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import OAI.Probability.FreeIsing.Posterior.ProgressiveIsingPosterior
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped Topology NNReal
namespace Problem367
private theorem measurable_artanh_field : Measurable Real.artanh := by
unfold Real.artanh
fun_prop
theorem tendsto_integral_sq_sub_posterior_fields
{Ω : Type*} [m : MeasurableSpace Ω] (μ : Measure Ω) [IsFiniteMeasure μ]
(F : Filtration ℕ m) (f X : Ω → ℝ) {K : ℝ}
(hb : ∀ n, ∀ᵐ ω ∂μ,
Real.tanh (X ω - K) ≤ (μ[f | F n]) ω ∧
(μ[f | F n]) ω ≤ Real.tanh (X ω + K)) :
Tendsto (fun n => ∫ ω,
(Real.artanh ((μ[f | F n]) ω) -
Real.artanh ((μ[f | ⨆ n, F n]) ω)) ^ 2 ∂μ) atTop (𝓝 0) := by
have hs : (⨆ n, F n) ≤ m := iSup_le (fun n => F.le n)
have hm (n : ℕ) : AEStronglyMeasurable
(fun ω => Real.artanh ((μ[f | F n]) ω)) μ :=
(measurable_artanh_field.comp
(stronglyMeasurable_condExp.mono (F.le n)).measurable).aestronglyMeasurable
have hm' : AEStronglyMeasurable
(fun ω => Real.artanh ((μ[f | ⨆ n, F n]) ω)) μ :=
(measurable_artanh_field.comp
(stronglyMeasurable_condExp.mono hs).measurable).aestronglyMeasurable
have hlim := tendsto_posterior_fields_ae μ F f X K hb
have hbfield (n : ℕ) : ∀ᵐ ω ∂μ,
|Real.artanh ((μ[f | F n]) ω) - X ω| ≤ K := by
filter_upwards [hb n] with ω hω
exact (posterior_field_limit_of_tanh_bounds
(u := fun _ : ℕ => (μ[f | F n]) ω) tendsto_const_nhds
(fun _ => hω)).2.1
have hbound (n : ℕ) : ∀ᵐ ω ∂μ,
‖(Real.artanh ((μ[f | F n]) ω) -
Real.artanh ((μ[f | ⨆ n, F n]) ω)) ^ 2‖ ≤ (2 * K) ^ 2 := by
filter_upwards [hbfield n, hlim] with ω hn hω
have hd : |Real.artanh ((μ[f | F n]) ω) -
Real.artanh ((μ[f | ⨆ n, F n]) ω)| ≤ 2 * K := by
calc
_ = |(Real.artanh ((μ[f | F n]) ω) - X ω) -
(Real.artanh ((μ[f | ⨆ n, F n]) ω) - X ω)| := by congr 1; ring
_ ≤ |Real.artanh ((μ[f | F n]) ω) - X ω| +
|Real.artanh ((μ[f | ⨆ n, F n]) ω) - X ω| := abs_sub _ _
_ ≤ 2 * K := by linarith [hω.2.1]
rw [Real.norm_of_nonneg (sq_nonneg _)]
simpa only [sq_abs] using pow_le_pow_left₀ (abs_nonneg _) hd 2
have hzero : ∀ᵐ ω ∂μ, Tendsto
(fun n => (Real.artanh ((μ[f | F n]) ω) -
Real.artanh ((μ[f | ⨆ n, F n]) ω)) ^ 2) atTop (𝓝 (0 : ℝ)) := by
filter_upwards [hlim] with ω hω
simpa using (hω.2.2.sub_const (Real.artanh ((μ[f | ⨆ n, F n]) ω))).pow 2
simpa using tendsto_integral_of_dominated_convergence
(fun _ : Ω => (2 * K) ^ 2) (fun n => ((hm n).sub hm').pow 2)
(integrable_const _) hbound hzero
open PlantedSymmetry
theorem HasFreeIsingLaw.finiteGibbsField_tendsto_ae
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) (v : TreeVertex d) :
∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t),
Tendsto (fun n => Real.artanh (finiteGibbsSpinMean beta v n (observation t p)))
atTop (𝓝 (Real.artanh (infiniteGibbsSpinMean beta v (observation t p)))) := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have hb (n : ℕ) := hμ.finiteObservation_spin_tanh_bounds hbeta
(connectedExhaustion v n) (connectedExhaustion_connected v n) t
⟨v, mem_connectedExhaustion v n⟩
have hlim := tendsto_posterior_fields_ae
