| import OAI.Probability.InvariantIsing.Magnetic.MagneticLevelModulus |
| import OAI.Probability.InvariantIsing.Magnetic.MagneticBlockPath |
|
|
| namespace OAI |
|
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| /-! 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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|
