| import OAI.Probability.InvariantIsing.Haar.MatrixRotation |
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| namespace OAI |
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| /-! |
| # Frobenius continuity of Haar pressure |
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| 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. |
| -/ |
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| noncomputable section |
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| open MeasureTheory |
| open scoped BigOperators Matrix |
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| namespace InvariantIsing |
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| 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 |
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| 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) |
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| 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 |
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| end InvariantIsing |
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| end |
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| end OAI |
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