(μ.prod (gaussianNoiseLaw (TreeVertex d) t))
(finiteObservationFiltration (connectedExhaustion v)
(connectedExhaustion_monotone v) t)
(fun p => spinSign (p.1 v)) (fun p => observation t p v) ((d : ℝ) * beta) hb
rw [finiteObservationFiltration_iSup (connectedExhaustion v)
(connectedExhaustion_monotone v)
(fun w => (eventually_mem_connectedExhaustion v w).exists) t] at hlim
have hce (n : ℕ) := hμ.finiteObservation_condExp (connectedExhaustion v n)
(connectedExhaustion_connected v n) t
(fun eta => spinSign (eta ⟨v, mem_connectedExhaustion v n⟩))
filter_upwards [hlim, ae_all_iff.mpr hce,
hμ.infiniteGibbsSpinMean_eq_posterior t v] with p hp hce hinf
convert hp.2.2 using 1
· ext n
exact congrArg Real.artanh (hce n).symm
· exact congrArg (fun x : ℝ => 𝓝 (Real.artanh x)) hinf
theorem HasFreeIsingLaw.finiteGibbsField_tendsto_meanSquare
{d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) (v : TreeVertex d) :
Tendsto (fun n => ∫ p,
(Real.artanh (finiteGibbsSpinMean beta v n (observation t p)) -
Real.artanh (infiniteGibbsSpinMean beta v (observation t p))) ^ 2
∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) atTop (𝓝 0) := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have hb (n : ℕ) := hμ.finiteObservation_spin_tanh_bounds hbeta
(connectedExhaustion v n) (connectedExhaustion_connected v n) t
⟨v, mem_connectedExhaustion v n⟩
have hlim := tendsto_integral_sq_sub_posterior_fields
(μ.prod (gaussianNoiseLaw (TreeVertex d) t))
(finiteObservationFiltration (connectedExhaustion v)
(connectedExhaustion_monotone v) t)
(fun p => spinSign (p.1 v)) (fun p => observation t p v)
hb
rw [finiteObservationFiltration_iSup (connectedExhaustion v)
(connectedExhaustion_monotone v)
(fun w => (eventually_mem_connectedExhaustion v w).exists) t] at hlim
convert hlim using 1
ext n
apply integral_congr_ae
filter_upwards [hμ.finiteObservation_condExp (connectedExhaustion v n)
(connectedExhaustion_connected v n) t
(fun eta => spinSign (eta ⟨v, mem_connectedExhaustion v n⟩)),
hμ.infiniteGibbsSpinMean_eq_posterior t v] with p hfinite hinf
change _ = (Real.artanh (_) - Real.artanh
(posteriorMagnetization (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) t v p)) ^ 2
rw [hinf]
congr 2
exact congrArg Real.artanh hfinite.symm
theorem HasFreeIsingLaw.finiteGibbsField_brownian_tendsto_ae
{d : ℕ} {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(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) (v : TreeVertex d) :
∀ᵐ p ∂μ.prod (Measure.infinitePi ν),
let y := fun i => (t : ℝ) * spinSign (p.1 i) + B i t (p.2 i)
Tendsto (fun n => Real.artanh (finiteGibbsSpinMean beta v n y))
atTop (𝓝 (Real.artanh (infiniteGibbsSpinMean beta v y))) := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have (i : TreeVertex d) : IsProbabilityMeasure (ν i) :=
(hB i).isGaussianProcess.isProbabilityMeasure
let T : ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) →
(TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) :=
Prod.map id (fun omega i => B i t (omega i))
