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f61e547 | 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 | import OAI.Probability.FreeIsing.Estimates.CenteredLocalDensity
import OAI.Probability.FreeIsing.Estimates.LocalCorrelation
import OAI.Probability.FreeIsing.Spectral.RadialPowerGrowth
import OAI.Probability.FreeIsing.Spectral.SpectralTransfer
import OAI.Probability.FreeIsing.Trees.WordAction
namespace OAI
noncomputable section
open MeasureTheory
namespace Problem367.TreeKoopman
theorem local_average_power_bound {d : ℕ} (hd : 3 ≤ d)
(s : Finset (TreeVertex d)) (f : Lp ℝ 2 (labelMeasure d))
(F : (s → ℝ) → ℝ) (hF : Measurable F)
(hf : f =ᵐ[labelMeasure d] fun U => F (fun i => U i))
(hzero : (∫ U, f U ∂labelMeasure d) = 0) :
∃ C : ℝ, ∀ n : ℕ, ‖(average d ^ n) f‖ ≤ C * RadialWalk.rate d ^ n := by
obtain ⟨C, hC, hcov⟩ := exists_local_koopman_critical_bound hd s f F hF hf hzero
have hword (v : TreeVertex d) :
|inner ℝ f (RadialWalk.wordOperator generatorOperator v f)| ≤
C * RadialWalk.treeWeight d v := by
have hdepth : ((wordAut v.1).symm.toEquiv TreeEscape.root).1.length =
v.1.length := by
change ((wordAut v.1).toEquiv.symm TreeEscape.root).1.length = _
rw [TreeEscape.symm_root_depth]
rw [wordAut_root v.1 v.2]
simpa only [wordOperator_eq, wordRelabel, RadialWalk.treeWeight,
RadialWalk.weight, hdepth] using hcov (wordAut v.1).symm
have hsym : (RadialWalk.operatorAverage (generatorOperator (d := d)) :
Lp ℝ 2 (labelMeasure d) →ₗ[ℝ] Lp ℝ 2 (labelMeasure d)).IsSymmetric := by
rw [operatorAverage_eq]
exact (average_selfAdjoint d).isSymmetric
refine ⟨Real.sqrt C, fun n => ?_⟩
simpa only [operatorAverage_eq] using
RadialWalk.norm_power_le_of_radial_bound generatorOperator generatorOperator_sq
hsym hd f hC hword n
theorem centeredAverage_norm_le {d : ℕ} (hd : 3 ≤ d) :
‖centeredAverage d‖ ≤ 2 * Real.sqrt ((d : ℝ) - 1) / d := by
apply selfAdjoint_opNorm_le_of_dense_power_bound (centeredAverage d)
(centeredAverage_selfAdjoint d) (RadialWalk.rate_nonneg d)
(dense_centered_finite_coordinate (labelMeasure d))
intro f hf
obtain ⟨s, F, hF, hf⟩ := hf
obtain ⟨C, hC⟩ := local_average_power_bound hd s (f : Lp ℝ 2 (labelMeasure d))
F hF hf ((mem_centeredL2_iff _).mp f.property)
refine ⟨C, fun n => ?_⟩
change ‖((centeredAverage d ^ n) f : Lp ℝ 2 (labelMeasure d))‖ ≤ _
rw [centeredAverage_pow_apply_coe]
exact hC n
end Problem367.TreeKoopman
end
end OAI
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