| 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 | |