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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 | import OAI.Probability.InvariantIsing.Magnetic.MagneticLevelModulus
import OAI.Probability.InvariantIsing.Magnetic.MagneticBlockPath
namespace OAI
/-! Height continuity after block averaging, uniformly in block size
and in every optimizing root bias. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Set
open scoped BigOperators NNReal
namespace InvariantIsing
lemma magneticBlockLevel_abs_sub_le {N : ℕ} (hN : 0 < N) (h : FieldStep)
{V : ℝ} (hV : h.height (Fin.last h.depth) ≤ V) (m : Fin N → ℝ)
(i j : Fin (h.depth+1)) :
|magneticBlockLevel h m i-magneticBlockLevel h m j| ≤
|h.height i-h.height j| * fieldGaussianMomentCap (Real.sqrt V) := by
unfold magneticBlockLevel
rw [← mul_sub, ← Finset.sum_sub_distrib, abs_mul,
abs_of_nonneg (inv_nonneg.mpr (Nat.cast_nonneg N))]
calc
_ ≤ (N : ℝ)⁻¹ * ∑ k, |magneticFieldLevel h (m k) i-magneticFieldLevel h (m k) j| :=
mul_le_mul_of_nonneg_left (Finset.abs_sum_le_sum_abs _ _) (by positivity)
_ ≤ (N : ℝ)⁻¹ * ∑ _k : Fin N,
|h.height i-h.height j| * fieldGaussianMomentCap (Real.sqrt V) :=
mul_le_mul_of_nonneg_left
(Finset.sum_le_sum fun k _ => magneticFieldLevel_abs_sub_le h hV (m k) i j) (by positivity)
_ = _ := by
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]
rw [← mul_assoc, inv_mul_cancel₀ (Nat.cast_ne_zero.mpr hN.ne'), one_mul]
lemma magneticBlockPath_on_cell {N : ℕ} (hN : 0 < N) (h : FieldStep) (m : Fin N → ℝ)
(i : Fin (h.depth+1)) {s : ℝ} (hs : s ∈ Ioo (h.cut i.castSucc) (h.cut i.succ)) :
magneticBlockPath hN h m s = magneticBlockLevel h m i :=
fieldLevelPath_on_cell h _ _ _ i hs
end InvariantIsing
end
end OAI
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