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import OAI.Probability.InvariantIsing.Pressure.RandomCoefficientIBP
namespace OAI
/-! Actual frozen exponential integrability and selected-field log partition identities. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped BigOperators NNReal
namespace InvariantIsing
lemma tensorNamespacedCoefficients_variance_eq {N m k : ℕ} (U : Rotation N)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (v : Fin (n + 1) → SpinTensorIndex I degree → ℝ≥0) (x : Spin N × LabeledLeaf n) :
(tensorNamespacedCoefficients U I degree amplitude n v x).sum (fun _ c => c ^ 2) =
(tensorLeafCoefficients U I degree amplitude n v x).sum (fun _ c => c ^ 2) := by
have hi : Function.Injective (fun i : Fin (n + 1) × SpinTensorIndex I degree =>
tensorNamespaceTag I degree (i.1.1, (labeledAddress n x.2).take i.1, i.2)) := by
intro i j hij
apply treeFeatureTag_injective n x.2
exact congrArg Encodable.encode ((tensorNamespaceTag_injective I degree) hij)
rw [tensorNamespacedCoefficients, featureCoefficients_variance _ hi,
tensorLeafCoefficients, featureCoefficients_variance _ (treeFeatureTag_injective n x.2)]
lemma tensorNamespacedPath_variance_le {N m k : ℕ} (U : Rotation N)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (treeDegree : Fin k → ℕ) (h : ℕ → ℝ) (hh : Monotone h) (h0 : 0 ≤ h 0)
(x : Spin N × LabeledLeaf n) :
(tensorNamespacedCoefficients U I degree amplitude n
(fun i => tensorPathProfile I degree n treeDegree h i) x).sum (fun _ c => c ^ 2) ≤
h n * N + ∑ j, amplitude j ^ 2 := by
rw [tensorNamespacedCoefficients_variance_eq]
exact tensorPathProfile_variance_cap U I degree amplitude n treeDegree h hh h0 x
private lemma measurable_tensorBaseEnergy {N : ℕ} (eig c : Fin N → ℝ) (n : ℕ) :
Measurable (fun p : (SpecialOrthogonal N × LabeledTree n) × (Spin N × LabeledLeaf n) =>
rotatedEnergy eig (specialRotation p.1.1) p.2.1 + fieldEnergy c p.2.1) := by
apply measurable_from_prod_countable_left
intro x
have hrot : Measurable (fun U : SpecialOrthogonal N => rotatedEnergy eig (specialRotation U) x.1) := by
unfold rotatedEnergy
exact (Finset.measurable_sum _ fun i _ =>
((measurable_specialRotation_eval (spinVector x.1) i).pow_const 2).const_mul (eig i)).const_mul (1 / 2 : ℝ)
exact (hrot.comp measurable_fst).add_const (fieldEnergy c x.1)
private lemma tensorBase_exp_integrable {N : ℕ} (eig c : Fin N → ℝ) (n : ℕ)
(ω : SpecialOrthogonal N × LabeledTree n) :
Integrable (fun x : Spin N × LabeledLeaf n =>
Real.exp (rotatedEnergy eig (specialRotation ω.1) x.1 + fieldEnergy c x.1))
(labeledSpinReference n (uniformSpinPrior N : Measure (Spin N)) ω.2) :=
(Integrable.of_finite : Integrable
(fun σ : Spin N => Real.exp (rotatedEnergy eig (specialRotation ω.1) σ + fieldEnergy c σ))
(uniformSpinPrior N : Measure (Spin N))).comp_fst (labeledLeafLaw n ω.2)
private lemma measurable_namespacedPathField {N m k : ℕ}
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (treeDegree : Fin k → ℕ) (h : ℕ → ℝ) :
Measurable (fun p : ((SpecialOrthogonal N × LabeledTree n) × (ℕ → ℝ)) ×
(Spin N × LabeledLeaf n) =>
cylinderField (tensorNamespacedCoefficients (specialRotation p.1.1.1) I degree amplitude n
(fun i => tensorPathProfile I degree n treeDegree h i) p.2) p.1.2) := by
apply measurable_from_prod_countable_left
intro x
let v : Fin (n + 1) → SpinTensorIndex I degree → ℝ≥0 :=
fun i => tensorPathProfile I degree n treeDegree h i
have hm : Measurable (fun p : (SpecialOrthogonal N × LabeledTree n) × (ℕ → ℝ) =>
(p.1.1, p.2)) := measurable_fst.fst.prodMk measurable_snd
have hf := (measurable_tensorNamespacedFields_joint I degree amplitude n v x).comp hm
convert hf using 1
rfl
private lemma measurable_frozenSelectedField {N m k : ℕ}
