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a89d282 | 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 | import OAI.Probability.InvariantIsing.Haar.MatrixRotation
namespace OAI
/-!
# Frobenius continuity of Haar pressure
The Lipschitz constant is independent of dimension and of a deterministic
magnetic field. This is the finite-volume estimate required to apply the
published SO(N) concentration input.
-/
noncomputable section
open MeasureTheory
open scoped BigOperators Matrix
namespace InvariantIsing
lemma specialRotation_spin_dist_sq_le {N : ℕ} (U V : SpecialOrthogonal N)
(σ : Spin N) :
(∑ i, (specialRotation U (spinVector σ) i -
specialRotation V (spinVector σ) i) ^ 2) ≤
(N : ℝ) * frobeniusDistance U V ^ 2 := by
have he (i : Fin N) : specialRotation U (spinVector σ) i -
specialRotation V (spinVector σ) i =
∑ j, ((U : Matrix (Fin N) (Fin N) ℝ) i j -
(V : Matrix (Fin N) (Fin N) ℝ) i j) * spinValue (σ j) := by
simp only [specialRotation_apply, spinVector_apply, sub_mul, Finset.sum_sub_distrib]
have hrow (i : Fin N) := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ
(fun j => (U : Matrix (Fin N) (Fin N) ℝ) i j -
(V : Matrix (Fin N) (Fin N) ℝ) i j) (fun j => spinValue (σ j))
have hsum := Finset.sum_le_sum (s := Finset.univ) (fun i _ => hrow i)
simp only [spinValue_sq, Finset.sum_const, Finset.card_univ, Fintype.card_fin,
nsmul_eq_mul, mul_one, ← Finset.sum_mul] at hsum
simp_rw [he]
rw [frobeniusDistance, Real.sq_sqrt (Finset.sum_nonneg fun i _ =>
Finset.sum_nonneg fun j _ => sq_nonneg _)]
nlinarith
lemma abs_rotatedEnergy_sub_rotation_le {N : ℕ} (eig : Fin N → ℝ)
(U V : SpecialOrthogonal N) (K : ℝ) (hK : 0 ≤ K)
(heig : ∀ i, |eig i| ≤ K) (σ : Spin N) :
|rotatedEnergy eig (specialRotation U) σ -
rotatedEnergy eig (specialRotation V) σ| ≤
K * N * frobeniusDistance U V := by
let x := specialRotation U (spinVector σ)
let y := specialRotation V (spinVector σ)
let d := frobeniusDistance U V
have hd : 0 ≤ d := Real.sqrt_nonneg _
have hN : (0 : ℝ) ≤ N := Nat.cast_nonneg _
have hx : ∑ i, x i ^ 2 = N := rotated_spin_sq_sum _ _
have hy : ∑ i, y i ^ 2 = N := rotated_spin_sq_sum _ _
have hm : (∑ i, (x i - y i) ^ 2) ≤ (N : ℝ) * d ^ 2 :=
specialRotation_spin_dist_sq_le U V σ
have hp : (∑ i, (x i + y i) ^ 2) ≤ 4 * (N : ℝ) := by
have hi := Finset.sum_le_sum (s := Finset.univ) (fun i _ =>
show (x i + y i) ^ 2 ≤ 2 * x i ^ 2 + 2 * y i ^ 2 by
nlinarith [sq_nonneg (x i - y i)])
simp only [Finset.sum_add_distrib, ← Finset.mul_sum, hx, hy] at hi
linarith
have hw : (∑ i, (eig i * (x i - y i)) ^ 2) ≤
K ^ 2 * ∑ i, (x i - y i) ^ 2 := by
rw [Finset.mul_sum]
apply Finset.sum_le_sum
intro i _
rw [mul_pow]
have hb : eig i ^ 2 ≤ K ^ 2 := by
simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg _) hK).mpr (heig i)
exact mul_le_mul_of_nonneg_right hb (sq_nonneg _)
have hcs := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ
(fun i => eig i * (x i - y i)) (fun i => x i + y i)
have hs : (∑ i, (eig i * (x i - y i)) * (x i + y i)) ^ 2 ≤
(2 * K * (N : ℝ) * d) ^ 2 := by
calc
_ ≤ (∑ i, (eig i * (x i - y i)) ^ 2) * ∑ i, (x i + y i) ^ 2 := hcs
_ ≤ (K ^ 2 * ∑ i, (x i - y i) ^ 2) * (4 * (N : ℝ)) :=
mul_le_mul hw hp (Finset.sum_nonneg fun i _ => sq_nonneg _) (by positivity)
_ ≤ (K ^ 2 * ((N : ℝ) * d ^ 2)) * (4 * (N : ℝ)) :=
mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hm (sq_nonneg _))
(by positivity)
_ = _ := by ring
have ha : |∑ i, (eig i * (x i - y i)) * (x i + y i)| ≤
2 * K * (N : ℝ) * d := by
apply (sq_le_sq₀ (abs_nonneg _) (by positivity)).mp
simpa only [sq_abs] using hs
have he : rotatedEnergy eig (specialRotation U) σ -
rotatedEnergy eig (specialRotation V) σ =
(1 / 2 : ℝ) * ∑ i, (eig i * (x i - y i)) * (x i + y i) := by
rw [rotatedEnergy, rotatedEnergy, ← mul_sub, ← Finset.sum_sub_distrib]
change (1 / 2 : ℝ) * (∑ i, (eig i * x i ^ 2 - eig i * y i ^ 2)) = _
congr 1
apply Finset.sum_congr rfl
intro i _
ring
rw [he, abs_mul, abs_of_pos (by norm_num : (0 : ℝ) < 1 / 2)]
exact (mul_le_mul_of_nonneg_left ha (by norm_num)).trans_eq (by dsimp [d]; ring)
lemma abs_rotatedPressure_sub_rotation_le {N : ℕ} (hN : 0 < N)
(eig : Fin N → ℝ) (U V : SpecialOrthogonal N) (c : Fin N → ℝ)
(K : ℝ) (hK : 0 ≤ K) (heig : ∀ i, |eig i| ≤ K) :
|rotatedPressure eig (specialRotation U) c -
rotatedPressure eig (specialRotation V) c| ≤ K * frobeniusDistance U V := by
have h := abs_logPartition_sub_le
(fun σ => rotatedEnergy eig (specialRotation U) σ + fieldEnergy c σ)
(fun σ => rotatedEnergy eig (specialRotation V) σ + fieldEnergy c σ)
(K * N * frobeniusDistance U V) (fun σ => by
simpa only [add_sub_add_right_eq_sub] using
abs_rotatedEnergy_sub_rotation_le eig U V K hK heig σ)
have hn : (N : ℝ) ≠ 0 := by exact_mod_cast Nat.ne_of_gt hN
calc
_ = (N : ℝ)⁻¹ * |logPartition (fun σ =>
rotatedEnergy eig (specialRotation U) σ + fieldEnergy c σ) -
logPartition (fun σ => rotatedEnergy eig (specialRotation V) σ + fieldEnergy c σ)| := by
rw [rotatedPressure, rotatedPressure, ← mul_sub, abs_mul,
abs_of_nonneg (show (0 : ℝ) ≤ (N : ℝ)⁻¹ by positivity)]
_ ≤ (N : ℝ)⁻¹ * (K * N * frobeniusDistance U V) :=
mul_le_mul_of_nonneg_left h (by positivity)
_ = K * frobeniusDistance U V := by field_simp
end InvariantIsing
end
end OAI
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