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namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367.IsingCopySpectral
open IsingObservationInnovation IsingObservationFactor IsingCopyStationarity
variable {d : ℕ} {beta : ℝ}
variable {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ]
def cavityLogDifference (hbeta : 0 ≤ beta) (t : ℝ≥0) (v w : TreeVertex d)
(q : Paths d × (Paths d × Paths d)) : ℝ :=
cavityLogObservable hbeta t v w q.2.1 - cavityLogObservable hbeta t v w q.2.2
omit [IsProbabilityMeasure μ] in
theorem cavityLogObservable_relabel_ae
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(g : TreeAut d) (t : ℝ≥0) (v w : TreeVertex d) :
(fun x => cavityLogObservable hbeta t v w (relabel g x)) =ᵐ[observationLaw μ]
cavityLogObservable hbeta t (g.toEquiv.symm v) (g.toEquiv.symm w) := by
filter_upwards [cavityObservable_relabel_ae hμ hbeta g t v w] with x hx
exact congrArg (fun z => Real.log (responseAttenuation (Real.tanh beta) z)) hx
theorem cavityLogDifference_relabel_ae
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(g : TreeAut d) (t : ℝ≥0) (v w : TreeVertex d) :
(fun q => cavityLogDifference hbeta t v w (copyRelabel g q)) =ᵐ[copies beta μ]
cavityLogDifference hbeta t (g.toEquiv.symm v) (g.toEquiv.symm w) := by
obtain ⟨hl, hr⟩ := measurePreserving_observations (beta := beta) (μ := μ)
filter_upwards [hl.quasiMeasurePreserving.ae
(cavityLogObservable_relabel_ae hμ hbeta g t v w),
hr.quasiMeasurePreserving.ae
(cavityLogObservable_relabel_ae hμ hbeta g t v w)] with q hq hq'
exact congrArg₂ (fun x y : ℝ => x - y) hq hq'
theorem memLp_cavityLogDifference
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) (v w : TreeVertex d) :
MemLp (cavityLogDifference hbeta t v w) 2 (copies beta μ) := by
obtain ⟨hl, hr⟩ := measurePreserving_observations (beta := beta) (μ := μ)
exact ((memLp_cavityLogObservable hμ hbeta t v w).comp_measurePreserving hl).sub
((memLp_cavityLogObservable hμ hbeta t v w).comp_measurePreserving hr)
theorem integral_cavityLogDifference_eq_zero
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) (v w : TreeVertex d) :
(∫ q, cavityLogDifference hbeta t v w q ∂copies beta μ) = 0 := by
obtain ⟨hl, hr⟩ := measurePreserving_observations (beta := beta) (μ := μ)
have hf := memLp_cavityLogObservable hμ hbeta t v w
have hi₁ : Integrable (fun q => cavityLogObservable hbeta t v w q.2.1)
(copies beta μ) := (hf.comp_measurePreserving hl).integrable (by norm_num)
have hi₂ : Integrable (fun q => cavityLogObservable hbeta t v w q.2.2)
(copies beta μ) := (hf.comp_measurePreserving hr).integrable (by norm_num)
unfold cavityLogDifference
rw [integral_sub hi₁ hi₂]
have he₁ := hl.hasLaw.integral_comp hf.aestronglyMeasurable
have he₂ := hr.hasLaw.integral_comp hf.aestronglyMeasurable
simp only [Function.comp_def] at he₁ he₂
rw [he₁, he₂, sub_self]
def cavityLogVector
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) (v w : TreeVertex d) : differenceSpace beta μ :=
observableDifference (beta := beta) (memLp_cavityLogObservable hμ hbeta t v w)
theorem cavityLogVector_ae
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) (v w : TreeVertex d) :
(cavityLogVector hμ hbeta t v w : Lp ℝ 2 (copies beta μ)) =ᵐ[copies beta μ]
cavityLogDifference hbeta t v w :=
