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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 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 | import OAI.Probability.FreeIsing.Spins.Moments
import OAI.Probability.FreeIsing.Trees.Koopman
import OAI.Probability.FreeIsing.Trees.RootCorrelation
namespace OAI
noncomputable section
open MeasureTheory
namespace Problem367.FactorObservable
variable {d : ℕ}
variable {Phi : (TreeVertex d → ℝ) → (TreeVertex d → IsingSpin)}
def field (Phi : (TreeVertex d → ℝ) → (TreeVertex d → IsingSpin))
(U : TreeVertex d → ℝ) (v : TreeVertex d) : ℝ :=
(Phi U v).toReal
lemma measurable_field (hPhi : Measurable Phi) (v : TreeVertex d) :
Measurable (fun U => field Phi U v) :=
measurable_spinToReal.comp ((measurable_pi_apply v).comp hPhi)
lemma equivariant_field
(heq : ∀ g : TreeAut d,
∀ᵐ U ∂labelMeasure d, Phi (relabel g U) = relabel g (Phi U)) :
∀ g : TreeAut d,
∀ᵐ U ∂labelMeasure d, field Phi (relabel g U) = relabel g (field Phi U) := by
intro g
filter_upwards [heq g] with U hU
funext v
exact congrArg IsingSpin.toReal (congrFun hU v)
lemma memLp_field (hPhi : Measurable Phi) (v : TreeVertex d) (p : ENNReal) :
MemLp (fun U => field Phi U v) p (labelMeasure d) := by
apply MemLp.of_bound (measurable_field hPhi v).aestronglyMeasurable 1
exact Filter.Eventually.of_forall (fun U => (norm_spinToReal (Phi U v)).le)
def spinLp (hPhi : Measurable Phi) (v : TreeVertex d) : Lp ℝ 2 (labelMeasure d) :=
(memLp_field hPhi v 2).toLp (fun U => field Phi U v)
lemma coe_spinLp (hPhi : Measurable Phi) (v : TreeVertex d) :
(spinLp hPhi v : (TreeVertex d → ℝ) → ℝ) =ᵐ[labelMeasure d]
(fun U => field Phi U v) :=
(memLp_field hPhi v 2).coeFn_toLp
lemma norm_spinLp (hPhi : Measurable Phi) (v : TreeVertex d) :
‖spinLp hPhi v‖ = 1 := by
rw [spinLp, Lp.norm_toLp]
have heq : eLpNorm (fun U => field Phi U v) 2 (labelMeasure d) =
eLpNorm (fun _ : TreeVertex d → ℝ => (1 : ℝ)) 2 (labelMeasure d) := by
apply eLpNorm_congr_norm_ae (measurable_field hPhi v).aestronglyMeasurable
aestronglyMeasurable_const
exact Filter.Eventually.of_forall (fun U => by
change ‖(Phi U v).toReal‖ = ‖(1 : ℝ)‖
rw [norm_spinToReal, norm_one])
rw [heq, eLpNorm_const' _ (by norm_num) (by norm_num)]
simp
lemma integral_spinLp {theta : ℝ} (hPhi : Measurable Phi)
(hlaw : HasFreeIsingLaw theta (Measure.map Phi (labelMeasure d)))
(v : TreeVertex d) : ∫ U, spinLp hPhi v U ∂labelMeasure d = 0 := by
exact (integral_congr_ae (coe_spinLp hPhi v)).trans
(hlaw.integral_factor_spin hPhi v)
lemma spinLp_mem_centered {theta : ℝ} (hPhi : Measurable Phi)
(hlaw : HasFreeIsingLaw theta (Measure.map Phi (labelMeasure d)))
(v : TreeVertex d) : spinLp hPhi v ∈ centeredL2 (labelMeasure d) := by
rw [mem_centeredL2_iff]
exact integral_spinLp hPhi hlaw v
def centeredSpinLp {theta : ℝ} (hPhi : Measurable Phi)
(hlaw : HasFreeIsingLaw theta (Measure.map Phi (labelMeasure d)))
(v : TreeVertex d) : centeredL2 (labelMeasure d) :=
⟨spinLp hPhi v, spinLp_mem_centered hPhi hlaw v⟩
@[simp] lemma centeredSpinLp_coe {theta : ℝ} (hPhi : Measurable Phi)
(hlaw : HasFreeIsingLaw theta (Measure.map Phi (labelMeasure d))) (v : TreeVertex d) :
(centeredSpinLp hPhi hlaw v : Lp ℝ 2 (labelMeasure d)) = spinLp hPhi v := rfl
@[simp] lemma norm_centeredSpinLp {theta : ℝ} (hPhi : Measurable Phi)
(hlaw : HasFreeIsingLaw theta (Measure.map Phi (labelMeasure d))) (v : TreeVertex d) :
‖centeredSpinLp hPhi hlaw v‖ = 1 := norm_spinLp hPhi v
lemma koopman_spinLp (hPhi : Measurable Phi)
