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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 | import OAI.Probability.FreeIsing.Critical.CanonicalRowEnergy
import OAI.Probability.FreeIsing.Posterior.ArrayPosteriorCovarianceStationarity
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal
namespace Problem367
open PlantedSymmetry CriticalCanonicalRowEnergy
theorem HasFreeIsingLaw.gaussian_posteriorResponse_relabel_ae
{d : ℕ} {theta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw theta μ) (t : ℝ≥0) (g : TreeAut d)
(u v : TreeVertex d) :
gaussianPosteriorResponse μ t u v ∘ Prod.map (relabel g) (relabel g) =ᵐ[
μ.prod (gaussianNoiseLaw (TreeVertex d) t)]
gaussianPosteriorResponse μ t (g.toEquiv.symm u) (g.toEquiv.symm v) := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
let F : IsingSpin → ℝ → ℝ := fun s z => (t : ℝ) * spinSign s + z
have hfield : (observationSigma (TreeVertex d) (t : ℝ)).comap
(Prod.map (relabel g) (relabel g)) = observationSigma (TreeVertex d) (t : ℝ) := by
change (selectedObservationField F id).comap
(Prod.map (relabel g) (relabel g)) = selectedObservationField F id
rw [selectedObservationField_comap_relabel]
apply selectedObservationField_eq_of_range_eq
simp only [id_eq, Equiv.range_eq_univ, Set.range_id]
have hs : Measurable spinSign := by intro s hs; trivial
have hm (w : TreeVertex d) : Measurable
(fun p : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) => spinSign (p.1 w)) :=
hs.comp ((measurable_pi_apply w).comp measurable_fst)
have h := posteriorResponse_comp_measurePreserving
(PlantedStationarity.measurePreserving_planted_relabel hμ (gaussianReal 0 t) g)
(hm u) (hm v) (integrable_plantedSpin _ u) (integrable_plantedSpin _ v)
(ArrayPosteriorStationarity.integrable_spin_pair _ u v)
(observationSigma_le (TreeVertex d) (t : ℝ))
rw [hfield] at h
exact h.symm
theorem HasFreeIsingLaw.gaussian_response_critical_intrinsic_bound
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d)
(hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1)
{t : ℝ≥0} (ht : 0 < t) (u v : TreeVertex d) :
(∫ p, gaussianPosteriorResponse μ t u v p ^ 2
∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ≤
(Real.tanh beta ^ 2 * Real.exp (-rate beta t)) ^ reducedWordDistance u.1 v.1 := by
obtain ⟨g, hgu, hdist⟩ := exists_treeAut_to_root_distance u
change g.toEquiv u = ConnectedHull.root d at hgu
have hsu : g.toEquiv.symm (ConnectedHull.root d) = u := by
rw [← hgu, Equiv.symm_apply_apply]
have hn := hμ.gaussian_posteriorResponse_relabel_ae t g (ConnectedHull.root d) (g.toEquiv v)
simp only [hsu, Equiv.symm_apply_apply] at hn
have hmp : MeasurePreserving (Prod.map (relabel g) (relabel g))
(μ.prod (gaussianNoiseLaw (TreeVertex d) t))
(μ.prod (gaussianNoiseLaw (TreeVertex d) t)) :=
PlantedStationarity.measurePreserving_planted_relabel hμ (gaussianReal 0 t) g
have hi := (hμ.gaussian_posteriorResponse_sq_decay hb ht
(ConnectedHull.root d) (g.toEquiv v) rfl).1
calc
_ = ∫ p, gaussianPosteriorResponse μ t (ConnectedHull.root d) (g.toEquiv v)
(Prod.map (relabel g) (relabel g) p) ^ 2
