| import OAI.Probability.InvariantIsing.Gaussian.GaussianColumnProductLaw |
| import OAI.Probability.InvariantIsing.Gaussian.GaussianGramResolventEquation |
| import OAI.Probability.InvariantIsing.Gaussian.HermitianGaussianQuadratic |
|
|
| namespace OAI |
|
|
| /-! Deleting a column removes all dependence on that Gaussian coordinate. -/ |
| noncomputable section |
| open Matrix |
| open scoped BigOperators RealInnerProductSpace |
| namespace InvariantIsing |
|
|
| lemma gaussianGramLeaveOneOut_congr {N m : ℕ} (g h : Fin m → Fin N → ℝ) (j : Fin m) |
| (he : ∀ l, l ≠ j → g l = h l) : |
| gaussianGramLeaveOneOut (gaussianColumnArray g) j = |
| gaussianGramLeaveOneOut (gaussianColumnArray h) j := by |
| unfold gaussianGramLeaveOneOut |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro l hl |
| rw [gaussianColumnArray_column,gaussianColumnArray_column,he l (Finset.mem_erase.mp hl).1] |
|
|
| lemma gaussianGramLeaveOneOut_insertNth {N m : ℕ} (j : Fin (m+1)) |
| (x y : Fin N → ℝ) (g : Fin m → Fin N → ℝ) : |
| gaussianGramLeaveOneOut (gaussianColumnArray (j.insertNth x g)) j = |
| gaussianGramLeaveOneOut (gaussianColumnArray (j.insertNth y g)) j := by |
| apply gaussianGramLeaveOneOut_congr |
| intro l hl |
| obtain ⟨k,rfl⟩ := Fin.exists_succAbove_eq hl |
| simp only [Fin.insertNth_apply_succAbove] |
|
|
| lemma gaussianGramColumnQuadratic_eq {N m : ℕ} (t : ℝ) (g : Fin m → Fin N → ℝ) (j : Fin m) : |
| gaussianGramColumnQuadratic t (gaussianColumnArray g) j = |
| (1/(N : ℝ))*gaussianQuadraticForm |
| (gaussianGramLeaveResolvent t (gaussianColumnArray g) j) (WithLp.toLp 2 (g j)) := by |
| have he : ((Real.sqrt (N : ℝ))⁻¹)^2 = 1/(N : ℝ) := by |
| rw [inv_pow,Real.sq_sqrt (Nat.cast_nonneg N),one_div] |
| unfold gaussianGramColumnQuadratic gaussianPatternScaledColumn |
| rw [gaussianColumnArray_column,Matrix.mulVec_smul,dotProduct_smul,smul_dotProduct] |
| simp only [smul_eq_mul] |
| unfold gaussianQuadraticForm |
| rw [EuclideanSpace.inner_eq_star_dotProduct] |
| simp only [star_trivial] |
| rw [dotProduct_comm] |
| change _ = (1/(N : ℝ))*((gaussianGramLeaveResolvent t (gaussianColumnArray g) j *ᵥ g j) ⬝ᵥ g j) |
| calc |
| _ = ((Real.sqrt (N : ℝ))⁻¹)^2*((gaussianGramLeaveResolvent t (gaussianColumnArray g) j *ᵥ g j) ⬝ᵥ g j) := by ring |
| _ = _ := by rw [he] |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
|
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