| 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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|
