File size: 3,724 Bytes
b317718 | 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 61 62 63 64 65 66 67 68 69 70 71 72 73 74 | import OAI.Probability.FreeIsing.Energy.FreeLawLocalCovariance
import OAI.Probability.FreeIsing.Trees.LawDecay
namespace OAI
noncomputable section
open MeasureTheory
namespace Problem367
theorem HasFreeIsingLaw.local_koopman_far_decay {d : ℕ} {theta : ℝ}
{μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ]
(hμ : HasFreeIsingLaw theta μ) (h0 : 0 ≤ theta) (h1 : theta ≤ 1)
(s : Finset (TreeVertex d)) (f : Lp ℝ 2 μ) (F : (s → IsingSpin) → ℝ)
(hF : Measurable F) (hf : f =ᵐ[μ] fun x => F (fun i => x i))
(hzero : ∫ x, f x ∂μ = 0) :
∃ N : ℕ, ∀ g : TreeAut d, N < (g.toEquiv TreeEscape.root).val.length →
|inner ℝ f (koopmanL2 (relabel g) (hμ.measurePreserving_relabel g) f)| ≤
‖f‖^2 * theta^((g.toEquiv TreeEscape.root).val.length - N) := by
let R := s.sup (fun v => v.val.length)
have hR : ∀ i : s, i.val.val.length ≤ R := fun i =>
Finset.le_sup (f := fun v : TreeVertex d => v.val.length) i.property
let f0 : (TreeVertex d → IsingSpin) → ℝ := fun x => F (fun i => x i)
have hf0 : MemLp f0 2 μ := (memLp_congr_ae hf).mp (Lp.memLp f)
have hz : ∫ x, f0 x ∂μ = 0 := (integral_congr_ae hf).symm.trans hzero
have hn : (∫ x, (f0 x)^2 ∂μ) = ‖f‖^2 := by
rw [← real_inner_self_eq_norm_sq f, L2.inner_def]
apply integral_congr_ae
filter_upwards [hf] with x hx
simp [hx, f0, pow_two]
refine ⟨R + R, fun g hg => ?_⟩
have hdepth : (g.symm.toEquiv TreeEscape.root).val.length =
(g.toEquiv TreeEscape.root).val.length := TreeEscape.symm_root_depth g
have hgz : ∫ x, f0 (relabel g x) ∂μ = 0 :=
(hμ.integral_relabel g f0).trans hz
have hgn : (∫ x, (f0 (relabel g x))^2 ∂μ) = ‖f‖^2 :=
(hμ.integral_relabel g (fun x => (f0 x)^2)).trans hn
have hgf : MemLp (fun x => f0 (relabel g x)) 2 μ :=
hf0.comp_measurePreserving (hμ.measurePreserving_relabel g)
have hb := hμ.abs_correlation_separated_local_blocks h0 h1
(fun i : s => i.val) (fun i : s => i.val) R hR hR g.symm
(hdepth ▸ hg) F F hF hF hz hgz hf0 hgf
rw [TreeLawKoopman.inner_koopman_eq_integral_of_local μ
hμ.measurePreserving_relabel s f F hf g]
change |∫ x, f0 x * f0 (relabel g x) ∂μ| ≤ _
change |∫ x, f0 x * f0 (relabel g x) ∂μ| ≤
Real.sqrt (∫ x, (f0 x)^2 ∂μ) * Real.sqrt (∫ x, (f0 (relabel g x))^2 ∂μ) *
theta ^ ((g.symm.toEquiv TreeEscape.root).val.length - (R + R)) at hb
simpa only [hn, hgn, Real.sqrt_sq_eq_abs, abs_norm, ← pow_two, hdepth] using hb
theorem HasFreeIsingLaw.localCriticalDecay {d : ℕ} {theta : ℝ}
{μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ]
(hμ : HasFreeIsingLaw theta μ) (hd : 3 ≤ d) (h0 : 0 ≤ theta) (h1 : theta ≤ 1)
(hthreshold : theta ≤ (Real.sqrt ((d : ℝ) - 1))⁻¹) :
TreeLawKoopman.LocalCriticalDecay μ hμ.measurePreserving_relabel := by
apply TreeLawKoopman.localCriticalDecay_of_far_decay μ hμ.measurePreserving_relabel
hd h0 hthreshold
intro s f F hF hf hzero
exact hμ.local_koopman_far_decay h0 h1 s f F hF hf hzero
theorem HasFreeIsingLaw.centeredAverage_norm_le_critical {d : ℕ} {theta : ℝ}
{μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ]
(hμ : HasFreeIsingLaw theta μ) (hd : 3 ≤ d) (h0 : 0 ≤ theta) (h1 : theta ≤ 1)
(hthreshold : theta ≤ (Real.sqrt ((d : ℝ) - 1))⁻¹) :
‖TreeLawKoopman.centeredAverage μ hμ.measurePreserving_relabel‖ ≤
2 * Real.sqrt ((d : ℝ) - 1) / d :=
TreeLawKoopman.centeredAverage_norm_le μ hμ.measurePreserving_relabel hd
(hμ.localCriticalDecay hd h0 h1 hthreshold)
end Problem367
end
end OAI
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