| import OAI.Probability.InvariantIsing.Gaussian.GaussianPatternModel |
| import Mathlib.LinearAlgebra.Matrix.PosDef |
|
|
| namespace OAI |
|
|
| /-! The positive real resolvent of the physical Gaussian Gram matrix. -/ |
| noncomputable section |
| open Matrix |
| namespace InvariantIsing |
|
|
| def gaussianGramResolvent {N m : ℕ} (t : ℝ) |
| (z : EuclideanSpace ℝ (Fin N × Fin m)) : Matrix (Fin N) (Fin N) ℝ := |
| (t • (1 : Matrix (Fin N) (Fin N) ℝ)+gaussianPatternCoupling 1 z)⁻¹ |
|
|
| def gaussianGramStieltjes {N m : ℕ} (t : ℝ) |
| (z : EuclideanSpace ℝ (Fin N × Fin m)) : ℝ := (gaussianGramResolvent t z).trace/N |
|
|
| lemma gaussianPatternCoupling_posSemidef {N m : ℕ} (z : EuclideanSpace ℝ (Fin N × Fin m)) : |
| (gaussianPatternCoupling 1 z).PosSemidef := by |
| have h := Matrix.posSemidef_self_mul_conjTranspose (gaussianPatternArray z) |
| simpa only [gaussianPatternCoupling,Matrix.conjTranspose_eq_transpose_of_trivial] using |
| h.smul (show 0 ≤ (1 : ℝ)/N by positivity) |
|
|
| lemma gaussianGramShift_posDef {N m : ℕ} {t : ℝ} (ht : 0 < t) |
| (z : EuclideanSpace ℝ (Fin N × Fin m)) : |
| (t • (1 : Matrix (Fin N) (Fin N) ℝ)+gaussianPatternCoupling 1 z).PosDef := |
| ((Matrix.PosDef.one).smul ht).add_posSemidef (gaussianPatternCoupling_posSemidef z) |
|
|
| lemma gaussianGramResolvent_posDef {N m : ℕ} {t : ℝ} (ht : 0 < t) |
| (z : EuclideanSpace ℝ (Fin N × Fin m)) : (gaussianGramResolvent t z).PosDef := |
| (gaussianGramShift_posDef ht z).inv |
|
|
| lemma gaussianGramResolvent_identity {N m : ℕ} {t : ℝ} (ht : 0 < t) |
| (z : EuclideanSpace ℝ (Fin N × Fin m)) : |
| t • gaussianGramResolvent t z+gaussianPatternCoupling 1 z*gaussianGramResolvent t z = 1 := by |
| have hd := (gaussianGramShift_posDef ht z).det_pos |
| have h := Matrix.mul_nonsing_inv (t • (1 : Matrix (Fin N) (Fin N) ℝ)+gaussianPatternCoupling 1 z) |
| (isUnit_iff_ne_zero.mpr hd.ne') |
| simpa only [gaussianGramResolvent,add_mul,smul_mul_assoc,one_mul] using h |
|
|
| lemma gaussianGramResolvent_trace_identity {N m : ℕ} {t : ℝ} (ht : 0 < t) |
| (z : EuclideanSpace ℝ (Fin N × Fin m)) : |
| t*(gaussianGramResolvent t z).trace+ |
| (gaussianPatternCoupling 1 z*gaussianGramResolvent t z).trace = N := by |
| have h := congrArg Matrix.trace (gaussianGramResolvent_identity ht z) |
| simpa only [Matrix.trace_add,Matrix.trace_smul,smul_eq_mul,Matrix.trace_one,Fintype.card_fin] using h |
|
|
| lemma gaussianGramStieltjes_nonneg {N m : ℕ} {t : ℝ} (ht : 0 < t) |
| (z : EuclideanSpace ℝ (Fin N × Fin m)) : 0 ≤ gaussianGramStieltjes t z := |
| div_nonneg (gaussianGramResolvent_posDef ht z).posSemidef.trace_nonneg (Nat.cast_nonneg N) |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
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|