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ)
(n : ℕ) (treeDegree : Fin k → ℕ) (j : Fin k) :
Measurable (fun p : (TensorFrozenData N n j × (ℕ → ℝ)) × (Spin N × LabeledLeaf n) =>
cylinderField (jointSpectralMonomialCoefficients
(specialRotation p.1.1.1.1) I (degree j) n (treeDegree j) p.2) p.1.2) := by
apply measurable_from_prod_countable_left
intro x
have hp : Measurable (fun p : TensorFrozenData N n j × (ℕ → ℝ) => (p.1.1.1, p.2)) :=
measurable_fst.fst.fst.prodMk measurable_snd
have hf := (measurable_jointSpectralMonomialField I (degree j) n (treeDegree j) x).comp hp
convert hf using 1
rfl
/-- Exponentials of the tensor Hamiltonian are integrable for
almost every joint rotation/tree/Gaussian disorder. No concentration or
limiting-pressure hypothesis is used. -/
theorem tensorNamespaced_exp_integrable_ae {N m k : ℕ}
(μ : Measure (SpecialOrthogonal N)) [IsProbabilityMeasure μ] (eig c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (treeDegree : Fin k → ℕ) (h : ℕ → ℝ) (hh : Monotone h) (h0 : 0 ≤ h 0) :
∀ᵐ p : (SpecialOrthogonal N × LabeledTree n) × (ℕ → ℝ)
∂(μ.prod (labeledCascadeLaw n b : Measure (LabeledTree n))).prod gaussianCoordinates,
Integrable (fun x : Spin N × LabeledLeaf n => Real.exp
(rotatedEnergy eig (specialRotation p.1.1) x.1 + fieldEnergy c x.1 +
cylinderField (tensorNamespacedCoefficients (specialRotation p.1.1) I degree amplitude n
(fun i => tensorPathProfile I degree n treeDegree h i) x) p.2))
(labeledSpinReference n (uniformSpinPrior N : Measure (Spin N)) p.1.2) := by
let Ω := SpecialOrthogonal N × LabeledTree n
let ν := fun ω : Ω => labeledSpinReference n (uniformSpinPrior N : Measure (Spin N)) ω.2
let H := fun (ω : Ω) (x : Spin N × LabeledLeaf n) =>
rotatedEnergy eig (specialRotation ω.1) x.1 + fieldEnergy c x.1
let A := fun (ω : Ω) => tensorNamespacedCoefficients (specialRotation ω.1) I degree amplitude n
(fun i => tensorPathProfile I degree n treeDegree h i)
have : ∀ ω, IsProbabilityMeasure (ν ω) := fun ω => by
change IsProbabilityMeasure (labeledSpinReference n (uniformSpinPrior N : Measure (Spin N)) ω.2)
infer_instance
have hν : Measurable ν :=
(measurable_labeledSpinReference_general n (uniformSpinPrior N : Measure (Spin N))).comp measurable_snd
have hbase (ω : Ω) : Integrable (fun x => Real.exp (H ω x)) (ν ω) := tensorBase_exp_integrable eig c n ω
have : ∀ ω, IsProbabilityMeasure ((ν ω).tilted (H ω)) := fun ω => isProbabilityMeasure_tilted (hbase ω)
have htilt : Measurable (fun ω => (ν ω).tilted (H ω)) :=
measurable_random_tilted_measure (H := Function.uncurry H) hν (measurable_tensorBaseEnergy eig c n)
have hAm : Measurable (fun p : (Ω × (ℕ → ℝ)) × (Spin N × LabeledLeaf n) =>
cylinderField (A p.1.1 p.2) p.1.2) := measurable_namespacedPathField I degree amplitude n treeDegree h
have hA (ω : Ω) (x : Spin N × LabeledLeaf n) :
(A ω x).sum (fun _ c => c ^ 2) ≤ h n * N + ∑ j, amplitude j ^ 2 :=
tensorNamespacedPath_variance_le (specialRotation ω.1) I degree amplitude n treeDegree h hh h0 x
have he := randomCoefficient_all_exp_ae
(P := μ.prod (labeledCascadeLaw n b : Measure (LabeledTree n))) htilt A hAm hA
filter_upwards [he] with p hp
have hi := hp 1
rw [integrable_tilted_iff (hbase p.1)] at hi
simpa only [H, A, ν, one_mul, smul_eq_mul, ← Real.exp_add] using hi
/-- The frozen base energy is the actual Hamiltonian with the selected
amplitude set to zero, and is exponentially integrable almost surely. -/
theorem tensorFrozenBase_exp_integrable_ae {N m k : ℕ}
(μ : Measure (SpecialOrthogonal N)) [IsProbabilityMeasure μ] (eig c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (treeDegree : Fin k → ℕ) (h : ℕ → ℝ) (hh : Monotone h) (h0 : 0 ≤ h 0)
(j : Fin k) :
