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67aee1d | 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 | import OAI.Analysis.FreeEntropy.MicrostatesLogLower
namespace OAI
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
end
end OAI
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