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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 | import OAI.Probability.InvariantIsing.Arrays.ReplicaContractions
namespace OAI
/-! Exact finite spectral-block Ward identity for an arbitrary spin prior. -/
noncomputable section
open MeasureTheory IsingPerceptron
open scoped BigOperators
namespace InvariantIsing
def coordinateWardTerm {N : ℕ} (w : Spin N → ℝ) (eig c : Fin N → ℝ)
(i j : Fin N) (U : Orthogonal N) : ℝ :=
gibbsAverage w (orbitHamiltonian eig c U)
(fun σ => spinCoordinate U σ j ^ 2 - spinCoordinate U σ i ^ 2) +
(eig i - eig j) *
(gibbsAverage w (orbitHamiltonian eig c U) (fun σ => coordinateProduct i j U σ ^ 2) -
gibbsAverage w (orbitHamiltonian eig c U) (coordinateProduct i j U) ^ 2)
lemma measurable_coordinateWardTerm {N : ℕ} (w : Spin N → ℝ) (eig c : Fin N → ℝ)
(i j : Fin N) : Measurable (coordinateWardTerm w eig c i j) := by
have hH := measurable_orbitHamiltonian eig c
have hP (σ : Spin N) : Measurable (fun U : Orthogonal N => coordinateProduct i j U σ) :=
(measurable_spinCoordinate σ i).mul (measurable_spinCoordinate σ j)
exact (measurable_gibbsAverage w _ _ hH (fun σ =>
((measurable_spinCoordinate σ j).pow_const 2).sub
((measurable_spinCoordinate σ i).pow_const 2))).add
(((measurable_gibbsAverage w _ _ hH (fun σ => (hP σ).pow_const 2)).sub
((measurable_gibbsAverage w _ _ hH hP).pow_const 2)).const_mul _)
lemma abs_coordinateWardTerm_le {N : ℕ} {w : Spin N → ℝ} (hw : GibbsReference w)
(eig c : Fin N → ℝ) (i j : Fin N) (U : Orthogonal N) :
|coordinateWardTerm w eig c i j U| ≤ N + |eig i - eig j| * (2 * (N : ℝ) ^ 2) := by
let H := orbitHamiltonian eig c U
have hP := abs_gibbsAverage_le hw H (coordinateProduct i j U)
(abs_coordinateProduct_le i j U)
have hP2 : |gibbsAverage w H (coordinateProduct i j U)| ^ 2 ≤ (N : ℝ) ^ 2 :=
pow_le_pow_left₀ (abs_nonneg _) hP 2
have hQ := abs_gibbsAverage_le hw H (fun σ => coordinateProduct i j U σ ^ 2)
(fun σ => by
rw [abs_sq, ← sq_abs]
exact pow_le_pow_left₀ (abs_nonneg _) (abs_coordinateProduct_le i j U σ) 2)
have hD := abs_gibbsAverage_le hw H
(fun σ => spinCoordinate U σ j ^ 2 - spinCoordinate U σ i ^ 2)
(abs_coordinateSquareDifference_le i j U)
have hV : |gibbsAverage w H (fun σ => coordinateProduct i j U σ ^ 2) -
gibbsAverage w H (coordinateProduct i j U) ^ 2| ≤ 2 * (N : ℝ) ^ 2 := by
have hP2' : |gibbsAverage w H (coordinateProduct i j U) ^ 2| ≤ (N : ℝ) ^ 2 := by
simpa only [abs_sq, sq_abs] using hP2
have hs := (abs_sub _ _).trans (add_le_add hQ hP2')
linarith
exact (abs_add_le _ _).trans (add_le_add hD (by
rw [abs_mul]
exact mul_le_mul_of_nonneg_left hV (abs_nonneg _)))
lemma integrable_coordinateWardTerm {N : ℕ}
(μ : Measure (Orthogonal N)) [IsProbabilityMeasure μ]
{w : Spin N → ℝ} (hw : GibbsReference w) (eig c : Fin N → ℝ) (i j : Fin N) :
Integrable (coordinateWardTerm w eig c i j) μ :=
(integrable_const (N + |eig i - eig j| * (2 * (N : ℝ) ^ 2))).mono'
(measurable_coordinateWardTerm w eig c i j).aestronglyMeasurable
(ae_of_all _ fun U => by
simpa only [Real.norm_eq_abs] using abs_coordinateWardTerm_le hw eig c i j U)
lemma coordinateWardTerm_block_sum {N : ℕ} (w : Spin N → ℝ) (eig c : Fin N → ℝ)