∀ᵐ ω : TensorFrozenData N n j ∂tensorFrozenLaw μ n b j,
Integrable (fun x => Real.exp (tensorFrozenBaseEnergy eig c I degree amplitude n treeDegree h j ω x))
(labeledSpinReference n (uniformSpinPrior N : Measure (Spin N)) ω.1.2) := by
have hp := tensorFrozenInsertion_measurePreserving μ n b j
have he := hp.quasiMeasurePreserving.ae
(tensorNamespaced_exp_integrable_ae μ eig c I degree (Function.update amplitude j 0)
n b treeDegree h hh h0)
have he' : ∀ᵐ p : TensorFrozenData N n j × (ℕ → ℝ)
∂(tensorFrozenLaw μ n b j).prod gaussianCoordinates,
Integrable (fun x => Real.exp (tensorFrozenBaseEnergy eig c I degree amplitude n treeDegree h j p.1 x))
(labeledSpinReference n (uniformSpinPrior N : Measure (Spin N)) p.1.1.2) := by
filter_upwards [he] with p hp
convert hp using 1
funext x
congr 1
dsimp only [tensorFrozenBaseEnergy]
rw [tensorNamespacedField_frozen (specialRotation p.1.1.1) I degree amplitude n treeDegree h hh h0 j x p.2 p.1.2]
filter_upwards [Measure.ae_ae_of_ae_prod he'] with ω hω
obtain ⟨z, hz⟩ := hω.exists
exact hz
def tensorNamespacedLog {N m k : ℕ} (eig c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (treeDegree : Fin k → ℕ) (h : ℕ → ℝ)
(p : (SpecialOrthogonal N × LabeledTree n) × (ℕ → ℝ)) : ℝ :=
Real.log (∫ x : Spin N × LabeledLeaf n, Real.exp
(rotatedEnergy eig (specialRotation p.1.1) x.1 + fieldEnergy c x.1 +
cylinderField (tensorNamespacedCoefficients (specialRotation p.1.1) I degree amplitude n
(fun i => tensorPathProfile I degree n treeDegree h i) x) p.2)
∂labeledSpinReference n (uniformSpinPrior N : Measure (Spin N)) p.1.2)
def tensorFrozenPartitionLog {N m k : ℕ} (eig c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (treeDegree : Fin k → ℕ) (h : ℕ → ℝ) (j : Fin k) (t : ℝ)
(p : TensorFrozenData N n j × (ℕ → ℝ)) : ℝ :=
tensorNamespacedLog eig c I degree (Function.update amplitude j t) n treeDegree h
(p.1.1, gaussianNamespaceJoin (j + 1) (p.2, p.1.2))
def tensorFrozenCGF {N m k : ℕ} (eig c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (treeDegree : Fin k → ℕ) (h : ℕ → ℝ) (j : Fin k) (t : ℝ)
(p : TensorFrozenData N n j × (ℕ → ℝ)) : ℝ :=
cgf (fun x => cylinderField (jointSpectralMonomialCoefficients
(specialRotation p.1.1.1) I (degree j) n (treeDegree j) x) p.2)
(tensorFrozenReference eig c I degree amplitude n treeDegree h j p.1) t
lemma measurable_tensorFrozenCGF {N m k : ℕ} (eig c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (treeDegree : Fin k → ℕ) (h : ℕ → ℝ) (j : Fin k) (t : ℝ) :
Measurable (tensorFrozenCGF eig c I degree amplitude n treeDegree h j t) := by
let A := fun (ω : TensorFrozenData N n j) =>
jointSpectralMonomialCoefficients (specialRotation ω.1.1) I (degree j) n (treeDegree j)
let ν := tensorFrozenReference eig c I degree amplitude n treeDegree h j
have hm : Measurable (fun p : (TensorFrozenData N n j × (ℕ → ℝ)) × (Spin N × LabeledLeaf n) =>
cylinderField (A p.1.1 p.2) p.1.2) := measurable_frozenSelectedField I degree n treeDegree j
have hf := measurable_randomCoefficient_cgf (ν := ν)
(measurable_tensorFrozenReference eig c I degree amplitude n treeDegree h j) A hm t
convert hf using 1
rfl
/-- The selected-field CGF is an actual square-integrable observable on
the frozen external disorder and independent Gaussian factor. -/
theorem tensorFrozenCGF_memLp_two {N m k : ℕ}
(μ : Measure (SpecialOrthogonal N)) [IsProbabilityMeasure μ] (eig c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (treeDegree : Fin k → ℕ) (h : ℕ → ℝ) (j : Fin k) (t : ℝ) :
MemLp (tensorFrozenCGF eig c I degree amplitude n treeDegree h j t) 2
((tensorFrozenLaw μ n b j).prod gaussianCoordinates) := by
have hm := measurable_tensorFrozenCGF eig c I degree amplitude n treeDegree h j t
