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import OAI.Probability.InvariantIsing.Arrays.OverlapBounds
namespace OAI
/-! Actual bounded spin pair tests under the published marked-path law. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped BigOperators ENNReal NNReal
namespace InvariantIsing
abbrev TensorPairPathData {N m k : ℕ} (I : Fin m → Finset (Fin N))
(degree : Fin k → Fin m → ℕ) (n : ℕ) :=
(SpinTensorIndex I degree → ℝ) ×
(Fin n → (SpinTensorIndex I degree → ℝ) × (SpinTensorIndex I degree → ℝ))
def tensorPairPathState {N m k : ℕ} (I : Fin m → Finset (Fin N))
(degree : Fin k → Fin m → ℕ) (n : ℕ) (p : TensorPairPathData I degree n) (j : Fin 2) :
SpinTensorIndex I degree → ℝ :=
p.1 + ∑ i, if j = 0 then (p.2 i).1 else (p.2 i).2
lemma measurable_tensorPairPathState {N m k : ℕ} (I : Fin m → Finset (Fin N))
(degree : Fin k → Fin m → ℕ) (n : ℕ) (j : Fin 2) :
Measurable (fun p => tensorPairPathState I degree n p j) := by
have hm (i : Fin n) : Measurable (fun p : TensorPairPathData I degree n =>
if j = 0 then (p.2 i).1 else (p.2 i).2) := by
split_ifs
· exact ((measurable_pi_apply i).comp measurable_snd).fst
· exact ((measurable_pi_apply i).comp measurable_snd).snd
exact measurable_fst.add (Finset.measurable_sum _ fun i _ => hm i)
private lemma measurable_tensorPairSpinMeasures {N m k : ℕ}
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) :
Measurable (fun p : TensorPairPathData I degree n => Measure.pi (fun j : Fin 2 =>
gibbsProbability (uniformSpinPrior N : Measure (Spin N))
(tensorSpinBaseEnergy eig U c I degree amplitude (tensorPairPathState I degree n p j)))) := by
have hν (j : Fin 2) : Measurable (fun p : TensorPairPathData I degree n =>
gibbsProbability (uniformSpinPrior N : Measure (Spin N))
(tensorSpinBaseEnergy eig U c I degree amplitude (tensorPairPathState I degree n p j))) := by
let H : TensorPairPathData I degree n × Spin N → ℝ := fun p =>
tensorSpinBaseEnergy eig U c I degree amplitude (tensorPairPathState I degree n p.1 j) p.2
have hH : Measurable H := by
apply measurable_from_prod_countable_left
intro σ
have hm := (measurable_pi_apply σ).comp
((measurable_spinTensorEnergyMark U I degree amplitude).comp
(measurable_tensorPairPathState I degree n j))
exact hm.const_add (rotatedEnergy eig U σ + fieldEnergy c σ)
exact measurable_gibbsProbability
(ν := fun _ : TensorPairPathData I degree n => (uniformSpinPrior N : Measure (Spin N)))
(H := H) measurable_const hH
apply Measure.measurable_of_measurable_coe
intro s hs
have he (p : TensorPairPathData I degree n) :
(Measure.pi (fun j : Fin 2 => gibbsProbability (uniformSpinPrior N : Measure (Spin N))
(tensorSpinBaseEnergy eig U c I degree amplitude (tensorPairPathState I degree n p j)))) s =
∑' σ : Fin 2 → Spin N,
(∏ j : Fin 2, gibbsProbability (uniformSpinPrior N : Measure (Spin N))
(tensorSpinBaseEnergy eig U c I degree amplitude (tensorPairPathState I degree n p j)) {σ j}) *
Measure.dirac σ s := by
rw [← Measure.sum_smul_dirac (Measure.pi _), Measure.sum_apply _ hs]
simp only [Measure.smul_apply, smul_eq_mul, Measure.pi_singleton]
simp_rw [he]
apply Measurable.tsum
intro σ