∂μ.prod (gaussianNoiseLaw (TreeVertex d) t) :=
integral_congr_ae (hn.symm.fun_comp (fun x : ℝ => x ^ 2))
_ = ∫ p, gaussianPosteriorResponse μ t (ConnectedHull.root d) (g.toEquiv v) p ^ 2
∂μ.prod (gaussianNoiseLaw (TreeVertex d) t) :=
hmp.hasLaw.integral_comp hi.aestronglyMeasurable
_ ≤ _ := by
simpa only [hdist] using
hμ.gaussian_rootResponse_critical_radial_bound hb hd hc ht (g.toEquiv v)
theorem HasFreeIsingLaw.gaussian_response_critical_row_energy
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d)
(hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1)
{t : ℝ≥0} (ht : 0 < t) (u : TreeVertex d) :
(∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t),
Summable (fun v => gaussianPosteriorResponse μ t u v p ^ 2)) ∧
Integrable (fun p => ∑' v, gaussianPosteriorResponse μ t u v p ^ 2)
(μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ∧
(∫ p, ∑' v, gaussianPosteriorResponse μ t u v p ^ 2
∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ≤
1 + (d : ℝ) * Real.tanh beta ^ 2 / rate beta t := by
obtain ⟨g, hgu⟩ := exists_treeAut_to_root u
change g.toEquiv u = ConnectedHull.root d at hgu
have hsu : g.toEquiv.symm (ConnectedHull.root d) = u := by
rw [← hgu, Equiv.symm_apply_apply]
have hn := ae_all_iff.mpr (fun v =>
hμ.gaussian_posteriorResponse_relabel_ae t g (ConnectedHull.root d) v)
simp only [hsu] at hn
have hmp : MeasurePreserving (Prod.map (relabel g) (relabel g))
(μ.prod (gaussianNoiseLaw (TreeVertex d) t))
(μ.prod (gaussianNoiseLaw (TreeVertex d) t)) :=
PlantedStationarity.measurePreserving_planted_relabel hμ (gaussianReal 0 t) g
have hg := hμ.gaussian_rootResponse_critical_row_energy hb hd hc ht
have he : (fun p => ∑' v,
gaussianPosteriorResponse μ t (ConnectedHull.root d) v p ^ 2) ∘
Prod.map (relabel g) (relabel g) =ᵐ[
μ.prod (gaussianNoiseLaw (TreeVertex d) t)]
fun p => ∑' v, gaussianPosteriorResponse μ t u v p ^ 2 := by
filter_upwards [hn] with p hp
change (∑' v, gaussianPosteriorResponse μ t (ConnectedHull.root d) v
(Prod.map (relabel g) (relabel g) p) ^ 2) = _
exact (tsum_congr (fun v => congrArg (fun x : ℝ => x ^ 2) (hp v))).trans
(g.toEquiv.symm.tsum_eq (fun v => gaussianPosteriorResponse μ t u v p ^ 2))
refine ⟨?_, (hmp.integrable_comp_of_integrable hg.2.1).congr he, ?_⟩
· filter_upwards [hmp.quasiMeasurePreserving.ae hg.1, hn] with p hp hnat
exact g.toEquiv.symm.summable_iff.mp
(hp.congr (fun v => congrArg (fun x : ℝ => x ^ 2) (hnat v)))
· rw [← integral_congr_ae he, hmp.hasLaw.integral_comp hg.2.1.aestronglyMeasurable]
exact hμ.integral_gaussian_rootResponse_row_le_inv_rate hb hd hc ht
theorem HasFreeIsingLaw.infiniteGibbsResponse_eq_gaussian_response
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (t : ℝ≥0) (u v : TreeVertex d) :
(fun p => infiniteGibbsResponse beta u v (observation t p)) =ᵐ[
μ.prod (gaussianNoiseLaw (TreeVertex d) t)] gaussianPosteriorResponse μ t u v := by
filter_upwards [hμ.infiniteGibbsCovariance_eq_posterior t u v,
hμ.infiniteGibbsSpinMean_eq_posterior t u] with p hc hm
unfold infiniteGibbsResponse gaussianPosteriorResponse posteriorResponse
rw [hc, hm]
rfl
