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b317718 | 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 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 | import OAI.Probability.InvariantIsing.Core.PairGibbs
namespace OAI
/-! Exact two-replica off-diagonal Haar Ward identity. -/
noncomputable section
open MeasureTheory IsingPerceptron
open scoped BigOperators
namespace InvariantIsing
lemma abs_crossCoordinateProduct_le {N : ℕ} (i j : Fin N)
(U : Orthogonal N) (σ τ : Spin N) :
|spinCoordinate U σ i * spinCoordinate U τ j| ≤ N := by
have hi := spinCoordinate_sq_le U σ i
have hj := spinCoordinate_sq_le U τ j
rw [abs_mul]
nlinarith [sq_nonneg (|spinCoordinate U σ i| - |spinCoordinate U τ j|),
sq_abs (spinCoordinate U σ i), sq_abs (spinCoordinate U τ j)]
def offCoordinateProduct {N : ℕ} (i j : Fin N) (F : Spin N → Spin N → ℝ)
(U : Orthogonal N) (x : Spin N × Spin N) : ℝ :=
spinCoordinate U x.1 i * spinCoordinate U x.2 j * F x.1 x.2
def offCoordinateVariation {N : ℕ} (i j : Fin N) (F : Spin N → Spin N → ℝ)
(U : Orthogonal N) (x : Spin N × Spin N) : ℝ :=
(spinCoordinate U x.1 j * spinCoordinate U x.2 j -
spinCoordinate U x.1 i * spinCoordinate U x.2 i) * F x.1 x.2
lemma hasDerivAt_offCoordinateProduct {N : ℕ} (i j : Fin N) (hij : i ≠ j)
(F : Spin N → Spin N → ℝ) (U : Orthogonal N) (x : Spin N × Spin N) (t : ℝ) :
HasDerivAt (fun s => offCoordinateProduct i j F (planeRotation i j s * U) x)
(offCoordinateVariation i j F (planeRotation i j t * U) x) t := by
have hd := ((hasDerivAt_planeRotation_coordinate i j i U (spinVector x.1) t).mul
(hasDerivAt_planeRotation_coordinate i j j U (spinVector x.2) t)).mul_const (F x.1 x.2)
convert hd using 1
· rfl
· simp only [offCoordinateVariation, spinCoordinate, ite_true, hij, Ne.symm hij,
ite_false, sub_zero, zero_sub]
ring
/-- Exact finite off-diagonal rotation identity. The spin test is fixed during
rotation; in the main application it is a function of the rotation-invariant
total overlap. Arbitrary deterministic nonzero positive priors are allowed. -/
theorem offDiagonal_coordinate_ward_identity {N : ℕ} (i j : Fin N) (hij : i ≠ j)
(μ : Measure (Orthogonal N)) [IsProbabilityMeasure μ] [μ.IsMulLeftInvariant]
{w : Spin N → ℝ} (hw : GibbsReference w) (eig c : Fin N → ℝ)
(F : Spin N → Spin N → ℝ) (B : ℝ) (hB : 0 ≤ B) (hF : ∀ σ τ, |F σ τ| ≤ B) :
(∫ U,
gibbsPairAverage w (orbitHamiltonian eig c U)
(fun σ τ => (spinCoordinate U σ j * spinCoordinate U τ j -
spinCoordinate U σ i * spinCoordinate U τ i) * F σ τ) +
(eig i - eig j) *
(gibbsPairAverage w (orbitHamiltonian eig c U)
(fun σ τ => spinCoordinate U σ i * spinCoordinate U τ j * F σ τ *
(coordinateProduct i j U σ + coordinateProduct i j U τ)) -
2 * gibbsPairAverage w (orbitHamiltonian eig c U)
(fun σ τ => spinCoordinate U σ i * spinCoordinate U τ j * F σ τ) *
gibbsAverage w (orbitHamiltonian eig c U) (coordinateProduct i j U)) ∂μ) = 0 := by
let R := planeRotation i j
let H : Orthogonal N → Spin N × Spin N → ℝ := fun U x =>
orbitHamiltonian eig c U x.1 + orbitHamiltonian eig c U x.2
let A := offCoordinateProduct i j F
let dH : ℝ → Orthogonal N → Spin N × Spin N → ℝ := fun t U x =>
(eig i - eig j) * (coordinateProduct i j (R t * U) x.1 +
coordinateProduct i j (R t * U) x.2)
let dA : ℝ → Orthogonal N → Spin N × Spin N → ℝ := fun t U =>
offCoordinateVariation i j F (R t * U)
have hR : R 0 = 1 := planeRotation_zero i j
have hP (σ : Spin N) : Measurable (fun U : Orthogonal N => coordinateProduct i j U σ) :=
(measurable_spinCoordinate σ i).mul (measurable_spinCoordinate σ j)
have hAm (x : Spin N × Spin N) : Measurable (fun U : Orthogonal N => A U x) :=