observableDifference_ae (beta := beta) (memLp_cavityLogObservable hμ hbeta t v w)
theorem inner_cavityLogVector_eq_integral
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) (v w u z : TreeVertex d) :
inner ℝ (cavityLogVector hμ hbeta t v w) (cavityLogVector hμ hbeta t u z) =
∫ q, cavityLogDifference hbeta t v w q * cavityLogDifference hbeta t u z q
∂copies beta μ := by
change inner ℝ (cavityLogVector hμ hbeta t v w : Lp ℝ 2 (copies beta μ))
(cavityLogVector hμ hbeta t u z : Lp ℝ 2 (copies beta μ)) = _
rw [L2.inner_def]
apply integral_congr_ae
filter_upwards [cavityLogVector_ae hμ hbeta t v w,
cavityLogVector_ae hμ hbeta t u z] with q hq hq'
simp [hq, hq', mul_comm]
theorem covariance_cavityLogDifference_eq_inner
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) (v w u z : TreeVertex d) :
covariance (cavityLogDifference hbeta t v w) (cavityLogDifference hbeta t u z)
(copies beta μ) =
inner ℝ (cavityLogVector hμ hbeta t v w) (cavityLogVector hμ hbeta t u z) := by
rw [covariance_eq_sub (memLp_cavityLogDifference hμ hbeta t v w)
(memLp_cavityLogDifference hμ hbeta t u z),
integral_cavityLogDifference_eq_zero hμ hbeta t v w,
integral_cavityLogDifference_eq_zero hμ hbeta t u z,
zero_mul, sub_zero, inner_cavityLogVector_eq_integral]
rfl
theorem inner_cavityLogVector_map
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(g : TreeAut d) (t : ℝ≥0) (v w u z : TreeVertex d) :
inner ℝ (cavityLogVector hμ hbeta t (g.toEquiv v) (g.toEquiv w))
(cavityLogVector hμ hbeta t (g.toEquiv u) (g.toEquiv z)) =
inner ℝ (cavityLogVector hμ hbeta t v w) (cavityLogVector hμ hbeta t u z) := by
rw [inner_cavityLogVector_eq_integral, inner_cavityLogVector_eq_integral]
have hmem := (memLp_cavityLogDifference hμ hbeta t v w).integrable_mul
(memLp_cavityLogDifference hμ hbeta t u z)
calc
_ = ∫ q, cavityLogDifference hbeta t v w (copyRelabel g.symm q) *
cavityLogDifference hbeta t u z (copyRelabel g.symm q) ∂copies beta μ := by
apply integral_congr_ae
filter_upwards [cavityLogDifference_relabel_ae hμ hbeta g.symm t v w,
cavityLogDifference_relabel_ae hμ hbeta g.symm t u z] with q hq hq'
exact (congrArg₂ (fun x y : ℝ => x * y) hq hq').symm
_ = _ := (measurePreserving_copies_relabel hμ g.symm).hasLaw.integral_comp
hmem.aestronglyMeasurable
theorem differenceReflection_cavityLogVector
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(c : Fin d) (t : ℝ≥0) (v w : TreeVertex d) :
differenceReflection hμ c (cavityLogVector hμ hbeta t v w) =
cavityLogVector hμ hbeta t (prefixFlip c v) (prefixFlip c w) := by
apply Subtype.ext
change koopmanL2 (copyRelabel (prefixTreeAut c))
(measurePreserving_copies_relabel hμ (prefixTreeAut c))
(cavityLogVector hμ hbeta t v w : Lp ℝ 2 (copies beta μ)) = _
apply Lp.ext
have hp := measurePreserving_copies_relabel hμ (prefixTreeAut c)
filter_upwards [Lp.coeFn_compMeasurePreserving
(cavityLogVector hμ hbeta t v w : Lp ℝ 2 (copies beta μ)) hp,
hp.quasiMeasurePreserving.ae (cavityLogVector_ae hμ hbeta t v w),
cavityLogDifference_relabel_ae hμ hbeta (prefixTreeAut c) t v w,
cavityLogVector_ae hμ hbeta t (prefixFlip c v) (prefixFlip c w)]
with q hq hq' hq'' hq'''
exact hq.trans (hq'.trans (hq''.trans hq'''.symm))
end Problem367.IsingCopySpectral
end
end OAI
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