theorem HasFreeIsingLaw.infiniteGibbsResponse_critical_row_energy
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d)
(hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1)
{t : ℝ≥0} (ht : 0 < t) (u : TreeVertex d) :
(∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t),
Summable (fun v => infiniteGibbsResponse beta u v (observation t p) ^ 2)) ∧
Integrable (fun p => infiniteGibbsResponseRow beta u (observation t p))
(μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ∧
(∫ p, infiniteGibbsResponseRow beta u (observation t p)
∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ≤
1 + (d : ℝ) * Real.tanh beta ^ 2 / rate beta t := by
have hg := hμ.gaussian_response_critical_row_energy hb hd hc ht u
have hn := ae_all_iff.mpr (fun v => hμ.infiniteGibbsResponse_eq_gaussian_response t u v)
have he : (fun p => infiniteGibbsResponseRow beta u (observation t p)) =ᵐ[
μ.prod (gaussianNoiseLaw (TreeVertex d) t)]
fun p => ∑' v, gaussianPosteriorResponse μ t u v p ^ 2 := by
filter_upwards [hn] with p hp
exact tsum_congr fun v => congrArg (fun x : ℝ => x ^ 2) (hp v)
refine ⟨?_, hg.2.1.congr he.symm, ?_⟩
· filter_upwards [hg.1, hn] with p hp hnat
exact hp.congr fun v => congrArg (fun x : ℝ => x ^ 2) (hnat v).symm
· rw [integral_congr_ae he]
exact hg.2.2
theorem HasFreeIsingLaw.brownian_response_critical_row_energy
{d : ℕ} {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d)
(hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1)
(ν : TreeVertex d → Measure Ω) {B : TreeVertex d → ℝ≥0 → Ω → ℝ}
(hB : ∀ i, IsPreBrownianReal (B i) (ν i))
(hmB : ∀ i t, Measurable (B i t)) {t : ℝ≥0} (ht : 0 < t) (u : TreeVertex d) :
(∀ᵐ p ∂μ.prod (Measure.infinitePi ν),
Summable (fun v => brownianPosteriorResponse μ ν B t u v p ^ 2)) ∧
Integrable (fun p => ∑' v, brownianPosteriorResponse μ ν B t u v p ^ 2)
(μ.prod (Measure.infinitePi ν)) ∧
(∫ p, ∑' v, brownianPosteriorResponse μ ν B t u v p ^ 2
∂μ.prod (Measure.infinitePi ν)) ≤
1 + (d : ℝ) * Real.tanh beta ^ 2 / rate beta t := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have (i : TreeVertex d) : IsProbabilityMeasure (ν i) :=
(hB i).isGaussianProcess.isProbabilityMeasure
have hT := (MeasurePreserving.id μ).prod
(brownian_array_eval_measurePreserving ν hB hmB t)
have hg := hμ.gaussian_response_critical_row_energy hb hd hc ht u
have heq := ae_all_iff.mpr (fun v =>
brownian_past_posteriorResponse_eq_gaussian μ ν hB hmB t u v)
have he : (fun p => ∑' v, brownianPosteriorResponse μ ν B t u v p ^ 2) =ᵐ[
μ.prod (Measure.infinitePi ν)]
(fun q => ∑' v, gaussianPosteriorResponse μ t u v q ^ 2) ∘
Prod.map id (fun p i => B i t (p i)) := by
filter_upwards [heq] with p hp
exact tsum_congr fun v => congrArg (fun x : ℝ => x ^ 2) (hp v)
refine ⟨?_, (hT.integrable_comp_of_integrable hg.2.1).congr he.symm, ?_⟩
· filter_upwards [hT.quasiMeasurePreserving.ae hg.1, heq] with p hp hpeq
exact hp.congr fun v => congrArg (fun x : ℝ => x ^ 2) (hpeq v).symm
· rw [integral_congr_ae he, hT.hasLaw.integral_comp hg.2.1.aestronglyMeasurable]
exact hg.2.2
end Problem367
end
end OAI
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