| import OAI.Analysis.FreeEntropy.MicrostatesLogLower |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped Topology |
| namespace FiniteEntropySeparation |
| theorem gueMatrix_isHermitian {n d : ℕ} (x : MatrixIndex n d → ℝ) (i : Fin n) : (gueMatrix x i).IsHermitian := by |
| change (((Real.sqrt d:ℂ)⁻¹) • hermitianMatrix (WithLp.toLp 2 x) i).IsHermitian |
| simp only [Matrix.IsHermitian,Matrix.conjTranspose_smul,map_inv₀,Complex.star_def,Complex.conj_ofReal] |
| rw [show (hermitianMatrix (WithLp.toLp 2 x) i).conjTranspose=hermitianMatrix (WithLp.toLp 2 x) i from hermitianMatrix_isHermitian _ _] |
| theorem exists_free_matrix_models (n : ℕ) : |
| ∃ (D : ℕ → ℕ) (A : (k : ℕ) → Fin n → Matrix (Fin (D k)) (Fin (D k)) ℂ), |
| (∀ k,0<D k) ∧ Tendsto D atTop atTop ∧ |
| (∀ k i,matrixOpNorm (A k i)≤64) ∧ (∀ k i,(A k i).IsHermitian) ∧ |
| ∀ w : List (Fin n), Tendsto (fun k => matrixTraceWord (A k) w) atTop (𝓝 (FreeSeries.fockMoments n w)) := by |
| have hex (k : ℕ) := exists_gue_model n k k (ε := 1/((k:ℝ)+1)) (by positivity) |
| choose D hD x hx hm using hex |
| refine ⟨D,fun k => gueMatrix (x k),fun k => lt_of_le_of_lt (Nat.zero_le _) (hD k),?_,hx,fun k i => gueMatrix_isHermitian _ _,?_⟩ |
| · exact tendsto_atTop_mono (fun k => (hD k).le) tendsto_id |
| · intro w |
| rw [tendsto_iff_norm_sub_tendsto_zero] |
| apply squeeze_zero' (Filter.Eventually.of_forall (fun _ => norm_nonneg _)) _ (tendsto_one_div_add_atTop_nhds_zero_nat) |
| filter_upwards [eventually_ge_atTop w.length] with k hk |
| exact (hm k w hk).le |
| end FiniteEntropySeparation |
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| end |
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| end OAI |
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