have hT : MeasurePreserving T (μ.prod (Measure.infinitePi ν))
(μ.prod (gaussianNoiseLaw (TreeVertex d) t)) :=
(MeasurePreserving.id μ).prod (brownian_array_eval_measurePreserving ν hB hmB t)
have hlim := hμ.finiteGibbsField_tendsto_ae hbeta t v
rw [← hT.map_eq] at hlim
have hlim' := ae_of_ae_map hT.measurable.aemeasurable hlim
exact hlim'
theorem HasFreeIsingLaw.finiteGibbsField_brownian_tendsto_meanSquare
{d : ℕ} {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(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) (v : TreeVertex d) :
Tendsto (fun n => ∫ p,
let y := fun i => (t : ℝ) * spinSign (p.1 i) + B i t (p.2 i)
(Real.artanh (finiteGibbsSpinMean beta v n y) -
Real.artanh (infiniteGibbsSpinMean beta v y)) ^ 2
∂μ.prod (Measure.infinitePi ν)) atTop (𝓝 0) := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have (i : TreeVertex d) : IsProbabilityMeasure (ν i) :=
(hB i).isGaussianProcess.isProbabilityMeasure
let T : ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) →
(TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) :=
Prod.map id (fun omega i => B i t (omega i))
have hT : MeasurePreserving T (μ.prod (Measure.infinitePi ν))
(μ.prod (gaussianNoiseLaw (TreeVertex d) t)) :=
(MeasurePreserving.id μ).prod (brownian_array_eval_measurePreserving ν hB hmB t)
have hlim := hμ.finiteGibbsField_tendsto_meanSquare hbeta t v
convert hlim using 1
ext n
let f : ((TreeVertex d → IsingSpin) × (TreeVertex d → ℝ)) → ℝ := fun p =>
(Real.artanh (finiteGibbsSpinMean beta v n (observation t p)) -
Real.artanh (infiniteGibbsSpinMean beta v (observation t p))) ^ 2
have hf : Measurable f :=
((measurable_artanh_field.comp ((continuous_finiteGibbsSpinMean beta v n).measurable.comp
(measurable_observation t))).sub
(measurable_artanh_field.comp ((measurable_infiniteGibbsSpinMean beta v).comp
(measurable_observation t)))).pow_const 2
change (∫ p, f (T p) ∂μ.prod (Measure.infinitePi ν)) =
∫ p, f p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)
rw [← hT.map_eq, integral_map hT.measurable.aemeasurable hf.aestronglyMeasurable]
theorem HasFreeIsingLaw.finiteGibbsField_brownian_discrepancy_bound
{d : ℕ} {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(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) (v : TreeVertex d) (n : ℕ) :
∀ᵐ p ∂μ.prod (Measure.infinitePi ν),
let y := fun i => (t : ℝ) * spinSign (p.1 i) + B i t (p.2 i)
|Real.artanh (finiteGibbsSpinMean beta v n y) -
Real.artanh (infiniteGibbsSpinMean beta v y)| ≤ 2 * ((d : ℝ) * beta) := by
filter_upwards [hμ.infiniteGibbsSpinMean_brownian_field_bound hbeta ν hB hmB t v]
with p hp
let y := fun i => (t : ℝ) * spinSign (p.1 i) + B i t (p.2 i)
have hfin := finite_tree_gibbs_field_uniform_bound (connectedExhaustion v n) hbeta
((connectedExhaustion v n).restrict y) ⟨v, mem_connectedExhaustion v n⟩
change |Real.artanh (finiteGibbsSpinMean beta v n y) - y v| ≤ (d : ℝ) * beta at hfin
change |Real.artanh (finiteGibbsSpinMean beta v n y) -
Real.artanh (infiniteGibbsSpinMean beta v y)| ≤ _
have hinf : |Real.artanh (infiniteGibbsSpinMean beta v y) - y v| ≤ (d : ℝ) * beta := hp.2
