| import OAI.Analysis.FreeEntropy.GUEFock |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Finset Filter Set |
| open scoped Classical Topology |
| namespace FiniteEntropySeparation |
| theorem gueMatrix_norm_tail64 {n d : ℕ} (hd : 0<d) (i : Fin n) : |
| (productGaussian (MatrixIndex n d)).real {x | 64 < matrixOpNorm (gueMatrix x i)} ≤ 4*Real.exp (-64*(d:ℝ)) := by |
| have hlog := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<9) |
| calc |
| _ ≤ 4*9^(4*d)*Real.exp (-((d:ℝ)*64^2)/32) := gueMatrix_norm_tail hd i (by norm_num) |
| _ = 4*Real.exp ((4*(d:ℝ))*Real.log 9 - (d:ℝ)*64^2/32) := by |
| rw [show (9:ℝ)^(4*d)=Real.exp ((4*(d:ℝ))*Real.log 9) by rw [show 4*(d:ℝ)=((4*d:ℕ):ℝ) by push_cast; rfl, Real.exp_nat_mul, Real.exp_log (by norm_num : (0:ℝ)<9)]] |
| rw [mul_assoc, ← Real.exp_add]; congr 2; ring |
| _ ≤ _ := by |
| gcongr |
| have hd' : 0 ≤ (d:ℝ) := Nat.cast_nonneg _ |
| nlinarith [mul_nonneg hd' (show 0 ≤ 8-Real.log 9 by linarith)] |
| theorem gueMatrix_norm_probability_tendsto {n : ℕ} {D : ℕ → ℕ} (hD0 : ∀k,0<D k) (hD : Tendsto D atTop atTop) (i : Fin n) : |
| Tendsto (fun k => (productGaussian (MatrixIndex n (D k))).real {x | 64 < matrixOpNorm (gueMatrix x i)}) atTop (𝓝 0) := by |
| apply squeeze_zero (fun _ => measureReal_nonneg) (fun k => gueMatrix_norm_tail64 (hD0 k) i) |
| have ht : Tendsto (fun k => (64:ℝ)*(D k:ℝ)) atTop atTop := (tendsto_natCast_atTop_atTop.comp hD).const_mul_atTop (by norm_num) |
| simpa only [neg_mul, mul_zero,Function.comp_def] using (Real.tendsto_exp_neg_atTop_nhds_zero.comp ht).const_mul 4 |
| def gueBad (n m d : ℕ) (ε : ℝ) : Set (MatrixIndex n d → ℝ) := |
| {x | (∃ i,64 < matrixOpNorm (gueMatrix x i)) ∨ ∃ w : List (Fin n), w.length ≤ m ∧ ε ≤ ‖matrixTraceWord (gueMatrix x) w-FreeSeries.fockMoments n w‖} |
| theorem gueBad_probability_tendsto (n m : ℕ) {ε : ℝ} (hε : 0<ε) : |
| Tendsto (fun k => (productGaussian (MatrixIndex n (k+1))).real (gueBad n m (k+1) ε)) atTop (𝓝 0) := by |
| let W := (List.finite_length_le (Fin n) m).toFinset |
| have hD : Tendsto (fun k : ℕ => k+1) atTop atTop := tendsto_add_atTop_nat 1 |
| have hn : Tendsto (fun k => ∑ i : Fin n, (productGaussian (MatrixIndex n (k+1))).real {x | 64 < matrixOpNorm (gueMatrix x i)}) atTop (𝓝 0) := by |
| simpa using tendsto_finsetSum univ (fun i _ => gueMatrix_norm_probability_tendsto (fun _ => Nat.succ_pos _) hD i) |
| have hw : Tendsto (fun k => ∑ w ∈ W, (productGaussian (MatrixIndex n (k+1))).real {x | ε ≤ ‖matrixTraceWord (gueMatrix x) w-FreeSeries.fockMoments n w‖}) atTop (𝓝 0) := by |
| simpa using tendsto_finsetSum W (fun w _ => gueTraceWord_probability_tendsto (fun _ => Nat.succ_pos _) hD w hε) |
| apply squeeze_zero (fun k => measureReal_nonneg) _ (by simpa using hn.add hw) |
| intro k |
| have he : gueBad n m (k+1) ε = |
| (⋃ i ∈ (univ : Finset (Fin n)), {x | 64 < matrixOpNorm (gueMatrix x i)}) ∪ |
| (⋃ w ∈ W, {x | ε ≤ ‖matrixTraceWord (gueMatrix x) w-FreeSeries.fockMoments n w‖}) := by |
| ext x |
| simp [gueBad,W] |
| rw [he] |
| exact (measureReal_union_le _ _).trans (add_le_add (measureReal_biUnion_finset_le _ _) (measureReal_biUnion_finset_le _ _)) |
| theorem exists_gue_model (n m K : ℕ) {ε : ℝ} (hε : 0<ε) : |
| ∃ d, K<d ∧ ∃ x : MatrixIndex n d → ℝ, (∀ i,matrixOpNorm (gueMatrix x i) ≤ 64) ∧ |
| ∀ w : List (Fin n), w.length ≤ m → ‖matrixTraceWord (gueMatrix x) w-FreeSeries.fockMoments n w‖ < ε := by |
| have ht := (gueBad_probability_tendsto n m hε).eventually (gt_mem_nhds (by norm_num : (0:ℝ)<1)) |
| obtain ⟨k,hp⟩ := (eventually_atTop.mp ht) |
| let q := max k K |
| have hq : (productGaussian (MatrixIndex n (q+1))).real (gueBad n m (q+1) ε)<1 := hp q (le_max_left _ _) |
| have hx : ∃ x : MatrixIndex n (q+1) → ℝ, x ∉ gueBad n m (q+1) ε := by |
| by_contra hh |
| have he : gueBad n m (q+1) ε = univ := by ext x; simp only [mem_univ,iff_true]; exact not_not.mp (not_exists.mp hh x) |
| simp [he] at hq |
| obtain ⟨x,hx⟩ := hx |
| refine ⟨q+1,by dsimp [q]; omega,x,?_,?_⟩ |
| · intro i; by_contra hi; exact hx (Or.inl ⟨i,not_le.mp hi⟩) |
| · intro w hw; by_contra hh; exact hx (Or.inr ⟨w,hw,not_lt.mp hh⟩) |
| end FiniteEntropySeparation |
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| end |
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| end OAI |
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