filter_upwards [(ν : Measure ℝ).support_mem_ae] with x hx
exact ⟨haK.trans (hbound hx).1,(hbound hx).2⟩)
end InvariantIsing
end
end OAI
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