apply (memLp_two_iff_integrable_sq hm.aestronglyMeasurable).mpr
exact joint_integrable_square_of_uniform_conditionals
(P := tensorFrozenLaw μ n b j) hm
(fun ω => (cylinder_cgf_second
(tensorFrozenReference eig c I degree amplitude n treeDegree h j ω)
(jointSpectralMonomialCoefficients (specialRotation ω.1.1) I (degree j) n (treeDegree j))
(jointSpectralMonomialCoefficients_variance_le_one (specialRotation ω.1.1) I (degree j) n (treeDegree j)) t).1)
(fun ω => (cylinder_cgf_second
(tensorFrozenReference eig c I degree amplitude n treeDegree h j ω)
(jointSpectralMonomialCoefficients (specialRotation ω.1.1) I (degree j) n (treeDegree j))
(jointSpectralMonomialCoefficients_variance_le_one (specialRotation ω.1.1) I (degree j) n (treeDegree j)) t).2)
/-- The CGF is exactly the difference of the actual log partitions at
selected amplitude `t` and selected amplitude zero, almost surely. -/
theorem tensorFrozenCGF_eq_log_difference {N m k : ℕ}
(μ : Measure (SpecialOrthogonal N)) [IsProbabilityMeasure μ] (eig c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (treeDegree : Fin k → ℕ) (h : ℕ → ℝ) (hh : Monotone h) (h0 : 0 ≤ h 0)
(j : Fin k) (t : ℝ) :
tensorFrozenCGF eig c I degree amplitude n treeDegree h j t =ᵐ[
(tensorFrozenLaw μ n b j).prod gaussianCoordinates]
(fun p => tensorFrozenPartitionLog eig c I degree amplitude n treeDegree h j t p -
tensorFrozenPartitionLog eig c I degree amplitude n treeDegree h j 0 p) := by
have hp := tensorFrozenInsertion_measurePreserving μ n b j
have hbase := (measurePreserving_fst (μ := tensorFrozenLaw μ n b j) (ν := gaussianCoordinates)).quasiMeasurePreserving.ae
(tensorFrozenBase_exp_integrable_ae μ eig c I degree amplitude n b treeDegree h hh h0 j)
have hfull := hp.quasiMeasurePreserving.ae
(tensorNamespaced_exp_integrable_ae μ eig c I degree (Function.update amplitude j t)
n b treeDegree h hh h0)
filter_upwards [hbase, hfull] with p hb hf
let ν := labeledSpinReference n (uniformSpinPrior N : Measure (Spin N)) p.1.1.2
let H := tensorFrozenBaseEnergy eig c I degree amplitude n treeDegree h j p.1
let Y := fun x : Spin N × LabeledLeaf n => cylinderField (jointSpectralMonomialCoefficients
(specialRotation p.1.1.1) I (degree j) n (treeDegree j) x) p.2
have henergy (a : ℝ) (x : Spin N × LabeledLeaf n) :
rotatedEnergy eig (specialRotation p.1.1.1) x.1 + fieldEnergy c x.1 +
cylinderField (tensorNamespacedCoefficients (specialRotation p.1.1.1) I degree
(Function.update amplitude j a) n (fun i => tensorPathProfile I degree n treeDegree h i) x)
(gaussianNamespaceJoin (j + 1) (p.2, p.1.2)) = H x + a * Y x := by
have he := tensorFrozenEnergy_insert eig c I degree (Function.update amplitude j a)
n treeDegree h hh h0 j p.1 p.2 x
simpa only [tensorFrozenBaseEnergy, Function.update_idem, Function.update_self, H, Y] using he
have hpart (a : ℝ) : tensorFrozenPartitionLog eig c I degree amplitude n treeDegree h j a p =
Real.log (∫ x, Real.exp (H x + a * Y x) ∂ν) := by
apply congrArg Real.log
apply integral_congr_ae
exact ae_of_all _ fun x => congrArg Real.exp (henergy a x)
have hi : Integrable (fun x => Real.exp (H x + t * Y x)) ν := by
convert hf using 1
funext x
exact congrArg Real.exp (henergy t x).symm
have hz : 0 < ∫ x, Real.exp (H x) ∂ν := integral_exp_pos hb
have hz' : 0 < ∫ x, Real.exp (H x + t * Y x) ∂ν := integral_exp_pos hi
rw [hpart t, hpart 0]
simp only [zero_mul, add_zero]
change Real.log (∫ x, Real.exp (t * Y x) ∂gibbsProbability ν H) = _
rw [gibbsProbability_eq_tilted ν H hb, integral_exp_tilted H (fun x => t * Y x)]
change Real.log ((∫ x, Real.exp (H x + t * Y x) ∂ν) / ∫ x, Real.exp (H x) ∂ν) = _
exact Real.log_div hz'.ne' hz.ne'
end InvariantIsing
end
end OAI
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