calc
_ = |(Real.artanh (finiteGibbsSpinMean beta v n y) - y v) -
(Real.artanh (infiniteGibbsSpinMean beta v y) - y v)| := by congr 1; ring
_ ≤ |Real.artanh (finiteGibbsSpinMean beta v n y) - y v| +
|Real.artanh (infiniteGibbsSpinMean beta v y) - y v| := abs_sub _ _
_ ≤ _ := by linarith
theorem HasFreeIsingLaw.finiteGibbsField_brownian_tendsto_timeMeanSquare
{d : ℕ} {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(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))
(hcB : ∀ i omega, Continuous (fun t => B i t omega))
(τ : Measure ℝ≥0) [IsFiniteMeasure τ] (v : TreeVertex d) :
Tendsto (fun n => ∫ q,
let y := fun i => (q.1 : ℝ) * spinSign (q.2.1 i) + B i q.1 (q.2.2 i)
(Real.artanh (finiteGibbsSpinMean beta v n y) -
Real.artanh (infiniteGibbsSpinMean beta v y)) ^ 2
∂τ.prod (μ.prod (Measure.infinitePi ν))) atTop (𝓝 0) := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have (i : TreeVertex d) : IsProbabilityMeasure (ν i) :=
(hB i).isGaussianProcess.isProbabilityMeasure
let Y : ℝ≥0 × ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) →
TreeVertex d → ℝ := fun q i =>
(q.1 : ℝ) * spinSign (q.2.1 i) + B i q.1 (q.2.2 i)
have hY : Measurable Y := by
apply Measurable.of_eval
intro i
have hnoise : Measurable (Function.uncurry (B i)) :=
measurable_uncurry_of_continuous_of_measurable (hcB i) (hmB i)
have hspin : Measurable (fun q : ℝ≥0 ×
((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) => spinSign (q.2.1 i)) :=
(measurable_of_finite spinSign).comp
((measurable_pi_apply i).comp (measurable_fst.comp measurable_snd))
exact ((measurable_subtype_coe.comp measurable_fst).mul hspin).add
(hnoise.comp (measurable_fst.prodMk
((measurable_pi_apply i).comp (measurable_snd.comp measurable_snd))))
let Z n q := Real.artanh (finiteGibbsSpinMean beta v n (Y q)) -
Real.artanh (infiniteGibbsSpinMean beta v (Y q))
have hZ (n : ℕ) : Measurable (Z n) :=
(measurable_artanh_field.comp ((continuous_finiteGibbsSpinMean beta v n).measurable.comp hY)).sub
(measurable_artanh_field.comp ((measurable_infiniteGibbsSpinMean beta v).comp hY))
have hbound (n : ℕ) : ∀ᵐ t ∂τ, ∀ᵐ p ∂μ.prod (Measure.infinitePi ν),
‖Z n (t, p)‖ ≤ 2 * ((d : ℝ) * beta) := by
filter_upwards with t
simpa only [Real.norm_eq_abs] using
hμ.finiteGibbsField_brownian_discrepancy_bound hbeta ν hB hmB t v n
have hzero : ∀ᵐ t ∂τ, ∀ᵐ p ∂μ.prod (Measure.infinitePi ν),
Tendsto (fun n => Z n (t, p)) atTop (𝓝 (0 : ℝ)) := by
filter_upwards with t
filter_upwards [hμ.finiteGibbsField_brownian_tendsto_ae hbeta ν hB hmB t v]
with p hp
simpa only [Z, Y, sub_self] using hp.sub_const
(Real.artanh (infiniteGibbsSpinMean beta v (Y (t, p))))
have hlim := tendsto_integral_prod_sq_sub_of_bounded_ae τ
(μ.prod (Measure.infinitePi ν)) hZ (measurable_const (a := (0 : ℝ)))
(show 0 ≤ 2 * ((d : ℝ) * beta) by positivity) hbound
(by filter_upwards with t; filter_upwards with p; simp; positivity) hzero
simpa only [sub_zero] using hlim
end Problem367
end
end OAI
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