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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 | import OAI.Probability.InvariantIsing.Arrays.Overlaps
import OAI.Probability.InvariantIsing.Pressure.PressureLipschitz
namespace OAI
/-! Dimension-independent bounds for projected spectral overlap kernels. -/
noncomputable section
open MeasureTheory
open scoped BigOperators
namespace InvariantIsing
lemma projected_spin_sq_sum_le {N : ℕ} (U : Rotation N) (I : Finset (Fin N)) (σ : Spin N) :
(∑ i ∈ I, U (spinVector σ) i ^ 2) ≤ N := by
rw [← rotated_spin_sq_sum U σ]
exact Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ I) (fun _ _ _ => sq_nonneg _)
lemma projectedOverlap_abs_le_one {N : ℕ} (U : Rotation N) (I : Finset (Fin N)) (σ τ : Spin N) :
|projectedOverlap U I σ τ| ≤ 1 := by
have hcs := Finset.sum_mul_sq_le_sq_mul_sq I
(fun i => U (spinVector σ) i) (fun i => U (spinVector τ) i)
have hs : (∑ i ∈ I, U (spinVector σ) i * U (spinVector τ) i) ^ 2 ≤ (N : ℝ) ^ 2 := by
apply hcs.trans
simpa only [pow_two] using
(mul_le_mul (projected_spin_sq_sum_le U I σ) (projected_spin_sq_sum_le U I τ)
(Finset.sum_nonneg fun _ _ => sq_nonneg _) (Nat.cast_nonneg _))
have ha : |∑ i ∈ I, U (spinVector σ) i * U (spinVector τ) i| ≤ N :=
(sq_le_sq₀ (abs_nonneg _) (Nat.cast_nonneg _)).mp (by simpa only [sq_abs] using hs)
unfold projectedOverlap
rw [abs_mul, abs_of_nonneg (show (0 : ℝ) ≤ (N : ℝ)⁻¹ by positivity)]
by_cases hN : N = 0
· simp [hN]
· have hn : (N : ℝ) ≠ 0 := by exact_mod_cast hN
exact (mul_le_mul_of_nonneg_left ha (by positivity)).trans_eq (inv_mul_cancel₀ hn)
lemma projectedOverlap_self_le_one {N : ℕ} (U : Rotation N) (I : Finset (Fin N)) (σ : Spin N) :
projectedOverlap U I σ σ ≤ 1 :=
(le_abs_self _).trans (projectedOverlap_abs_le_one U I σ σ)
lemma abs_projected_cross_difference_sum_le {N : ℕ}
(U V W : SpecialOrthogonal N) (I : Finset (Fin N)) (σ τ : Spin N) :
|∑ i ∈ I, (specialRotation U (spinVector σ) i - specialRotation V (spinVector σ) i) *
specialRotation W (spinVector τ) i| ≤ N * frobeniusDistance U V := by
let d := frobeniusDistance U V
let x := fun i => specialRotation U (spinVector σ) i - specialRotation V (spinVector σ) i
let y := fun i => specialRotation W (spinVector τ) i
have hd : 0 ≤ d := Real.sqrt_nonneg _
have hx : (∑ i ∈ I, x i ^ 2) ≤ N * d ^ 2 := by
exact (Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ I)
(fun _ _ _ => sq_nonneg _)).trans (specialRotation_spin_dist_sq_le U V σ)
have hy : (∑ i ∈ I, y i ^ 2) ≤ N := projected_spin_sq_sum_le _ I τ
have hsq : (∑ i ∈ I, x i * y i) ^ 2 ≤ ((N : ℝ) * d) ^ 2 := by
calc
_ ≤ (∑ i ∈ I, x i ^ 2) * ∑ i ∈ I, y i ^ 2 := Finset.sum_mul_sq_le_sq_mul_sq I x y
_ ≤ ((N : ℝ) * d ^ 2) * N := mul_le_mul hx hy
(Finset.sum_nonneg fun _ _ => sq_nonneg _) (by positivity)
_ = _ := by ring
exact (sq_le_sq₀ (abs_nonneg _) (by positivity)).mp (by simpa only [sq_abs] using hsq)
/-- Every projected overlap has a uniform Frobenius Lipschitz constant. This
controls the covariances of the Gaussian GG perturbations. -/
theorem abs_projectedOverlap_sub_rotation_le {N : ℕ} (hN : 0 < N)
(U V : SpecialOrthogonal N) (I : Finset (Fin N)) (σ τ : Spin N) :
|projectedOverlap (specialRotation U) I σ τ - projectedOverlap (specialRotation V) I σ τ| ≤
2 * frobeniusDistance U V := by
let A := ∑ i ∈ I, (specialRotation U (spinVector σ) i - specialRotation V (spinVector σ) i) *
specialRotation U (spinVector τ) i
let B := ∑ i ∈ I, (specialRotation U (spinVector τ) i - specialRotation V (spinVector τ) i) *
specialRotation V (spinVector σ) i
have hA : |A| ≤ N * frobeniusDistance U V := abs_projected_cross_difference_sum_le U V U I σ τ
have hB : |B| ≤ N * frobeniusDistance U V := abs_projected_cross_difference_sum_le U V V I τ σ
have he : projectedOverlap (specialRotation U) I σ τ - projectedOverlap (specialRotation V) I σ τ =
(N : ℝ)⁻¹ * (A + B) := by
unfold projectedOverlap
rw [← mul_sub]
congr 1
dsimp only [A, B]
rw [← Finset.sum_sub_distrib, ← Finset.sum_add_distrib]
apply Finset.sum_congr rfl
intro i _
ring
have hb : |A + B| ≤ 2 * N * frobeniusDistance U V := by
have h := (abs_add_le A B).trans (add_le_add hA hB)
linarith
rw [he, abs_mul, abs_of_nonneg (show (0 : ℝ) ≤ (N : ℝ)⁻¹ by positivity)]
have hn : (N : ℝ) ≠ 0 := by exact_mod_cast hN.ne'
calc
_ ≤ (N : ℝ)⁻¹ * (2 * N * frobeniusDistance U V) := mul_le_mul_of_nonneg_left hb (by positivity)
_ = _ := by field_simp
end InvariantIsing
end
end OAI
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