File size: 3,400 Bytes
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 | import OAI.Probability.InvariantIsing.Spectral.SpectralLabelCounts
import OAI.Probability.InvariantIsing.Spectral.FiniteSpectralMeasure
namespace OAI
/-! Remove zero limiting weights without dropping their finite-volume labels. -/
noncomputable section
open MeasureTheory Filter Set
open scoped Topology Classical BigOperators
namespace InvariantIsing
def positiveSpectralLabel {ι : Type*} (w : ι → ℝ) (fallback : {i // 0 < w i})
(i : ι) : {i // 0 < w i} := if h : 0 < w i then ⟨i,h⟩ else fallback
lemma positiveSpectralLabel_self {ι : Type*} (w : ι → ℝ)
(fallback i : {i // 0 < w i}) : positiveSpectralLabel w fallback i=i := by
simp only [positiveSpectralLabel, dite_eq_left i.property]
lemma positiveSpectralWeights_sum {ι : Type*} [Fintype ι]
(w : ι → ℝ) (hw : ∀ i, 0 ≤ w i) (hsum : ∑ i, w i=1) :
∑ i : {i // 0 < w i}, w i=1 := by
have hz : (∑ i : {i // ¬0 < w i}, w i)=0 := Finset.sum_eq_zero fun i _ =>
le_antisymm (le_of_not_gt i.property) (hw i)
have hh := Fintype.sum_subtype_add_sum_subtype (fun i => 0 < w i) w
rw [hz,add_zero,hsum] at hh
exact hh
lemma positiveSpectralLabel_weight {ι : Type*} [Fintype ι]
(w : ι → ℝ) (hw : ∀ i, 0 ≤ w i) (fallback b : {i // 0 < w i}) :
(∑ a : {a // positiveSpectralLabel w fallback a=b}, w a)=w b := by
rw [← Finset.sum_subtype (Finset.univ.filter (fun a => positiveSpectralLabel w fallback a=b))
(by simp) w]
apply Finset.sum_eq_single b.val
· intro a ha hab
by_cases hp : 0 < w a
· have hh : positiveSpectralLabel w fallback a=b := (Finset.mem_filter.mp ha).2
have he : a=b.val := by
simpa only [positiveSpectralLabel,dite_eq_left hp] using congrArg Subtype.val hh
exact (hab he).elim
· exact le_antisymm (le_of_not_gt hp) (hw a)
· intro hb
exact (hb (by simp [positiveSpectralLabel_self])).elim
lemma positive_spectral_counts_tendsto {ι : Type*} [Fintype ι]
(N : ℕ → ℕ) (g : (k : ℕ) → Fin (N k) → ι)
(w : ι → ℝ) (hw : ∀ i, 0 ≤ w i) (fallback : {i // 0 < w i})
(hlim : Tendsto (fun k a => (spectralLabelCount (g k) a : ℝ)/N k) atTop (𝓝 w)) :
Tendsto (fun k b =>
(spectralLabelCount (positiveSpectralLabel w fallback ∘ g k) b : ℝ)/N k)
atTop (𝓝 (fun b : {i // 0 < w i} => w b)) := by
apply tendsto_pi_nhds.mpr
intro b
have hh := (tendsto_pi_nhds.mp
(spectral_label_counts_relabel N g w hlim (positiveSpectralLabel w fallback))) b
convert hh using 1
apply congrArg nhds
calc
w b = ∑ a : {a // positiveSpectralLabel w fallback a=b}, w a :=
(positiveSpectralLabel_weight w hw fallback b).symm
_ = _ := by
apply Finset.sum_congr
· ext a
simp only [Finset.mem_univ]
· intro a _
rfl
lemma finiteSpectralMeasure_positive {ι : Type*} [Fintype ι]
(w lam : ι → ℝ) (hw : ∀ i, 0 ≤ w i) :
finiteSpectralMeasure (fun i : {i // 0 < w i} => w i) (fun i => lam i)=
finiteSpectralMeasure w lam := by
simp only [finiteSpectralMeasure,Measure.sum_fintype]
apply Fintype.sum_of_injective Subtype.val Subtype.val_injective
· intro i hi
have hp : ¬0 < w i := fun hp => hi ⟨⟨i,hp⟩,rfl⟩
have hz : w i=0 := le_antisymm (le_of_not_gt hp) (hw i)
simp only [hz,ENNReal.ofReal_zero,zero_smul]
· intro i
rfl
end InvariantIsing
end
end OAI
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