File size: 3,837 Bytes
c4cdbbd | 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 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 | 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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