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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 | import OAI.Probability.InvariantIsing.Magnetic.MagneticBlockPath
namespace OAI
/-! Uniform replacement of the entire original constrained overlap path,
with no bound on the number of covariance levels. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Filter
open scoped BigOperators NNReal Topology
namespace InvariantIsing
theorem magneticBlockPath_l1_tendsto {A : Type*} [Fintype A] [DecidableEq A]
(N : ℕ → ℕ) (hN : ∀ n, 0 < N n) (hNlim : Tendsto N atTop atTop)
(group : ∀ n, Fin (N n) → A) (k : ℕ → A → ℕ)
(hk : ∀ n a, k n a ≤ spinGroupSize (group n) a) (m : ℕ → A → ℝ)
{r H : ℝ} (hr : r < 1) (hm : ∀ n a, |m n a| ≤ r)
(hc : ∀ n a, (k n a : ℝ) = spinGroupSize (group n) a * ((1 + m n a) / 2))
(h : ℕ → FieldStep) (hH : ∀ n, (h n).height (Fin.last (h n).depth) ≤ H) :
Tendsto (fun n => ∫ s, |restrictedBlockOverlapPath (hN n)
(spinGroupSlice (group n) (k n)) (spinGroupSlice_nonempty (group n) (k n) (hk n)) (h n) s -
magneticBlockPath (hN n) (h n) (fun j => m n (group n j)) s| ∂pathMeasure)
atTop (𝓝 0) := by
apply tendsto_zero_of_pair_entropy _
(fun n => (N n : ℝ)⁻¹ * (∑ j, constrainedFieldValue (h n) (m n (group n j))) -
constrainedBlockValue (spinGroupSlice (group n) (k n)) (h n))
(fun n => integral_nonneg (fun _ => abs_nonneg _))
(magneticBlockGap_tendsto N hN hNlim group k hk m hr hm hc h hH)
intro t ht n
exact magneticBlockPath_l1_comparison (hN n) (group n) (k n) (hk n) (m n)
(fun a => (hm n a).trans_lt hr) (hc n) (h n) ht
end InvariantIsing
end
end OAI
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