exact (Finset.measurable_prod _ fun j _ =>
(Measure.measurable_coe (measurableSet_singleton _)).comp (hν j)).mul_const _
def tensorPairPathSpinKernel {N m k : ℕ}
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) : Kernel (TensorPairPathData I degree n) (Fin 2 → Spin N) :=
⟨fun p => Measure.pi (fun j : Fin 2 =>
gibbsProbability (uniformSpinPrior N : Measure (Spin N))
(tensorSpinBaseEnergy eig U c I degree amplitude (tensorPairPathState I degree n p j))),
measurable_tensorPairSpinMeasures eig U c I degree amplitude n⟩
instance tensorPairPathSpinKernel_markov {N m k : ℕ}
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) : IsMarkovKernel (tensorPairPathSpinKernel eig U c I degree amplitude n) :=
⟨fun _ => inferInstanceAs (IsProbabilityMeasure (Measure.pi _))⟩
def tensorPairPathSpinMean {N m k : ℕ}
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (p : TensorPairPathData I degree n) (D : (Fin 2 → Spin N) → ℝ) : ℝ :=
∫ σ, D σ ∂tensorPairPathSpinKernel eig U c I degree amplitude n p
lemma measurable_tensorPairPathSpinMean {N m k : ℕ}
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (D : (Fin 2 → Spin N) → ℝ) :
Measurable (fun p => tensorPairPathSpinMean eig U c I degree amplitude n p D) := by
have hm : Measurable (fun p : TensorPairPathData I degree n × (Fin 2 → Spin N) => D p.2) :=
(measurable_of_countable D).comp measurable_snd
exact (hm.stronglyMeasurable.integral_kernel_prod_right'
(κ := tensorPairPathSpinKernel eig U c I degree amplitude n)).measurable
lemma tensorPairPathSpinMean_abs_le {N m k : ℕ}
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (p : TensorPairPathData I degree n) (D : (Fin 2 → Spin N) → ℝ)
{C : ℝ} (hD : ∀ σ, |D σ| ≤ C) :
|tensorPairPathSpinMean eig U c I degree amplitude n p D| ≤ C :=
abs_integral_le_const_of_bound (measurable_of_countable D) hD
def tensorPairSpinPathPayoff {N m k : ℕ}
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (ψ : ℕ → ℝ) (D : (Fin 2 → Spin N) → ℝ)
(p : (SpinTensorIndex I degree → ℝ) ×
(ℕ × (Fin n → (SpinTensorIndex I degree → ℝ) × (SpinTensorIndex I degree → ℝ)))) : ℝ :=
ψ p.2.1 * tensorPairPathSpinMean eig U c I degree amplitude n (p.1, p.2.2) D
lemma measurable_tensorPairSpinPathPayoff {N m k : ℕ}
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (ψ : ℕ → ℝ) (D : (Fin 2 → Spin N) → ℝ) :
Measurable (tensorPairSpinPathPayoff eig U c I degree amplitude n ψ D) :=
((measurable_of_countable ψ).comp measurable_snd.fst).mul
((measurable_tensorPairPathSpinMean eig U c I degree amplitude n D).comp
(measurable_fst.prodMk measurable_snd.snd))
lemma tensorPairSpinPathPayoff_abs_le {N m k : ℕ}
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (ψ : ℕ → ℝ) (D : (Fin 2 → Spin N) → ℝ)
{B C : ℝ} (hB : 0 ≤ B) (hψ : ∀ d, |ψ d| ≤ B) (hD : ∀ σ, |D σ| ≤ C)
(p : (SpinTensorIndex I degree → ℝ) ×
(ℕ × (Fin n → (SpinTensorIndex I degree → ℝ) × (SpinTensorIndex I degree → ℝ)))) :
|tensorPairSpinPathPayoff eig U c I degree amplitude n ψ D p| ≤ B * C := by
rw [tensorPairSpinPathPayoff, abs_mul]
exact mul_le_mul (hψ _) (tensorPairPathSpinMean_abs_le eig U c I degree amplitude n _ D hD)
(abs_nonneg _) hB
private lemma tensorPairPathState_edges {N m k : ℕ}
