File size: 4,370 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 75 76 77 78 79 80 81 82 83 84 85 86 87 88 | import OAI.Probability.IsingPerceptron.NormalizedRestrictionPi
namespace OAI
/-! Leaf exhaustion for bounded replica observables of an arbitrary
integrable Gibbs Hamiltonian. No Gaussian insertion remains in the Ward
terms to which this lemma will be applied. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Filter Set
open scoped BigOperators Topology
namespace InvariantIsing
lemma normalizedRestriction_prod {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]
(μ : Measure X) (ν : Measure Y) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
(S : Set X) (T : Set Y) :
(normalizedRestriction μ S).prod (normalizedRestriction ν T) =
normalizedRestriction (μ.prod ν) (S ×ˢ T) := by
unfold normalizedRestriction
rw [Measure.prod_smul_left, Measure.prod_smul_right, smul_smul, Measure.prod_restrict,
Measure.prod_prod, ENNReal.mul_inv]
· exact Or.inr (measure_ne_top ν T)
· exact Or.inl (measure_ne_top μ S)
lemma referenceReplicaMean_restriction_tendsto {X : Type*}
[MeasurableSpace X] [Countable X] [MeasurableSingletonClass X]
(ν : Measure X) [IsProbabilityMeasure ν] (H : X → ℝ)
(hH : Integrable (fun x => Real.exp (H x)) ν) {r : ℕ}
(D : (Fin r → X) → ℝ) {B : ℝ} (hD : ∀ σ, |D σ| ≤ B)
{S : ℕ → Set X} (hS : ∀ n, MeasurableSet (S n))
(hexh : ∀ x, ∀ᶠ n in atTop, x ∈ S n) (hpos : ∀ n, ν (S n) ≠ 0) :
Tendsto (fun n => referenceReplicaMean (normalizedRestriction ν (S n)) H D) atTop
(𝓝 (referenceReplicaMean ν H D)) := by
have : ∀ n, IsProbabilityMeasure (normalizedRestriction ν (S n)) :=
fun n => normalizedRestriction_probability (hpos n)
have he : Integrable (fun σ : Fin r → X => Real.exp (∑ i, H (σ i)))
(Measure.pi (fun _ : Fin r => ν)) := by
simpa only [Real.exp_sum] using
(Integrable.fintype_prod (fun _ : Fin r => hH))
have hi : Integrable (fun σ : Fin r → X => Real.exp (∑ i, H (σ i)) * D σ)
(Measure.pi (fun _ : Fin r => ν)) :=
he.mul_bdd (measurable_of_countable D).aestronglyMeasurable
(ae_of_all _ fun σ => by simpa only [Real.norm_eq_abs] using hD σ)
have hz : (∫ σ : Fin r → X, Real.exp (∑ i, H (σ i)) ∂Measure.pi (fun _ => ν)) ≠ 0 := by
rw [reference_replica_partition]
exact pow_ne_zero _ (integral_exp_pos hH).ne'
have ht := normalized_restriction_gibbs_tendsto
(fun n => MeasurableSet.univ_pi (fun _ : Fin r => hS n))
(replica_restriction_eventually hexh) he hi hz
simpa only [referenceReplicaMean_eq_ratio,
← normalizedRestriction_pi (fun _ : Fin r => S _) (fun _ => hpos _)] using ht
theorem expected_referenceReplicaMean_restriction_tendsto {Ω X : Type*}
[MeasurableSpace Ω] [MeasurableSpace X] [Countable X] [MeasurableSingletonClass X]
(P : Measure Ω) [IsProbabilityMeasure P] (ν : Measure X) [IsProbabilityMeasure ν]
(H : Ω × X → ℝ) (hH : Measurable H)
(he : ∀ᵐ ω ∂P, Integrable (fun x => Real.exp (H (ω,x))) ν)
{r : ℕ} (D : Ω × (Fin r → X) → ℝ) (hD : Measurable D)
{B : ℝ} (hB : 0 ≤ B) (hbound : ∀ z, |D z| ≤ B)
{S : ℕ → Set X} (hS : ∀ n, MeasurableSet (S n))
(hexh : ∀ x, ∀ᶠ n in atTop, x ∈ S n) (hpos : ∀ n, ν (S n) ≠ 0) :
Tendsto (fun n => ∫ ω, referenceReplicaMean (normalizedRestriction ν (S n))
(fun x => H (ω,x)) (fun σ => D (ω,σ)) ∂P) atTop
(𝓝 (∫ ω, referenceReplicaMean ν (fun x => H (ω,x)) (fun σ => D (ω,σ)) ∂P)) := by
have : ∀ n, IsProbabilityMeasure (normalizedRestriction ν (S n)) :=
fun n => normalizedRestriction_probability (hpos n)
apply tendsto_integral_of_dominated_convergence (fun _ => B)
· intro n
exact (measurable_random_referenceReplicaMean
(ν := fun _ : Ω => normalizedRestriction ν (S n)) measurable_const hH hD).aestronglyMeasurable
· exact integrable_const _
· intro n
exact ae_of_all _ fun ω => by
rw [Real.norm_eq_abs]
exact referenceReplicaMean_abs_le _ _ _
(hD.comp (measurable_const.prodMk measurable_id)) hB (fun σ => hbound (ω,σ))
· filter_upwards [he] with ω hω
exact referenceReplicaMean_restriction_tendsto ν (fun x => H (ω,x)) hω
(fun σ => D (ω,σ)) (fun σ => hbound (ω,σ)) hS hexh hpos
end InvariantIsing
end
end OAI
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