| import OAI.Probability.InvariantIsing.Cavity.CavityRetainedFrame |
| import OAI.Probability.InvariantIsing.Cavity.CavityPhysicalFactor |
|
|
| namespace OAI |
|
|
| /-! Exact removal and replacement of the small spectral block. The base |
| matrix is defined without choosing bases in the retained eigenspaces. -/ |
|
|
| noncomputable section |
| open scoped BigOperators Matrix |
|
|
| namespace InvariantIsing |
|
|
| lemma cavityPhysicalChange_sub_special {r d n : ℕ} |
| (V : Matrix (Fin r) (Fin d) ℝ) (E : Matrix (Fin r) (Fin n) ℝ) |
| (A A₀ : Matrix (Fin d) (Fin d) ℝ) (L : Matrix (Fin d) (Fin n) ℝ) |
| (C : Matrix (Fin n) (Fin n) ℝ) : |
| cavityPhysicalChange V E (A - A₀) L C = |
| cavityPhysicalChange V E A L C - V * A₀ * V.transpose := by |
| simp only [cavityPhysicalChange, Matrix.mul_sub, Matrix.sub_mul] |
| abel |
|
|
| lemma cavityPhysicalChange_pullback {r s d n : ℕ} |
| (W : Matrix (Fin r) (Fin s) ℝ) |
| (B : Matrix (Fin s) (Fin d) ℝ) (E : Matrix (Fin s) (Fin n) ℝ) |
| (A : Matrix (Fin d) (Fin d) ℝ) (L : Matrix (Fin d) (Fin n) ℝ) |
| (C : Matrix (Fin n) (Fin n) ℝ) : |
| cavityPhysicalChange (W * B) (W * E) A L C = |
| W * cavityPhysicalChange B E A L C * W.transpose := by |
| simp only [cavityPhysicalChange, Matrix.transpose_mul, Matrix.transpose_transpose, Matrix.mul_add, |
| Matrix.add_mul, Matrix.mul_assoc] |
|
|
| lemma cavityPhysicalChange_reconstruct {r d n : ℕ} |
| (D : Matrix (Fin r) (Fin r) ℝ) (hD : D.transpose = D) |
| (B : Matrix (Fin r) (Fin d) ℝ) (E : Matrix (Fin r) (Fin n) ℝ) |
| (hBE : B * B.transpose + E * E.transpose = 1) : |
| cavityPhysicalChange B E (B.transpose * D * B) (B.transpose * D * E) |
| (E.transpose * D * E) = D := by |
| calc |
| _ = (B * B.transpose + E * E.transpose) * D * |
| (B * B.transpose + E * E.transpose) := by |
| simp only [cavityPhysicalChange, Matrix.transpose_mul, Matrix.transpose_transpose, |
| hD, Matrix.mul_add, Matrix.add_mul, Matrix.mul_assoc] |
| abel |
| _ = D := by rw [hBE, Matrix.one_mul, Matrix.mul_one] |
|
|
| def cavityBaseReplacement {r s d : ℕ} |
| (J : Matrix (Fin r) (Fin r) ℝ) (W : Matrix (Fin r) (Fin s) ℝ) |
| (D : Matrix (Fin s) (Fin s) ℝ) (B : Matrix (Fin s) (Fin d) ℝ) |
| (A₀ : Matrix (Fin d) (Fin d) ℝ) : Matrix (Fin r) (Fin r) ℝ := |
| J - W * D * W.transpose + (W * B) * A₀ * (W * B).transpose |
|
|
| theorem cavityBaseReplacement_difference {r s d n : ℕ} |
| (J : Matrix (Fin r) (Fin r) ℝ) (W : Matrix (Fin r) (Fin s) ℝ) |
| (D : Matrix (Fin s) (Fin s) ℝ) (hD : D.transpose = D) |
| (B : Matrix (Fin s) (Fin d) ℝ) (E : Matrix (Fin s) (Fin n) ℝ) |
| (hBE : B * B.transpose + E * E.transpose = 1) |
| (A₀ : Matrix (Fin d) (Fin d) ℝ) : |
| J - cavityBaseReplacement J W D B A₀ = |
| cavityPhysicalChange (W * B) (W * E) (B.transpose * D * B - A₀) |
| (B.transpose * D * E) (E.transpose * D * E) := by |
| rw [cavityPhysicalChange_sub_special, cavityPhysicalChange_pullback, |
| cavityPhysicalChange_reconstruct D hD B E hBE, cavityBaseReplacement] |
| abel |
|
|
| theorem cavityBaseReplacement_energy_difference {r s d n : ℕ} |
| (J : Matrix (Fin r) (Fin r) ℝ) (W : Matrix (Fin r) (Fin s) ℝ) |
| (D : Matrix (Fin s) (Fin s) ℝ) (hD : D.transpose = D) |
| (B : Matrix (Fin s) (Fin d) ℝ) (E : Matrix (Fin s) (Fin n) ℝ) |
| (hBE : B * B.transpose + E * E.transpose = 1) |
| (A₀ : Matrix (Fin d) (Fin d) ℝ) (x : Fin r → ℝ) (ε : Spin n) |
| (hε : (W * E).transpose *ᵥ x = fun j => spinValue (ε j)) : |
| cavityQuadratic J x - cavityQuadratic (cavityBaseReplacement J W D B A₀) x = |
| cavityLogFactor (B.transpose * D * B - A₀) (B.transpose * D * E) |
| (E.transpose * D * E) ((W * B).transpose *ᵥ x) ε := by |
| rw [← cavityQuadratic_sub, cavityBaseReplacement_difference J W D hD B E hBE A₀] |
| exact cavityPhysicalChange_factor _ _ _ _ _ x ε hε |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
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|
