| import OAI.Analysis.FreeEntropy.MatrixHeatLimit |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Finset Filter |
| open scoped Classical Topology |
| namespace FiniteEntropySeparation |
| open FreeSeries |
| variable {n d : ℕ} |
| theorem matrixTraceWord_gue_map (x : MatrixIndex n d → ℝ) (w : List (Fin n)) : |
| matrixTraceWord (jointMatrix (fun _ : Empty => (0 : Matrix (Fin d) (Fin d) ℂ)) x) (w.map Sum.inr) = matrixTraceWord (gueMatrix x) w := by |
| rw [matrixTraceWord_map] |
| rfl |
| theorem gueTraceWord_stein_pure (hd : 0 < d) (i : Fin n) (w : List (Fin n)) : |
| (∫ x, matrixTraceWord (gueMatrix x) (i::w) ∂productGaussian (MatrixIndex n d)) = |
| ∑ r : Fin w.length, if w[r.val]=i then |
| ∫ x, matrixTraceWord (gueMatrix x) (w.take r.val)*matrixTraceWord (gueMatrix x) (w.drop (r.val+1)) ∂productGaussian (MatrixIndex n d) else 0 := by |
| have h := gueTraceWord_stein hd (fun _ : Empty => (0 : Matrix (Fin d) (Fin d) ℂ)) i (w.map Sum.inr) |
| have he : Sum.inr i::w.map (Sum.inr (α := Empty)) = (i::w).map Sum.inr := rfl |
| rw [he] at h |
| simp only [matrixTraceWord_gue_map] at h |
| rw [h] |
| apply Fintype.sum_equiv (finCongr (List.length_map (as := w) (Sum.inr (α := Empty)))) |
| intro r |
| simp only [finCongr_apply,Fin.val_cast,List.getElem_map,Sum.inr.injEq,← List.map_take,← List.map_drop,matrixTraceWord_gue_map] |
| theorem gueTraceWord_expected_tendsto {D : ℕ → ℕ} (hD0 : ∀ k,0<D k) (hD : Tendsto D atTop atTop) (w : List (Fin n)) : |
| Tendsto (fun k => ∫ x, matrixTraceWord (gueMatrix x) w ∂productGaussian (MatrixIndex n (D k))) atTop (𝓝 (fockMoments n w)) := by |
| induction w using FreeSeries.length_induction with |
| | h w ih => |
| cases w with |
| | nil => |
| have he (k : ℕ) (x : MatrixIndex n (D k) → ℝ) : matrixTraceWord (gueMatrix x) [] = 1 := by |
| simp [matrixTraceWord,matrixWord,ne_of_gt (hD0 k)] |
| simp [he,fockMoments] |
| | cons i w => |
| have hf : fockMoments n (i::w) = ∑ r : Fin w.length, if w[r.val]=i then fockMoments n (w.take r.val)*fockMoments n (w.drop (r.val+1)) else 0 := |
| Fock.semicircular_integration_by_parts i w |
| simp only [gueTraceWord_stein_pure (hD0 _), hf] |
| apply tendsto_finsetSum |
| intro r _ |
| split_ifs with hr |
| · have hc := matrixTraceWord_covariance_tendsto hD0 hD (fun k (_ : Empty) => (0 : Matrix (Fin (D k)) (Fin (D k)) ℂ)) (R := 0) le_rfl (by intro k a; exact a.elim) ((w.take r.val).map Sum.inr) ((w.drop (r.val+1)).map Sum.inr) |
| simp only [matrixTraceWord_gue_map,expectedMatrixWord] at hc |
| apply tendsto_of_sub_zero hc |
| exact (ih (w.take r.val) (by simp)).mul (ih (w.drop (r.val+1)) (by simp)) |
| · exact tendsto_const_nhds |
| theorem gueTraceWord_probability_tendsto {D : ℕ → ℕ} (hD0 : ∀ k,0<D k) (hD : Tendsto D atTop atTop) |
| (w : List (Fin n)) {ε : ℝ} (hε : 0 < ε) : |
| Tendsto (fun k => (productGaussian (MatrixIndex n (D k))).real {x | ε ≤ ‖matrixTraceWord (gueMatrix x) w-fockMoments n w‖}) atTop (𝓝 0) := by |
| have hm : Tendsto (fun k => expectedMatrixWord (fun k (_ : Empty) => (0 : Matrix (Fin (D k)) (Fin (D k)) ℂ)) k (w.map Sum.inr)) atTop (𝓝 (fockMoments n w)) := by |
| simpa only [expectedMatrixWord,matrixTraceWord_gue_map] using gueTraceWord_expected_tendsto hD0 hD w |
| simpa only [matrixTraceWord_gue_map] using matrixTraceWord_probability_tendsto hD0 hD _ (R := 0) le_rfl (by intro k a; exact a.elim) (w.map Sum.inr) hm hε |
| end FiniteEntropySeparation |
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| end |
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| end OAI |
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