(heq : ∀ g : TreeAut d,
∀ᵐ U ∂labelMeasure d, Phi (relabel g U) = relabel g (Phi U))
(g : TreeAut d) (v : TreeVertex d) :
koopmanL2 (relabel g) (measurePreserving_relabel g) (spinLp hPhi v) =
spinLp hPhi (g.toEquiv.symm v) := by
change Lp.compMeasurePreserving (relabel g) (measurePreserving_relabel g)
((memLp_field hPhi v 2).toLp _) = _
rw [Lp.toLp_compMeasurePreserving]
apply MemLp.toLp_congr
filter_upwards [heq g] with U hU
exact congrArg IsingSpin.toReal (congrFun hU v)
lemma inner_spinLp (hPhi : Measurable Phi) (u v : TreeVertex d) :
inner ℝ (spinLp hPhi u) (spinLp hPhi v) =
∫ U, field Phi U u * field Phi U v ∂labelMeasure d := by
rw [L2.inner_def]
apply integral_congr_ae
filter_upwards [coe_spinLp hPhi u, coe_spinLp hPhi v] with U hu hv
rw [hu, hv]
simp [mul_comm]
lemma inner_spinLp_root {theta : ℝ} (hPhi : Measurable Phi)
(hlaw : HasFreeIsingLaw theta (Measure.map Phi (labelMeasure d)))
(hlo : -1 ≤ theta) (hhi : theta ≤ 1) (v : TreeVertex d) :
inner ℝ (spinLp hPhi (ConnectedHull.root d)) (spinLp hPhi v) =
theta ^ v.1.length := by
rw [inner_spinLp]
simpa only [field, IsingSpin.toReal_eq_spinSign] using
hlaw.factor_root_correlation hPhi hlo hhi v
lemma inner_centeredSpinLp_root {theta : ℝ} (hPhi : Measurable Phi)
(hlaw : HasFreeIsingLaw theta (Measure.map Phi (labelMeasure d)))
(hlo : -1 ≤ theta) (hhi : theta ≤ 1) (v : TreeVertex d) :
inner ℝ (centeredSpinLp hPhi hlaw (ConnectedHull.root d))
(centeredSpinLp hPhi hlaw v) = theta ^ v.1.length :=
inner_spinLp_root hPhi hlaw hlo hhi v
lemma inner_koopman_spinLp_root {theta : ℝ} (hPhi : Measurable Phi)
(hlaw : HasFreeIsingLaw theta (Measure.map Phi (labelMeasure d)))
(heq : ∀ g : TreeAut d,
∀ᵐ U ∂labelMeasure d, Phi (relabel g U) = relabel g (Phi U))
(hlo : -1 ≤ theta) (hhi : theta ≤ 1) (g : TreeAut d) :
inner ℝ (spinLp hPhi (ConnectedHull.root d))
(koopmanL2 (relabel g) (measurePreserving_relabel g)
(spinLp hPhi (ConnectedHull.root d))) =
theta ^ (g.toEquiv.symm (ConnectedHull.root d)).1.length := by
rw [koopman_spinLp hPhi heq]
exact inner_spinLp_root hPhi hlaw hlo hhi _
lemma centeredAverage_centeredSpinLp {theta : ℝ} (hPhi : Measurable Phi)
(hlaw : HasFreeIsingLaw theta (Measure.map Phi (labelMeasure d)))
(heq : ∀ g : TreeAut d,
∀ᵐ U ∂labelMeasure d, Phi (relabel g U) = relabel g (Phi U))
(v : TreeVertex d) :
TreeKoopman.centeredAverage d (centeredSpinLp hPhi hlaw v) =
(d : ℝ)⁻¹ • ∑ c : Fin d, centeredSpinLp hPhi hlaw (prefixFlip c v) := by
apply Subtype.ext
simp only [TreeKoopman.centeredAverage_apply_coe, Submodule.coe_smul,
Submodule.coe_sum, centeredSpinLp_coe]
exact TreeKoopman.average_toLp_eq (field Phi) (equivariant_field heq)
(fun v => memLp_field hPhi v 2) v
end Problem367.FactorObservable
namespace Problem367
theorem IsFactorOfIID.exists_centered_observable {d : ℕ} {theta : ℝ}
(hfactor : IsFactorOfIID d theta) (hlo : -1 ≤ theta) (hhi : theta ≤ 1) :
∃ f : centeredL2 (labelMeasure d), ‖f‖ = 1 ∧
∀ g : TreeAut d,
inner ℝ (f : Lp ℝ 2 (labelMeasure d))
(koopmanL2 (relabel g) (measurePreserving_relabel g)
(f : Lp ℝ 2 (labelMeasure d))) =
theta ^ (g.toEquiv.symm (ConnectedHull.root d)).1.length := by
obtain ⟨Phi, hPhi, hlaw, heq⟩ := hfactor
refine ⟨FactorObservable.centeredSpinLp hPhi hlaw (ConnectedHull.root d),
FactorObservable.norm_centeredSpinLp hPhi hlaw _, ?_⟩
intro g
exact FactorObservable.inner_koopman_spinLp_root hPhi hlaw heq hlo hhi g
end Problem367
end
end OAI
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