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ)
(n : ℕ) (z : SpinTensorIndex I degree → ℝ) (p : TensorCoordinateData I degree n)
(α : Fin 2 → LabeledLeaf n) (j : Fin 2) :
tensorPairPathState I degree n
(z, fun i => (p.2 (edgeAt n (α 0) i), p.2 (edgeAt n (α 1) i))) j =
labeledEnergy n (markForestOfCoords (SpinTensorIndex I degree → ℝ) n p.2) (α j) z := by
funext a
rw [labeledEnergy_edge_sum]
fin_cases j <;> simp [tensorPairPathState, Pi.add_apply, Finset.sum_apply]
private lemma tensorPairPathSpinKernel_edges {N m k : ℕ}
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (z : SpinTensorIndex I degree → ℝ) (p : TensorCoordinateData I degree n)
(α : Fin 2 → LabeledLeaf n) :
tensorPairPathSpinKernel eig U c I degree amplitude n
(z, fun i => (p.2 (edgeAt n (α 0) i), p.2 (edgeAt n (α 1) i))) =
spinLeafPairResidualKernel (tensorLeafSpinHamiltonian eig U c I degree amplitude n z p) α := by
change Measure.pi _ = Measure.pi _
congr 1
funext j
rw [tensorPairPathState_edges]
rfl
def tensorSpinPairAverage {N m k : ℕ}
(eig : Fin N → ℝ) (U : Rotation N) (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) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(z : SpinTensorIndex I degree → ℝ) (ψ : ℕ → ℝ) (D : (Fin 2 → Spin N) → ℝ) : ℝ :=
∫ p : TensorCoordinateData I degree n,
referenceReplicaMean ((uniformSpinPrior N : Measure (Spin N)).prod (labeledLeafLaw n p.1))
(tensorLeafSpinHamiltonian eig U c I degree amplitude n z p)
(fun x : Fin 2 → Spin N × LabeledLeaf n =>
ψ (labeledCommonDepth n (x 0).2 (x 1).2) * D (fun j => (x j).1))
∂tensorCoordinateLaw I degree n b v
theorem tensorSpinPairAverage_eq_tiltedPath {N m k : ℕ} (hN : 0 < N)
(eig : Fin N → ℝ) (U : Rotation N) (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) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(hb : CascadeExponents n b) (z : SpinTensorIndex I degree → ℝ)
(ψ : ℕ → ℝ) (D : (Fin 2 → Spin N) → ℝ)
{B C : ℝ} (hB : 0 ≤ B) (hψ : ∀ d, |ψ d| ≤ B) (hD : ∀ σ, |D σ| ≤ C) :
tensorSpinPairAverage eig U c I degree amplitude n b v z ψ D =
tensorTiltedPairPathMean eig U c I degree amplitude n b v z
(fun q => tensorPairSpinPathPayoff eig U c I degree amplitude n ψ D (z, q)) := by
apply integral_congr_ae
filter_upwards [tensorLeafSpinPair_conditioning_ae hN eig U c I degree amplitude n b v hb z] with p hp
let D' := fun x : Fin 2 → Spin N × LabeledLeaf n =>
ψ (labeledCommonDepth n (x 0).2 (x 1).2) * D (fun j => (x j).1)
have hD' : ∀ x, |D' x| ≤ B * C := by
intro x
dsimp only [D']
rw [abs_mul]
exact mul_le_mul (hψ _) (hD _) (abs_nonneg _) hB
rw [hp D' ⟨B * C, hD'⟩]
congr 1
funext α
dsimp only [tensorPairSpinPathPayoff, tensorPairPathSpinMean]
rw [tensorPairPathSpinKernel_edges]
rw [← integral_const_mul]
rfl
/-- The actual spin/leaf pair expectation is a Markov ancestor-path
expectation conditional only on the precise published cascade input. -/
theorem tensorSpinPair_markov_identity (hpub : PanchenkoTalagrandTensorPairInput)
{N m k : ℕ} (hN : 0 < N)
(eig : Fin N → ℝ) (U : Rotation N) (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) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(hb : CascadeExponents n b) (z : SpinTensorIndex I degree → ℝ)