(I J : Finset (Fin N)) (a b : ℝ)
(hI : ∀ i ∈ I, eig i = a) (hJ : ∀ j ∈ J, eig j = b) (U : Orthogonal N) :
(∑ i ∈ I, ∑ j ∈ J, coordinateWardTerm w eig c i j U) =
(I.card : ℝ) * gibbsAverage w (orbitHamiltonian eig c U)
(fun σ => blockOverlap (spinCoordinate U) J σ σ) -
(J.card : ℝ) * gibbsAverage w (orbitHamiltonian eig c U)
(fun σ => blockOverlap (spinCoordinate U) I σ σ) +
(a - b) *
(gibbsAverage w (orbitHamiltonian eig c U)
(fun σ => blockOverlap (spinCoordinate U) I σ σ * blockOverlap (spinCoordinate U) J σ σ) -
gibbsPairAverage w (orbitHamiltonian eig c U)
(fun σ τ => blockOverlap (spinCoordinate U) I σ τ * blockOverlap (spinCoordinate U) J σ τ)) := by
have he : (∑ i ∈ I, ∑ j ∈ J, coordinateWardTerm w eig c i j U) =
∑ i ∈ I, ∑ j ∈ J, (
gibbsAverage w (orbitHamiltonian eig c U)
(fun σ => spinCoordinate U σ j ^ 2 - spinCoordinate U σ i ^ 2) +
(a - b) *
(gibbsAverage w (orbitHamiltonian eig c U) (fun σ => coordinateProduct i j U σ ^ 2) -
gibbsAverage w (orbitHamiltonian eig c U) (coordinateProduct i j U) ^ 2)) := by
apply Finset.sum_congr rfl
intro i hi
apply Finset.sum_congr rfl
intro j hj
rw [coordinateWardTerm, hI i hi, hJ j hj]
rw [he]
simp only [Finset.sum_add_distrib, ← Finset.mul_sum, Finset.sum_sub_distrib]
simp only [coordinateProduct]
rw [coordinate_difference_contraction, coordinate_fourth_contraction]
have hm := coordinate_mean_square_contraction w (orbitHamiltonian eig c U)
(spinCoordinate U) I J
change (∑ i ∈ I, ∑ j ∈ J,
gibbsAverage w (orbitHamiltonian eig c U) (coordinateProduct i j U) ^ 2) = _ at hm
rw [hm]
/-- The exact unperturbed spectral-block rotation identity. The separate
small-perturbation and limiting GG arguments are not assumed here. -/
theorem block_ward_identity {N : ℕ}
(μ : Measure (Orthogonal N)) [IsProbabilityMeasure μ] [μ.IsMulLeftInvariant]
{w : Spin N → ℝ} (hw : GibbsReference w) (eig c : Fin N → ℝ)
(K : ℝ) (hK : 0 ≤ K) (heig : ∀ k, |eig k| ≤ K)
(I J : Finset (Fin N)) (hIJ : Disjoint I J) (a b : ℝ)
(hI : ∀ i ∈ I, eig i = a) (hJ : ∀ j ∈ J, eig j = b) :
(∫ U,
(I.card : ℝ) * gibbsAverage w (orbitHamiltonian eig c U)
(fun σ => blockOverlap (spinCoordinate U) J σ σ) -
(J.card : ℝ) * gibbsAverage w (orbitHamiltonian eig c U)
(fun σ => blockOverlap (spinCoordinate U) I σ σ) +
(a - b) *
(gibbsAverage w (orbitHamiltonian eig c U)
(fun σ => blockOverlap (spinCoordinate U) I σ σ * blockOverlap (spinCoordinate U) J σ σ) -
gibbsPairAverage w (orbitHamiltonian eig c U)
(fun σ τ => blockOverlap (spinCoordinate U) I σ τ * blockOverlap (spinCoordinate U) J σ τ)) ∂μ) = 0 := by
simp_rw [← coordinateWardTerm_block_sum w eig c I J a b hI hJ]
rw [integral_finsetSum I (fun i _ => integrable_finsetSum J
(fun j _ => integrable_coordinateWardTerm μ hw eig c i j))]
apply Finset.sum_eq_zero
intro i hi
rw [integral_finsetSum J (fun j _ => integrable_coordinateWardTerm μ hw eig c i j)]
apply Finset.sum_eq_zero
intro j hj
have hij : i ≠ j := by
intro hij
subst j
exact Finset.disjoint_left.mp hIJ hi hj
exact coordinate_ward_identity i j hij μ hw eig c K hK heig
end InvariantIsing
end
end OAI
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