((measurable_spinCoordinate x.1 i).mul (measurable_spinCoordinate x.2 j)).mul_const _
have hHm (x : Spin N × Spin N) : Measurable (fun U => H U x) :=
(measurable_orbitHamiltonian eig c x.1).add (measurable_orbitHamiltonian eig c x.2)
have hdHm (x : Spin N × Spin N) : Measurable (fun U => dH 0 U x) := by
simp only [dH, hR, one_mul]
exact ((hP x.1).add (hP x.2)).const_mul _
have hdAm (x : Spin N × Spin N) : Measurable (fun U => dA 0 U x) := by
simp only [dA, hR, one_mul, offCoordinateVariation]
exact (((measurable_spinCoordinate x.1 j).mul (measurable_spinCoordinate x.2 j)).sub
((measurable_spinCoordinate x.1 i).mul (measurable_spinCoordinate x.2 i))).mul_const _
have hHd : ∀ U t, t ∈ Set.Ioo (-1 : ℝ) 1 → ∀ x,
HasDerivAt (fun s => H (R s * U) x) (dH t U x) t := by
intro U t _ x
have h1 := (hasDerivAt_planeRotation_energy i j U eig x.1 t).add_const (fieldEnergy c x.1)
have h2 := (hasDerivAt_planeRotation_energy i j U eig x.2 t).add_const (fieldEnergy c x.2)
convert h1.add h2 using 1
· rfl
· dsimp only [dH, R, coordinateProduct, spinCoordinate]
ring
have hAd : ∀ U t, t ∈ Set.Ioo (-1 : ℝ) 1 → ∀ x,
HasDerivAt (fun s => A (R s * U) x) (dA t U x) t := by
intro U t _ x
exact hasDerivAt_offCoordinateProduct i j hij F U x t
have hAb : ∀ U x, |A U x| ≤ N * B := by
intro U x
rw [show A U x = spinCoordinate U x.1 i * spinCoordinate U x.2 j * F x.1 x.2 from rfl,
abs_mul]
exact mul_le_mul (abs_crossCoordinateProduct_le i j U x.1 x.2) (hF x.1 x.2)
(abs_nonneg _) (Nat.cast_nonneg _)
have hHb : ∀ U t, t ∈ Set.Ioo (-1 : ℝ) 1 → ∀ x,
|dH t U x| ≤ |eig i - eig j| * (2 * N) := by
intro U t _ x
dsimp only [dH]
rw [abs_mul]
apply mul_le_mul_of_nonneg_left _ (abs_nonneg _)
have hs := (abs_add_le _ _).trans (add_le_add
(abs_coordinateProduct_le i j (R t * U) x.1)
(abs_coordinateProduct_le i j (R t * U) x.2))
linarith
have hAdb : ∀ U t, t ∈ Set.Ioo (-1 : ℝ) 1 → ∀ x, |dA t U x| ≤ 2 * N * B := by
intro U t _ x
dsimp only [dA, offCoordinateVariation]
rw [abs_mul]
apply mul_le_mul _ (hF x.1 x.2) (abs_nonneg _) (by positivity)
have hs := (abs_sub _ _).trans (add_le_add
(abs_crossCoordinateProduct_le j j (R t * U) x.1 x.2)
(abs_crossCoordinateProduct_le i i (R t * U) x.1 x.2))
linarith
have hwrd := integral_gibbs_rotation_identity μ R hR (GibbsReference_pair hw)
H A hHm hAm dH dA hdHm hdAm hHd hAd
(N * B) (|eig i - eig j| * (2 * N)) (2 * N * B) (by positivity) hAb hHb hAdb
simp only [dH, dA, H, A, hR, one_mul, offCoordinateProduct] at hwrd
convert hwrd using 1
congr 1
funext U
unfold offCoordinateVariation offCoordinateProduct
rw [gibbsAverage_pair w (orbitHamiltonian eig c U)
(fun σ τ => (spinCoordinate U σ j * spinCoordinate U τ j -
spinCoordinate U σ i * spinCoordinate U τ i) * F σ τ)]
have hfactor : (fun x : Spin N × Spin N =>
spinCoordinate U x.1 i * spinCoordinate U x.2 j * F x.1 x.2 *
((eig i - eig j) * (coordinateProduct i j U x.1 + coordinateProduct i j U x.2))) =
(fun x => (eig i - eig j) *
(spinCoordinate U x.1 i * spinCoordinate U x.2 j * F x.1 x.2 *
(coordinateProduct i j U x.1 + coordinateProduct i j U x.2))) := by
funext x
ring
rw [hfactor, gibbsAverage_const_mul, gibbsAverage_const_mul,
gibbsAverage_pair_sum hw]
rw [gibbsAverage_pair w (orbitHamiltonian eig c U)
(fun σ τ => spinCoordinate U σ i * spinCoordinate U τ j * F σ τ *
(coordinateProduct i j U σ + coordinateProduct i j U τ)),
gibbsAverage_pair w (orbitHamiltonian eig c U)
(fun σ τ => spinCoordinate U σ i * spinCoordinate U τ j * F σ τ)]
ring
end InvariantIsing
end
end OAI
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