(ψ : ℕ → ℝ) (D : (Fin 2 → Spin N) → ℝ)
{B C : ℝ} (hB : 0 ≤ B) (hψ : ∀ d, |ψ d| ≤ B) (hD : ∀ σ, |D σ| ≤ C) :
tensorSpinPairAverage eig U c I degree amplitude n b v z ψ D =
∫ α : ℕ → LabeledLeaf n,
tensorAncestorPairPathMean eig U c I degree amplitude n b v
(labeledCommonDepth n (α 0) (α 1)) n 0 z z
(fun w => tensorPairSpinPathPayoff eig U c I degree amplitude n ψ D
(z, labeledCommonDepth n (α 0) (α 1), w))
∂cascadeReplicaLaw n b := by
rw [tensorSpinPairAverage_eq_tiltedPath hN eig U c I degree amplitude n b v hb z ψ D hB hψ hD]
apply hpub N m k hN eig U c I degree amplitude n b v hb z
· exact (measurable_tensorPairSpinPathPayoff eig U c I degree amplitude n ψ D).comp
(measurable_const.prodMk measurable_id)
· exact ⟨B * C, fun p => tensorPairSpinPathPayoff_abs_le eig U c I degree amplitude n ψ D hB hψ hD (z, p)⟩
theorem tensorProjectedOverlap_markov_identity (hpub : PanchenkoTalagrandTensorPairInput)
{N m k : ℕ} (hN : 0 < N)
(eig : Fin N → ℝ) (U : Rotation N) (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) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(hb : CascadeExponents n b) (z : SpinTensorIndex I degree → ℝ)
(J : Finset (Fin N)) (ψ : ℕ → ℝ) {B : ℝ} (hB : 0 ≤ B) (hψ : ∀ d, |ψ d| ≤ B) :
tensorSpinPairAverage eig U c I degree amplitude n b v z ψ
(fun σ => projectedOverlap U J (σ 0) (σ 1)) =
∫ α : ℕ → LabeledLeaf n,
tensorAncestorPairPathMean eig U c I degree amplitude n b v
(labeledCommonDepth n (α 0) (α 1)) n 0 z z
(fun w => tensorPairSpinPathPayoff eig U c I degree amplitude n ψ
(fun σ => projectedOverlap U J (σ 0) (σ 1))
(z, labeledCommonDepth n (α 0) (α 1), w))
∂cascadeReplicaLaw n b :=
tensorSpinPair_markov_identity hpub hN eig U c I degree amplitude n b v hb z ψ _ hB hψ
(fun σ => projectedOverlap_abs_le_one U J (σ 0) (σ 1))
/-- The root Gaussian retains its original law and is averaged after the
conditional spin/ancestor-path identity. -/
theorem tensorRootedSpinPair_markov_identity (hpub : PanchenkoTalagrandTensorPairInput)
{N m k : ℕ} (hN : 0 < N)
(eig : Fin N → ℝ) (U : Rotation N) (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) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(hb : CascadeExponents n b) (ψ : ℕ → ℝ) (D : (Fin 2 → Spin N) → ℝ)
{B C : ℝ} (hB : 0 ≤ B) (hψ : ∀ d, |ψ d| ≤ B) (hD : ∀ σ, |D σ| ≤ C) :
(∫ z, tensorSpinPairAverage eig U c I degree amplitude n b (fun i => v (i + 1)) z ψ D
∂(tensorGaussianLaw I degree (v 0) : Measure (SpinTensorIndex I degree → ℝ))) =
∫ z, ∫ α : ℕ → LabeledLeaf n,
tensorAncestorPairPathMean eig U c I degree amplitude n b (fun i => v (i + 1))
(labeledCommonDepth n (α 0) (α 1)) n 0 z z
(fun w => tensorPairSpinPathPayoff eig U c I degree amplitude n ψ D
(z, labeledCommonDepth n (α 0) (α 1), w))
∂cascadeReplicaLaw n b
∂(tensorGaussianLaw I degree (v 0) : Measure (SpinTensorIndex I degree → ℝ)) := by
exact integral_congr_ae (ae_of_all _ fun z => tensorSpinPair_markov_identity hpub hN eig U c I degree
amplitude n b (fun i => v (i + 1)) hb z ψ D hB hψ hD)
end InvariantIsing
end
end OAI
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