| import OAI.Probability.FreeIsing.Ising.CopyEdgeCorrelations |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory |
| open scoped NNReal |
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| namespace Problem367.IsingCopySpectral |
| open IsingObservationInnovation IsingObservationFactor IsingCopyStationarity |
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| variable {d : ℕ} {beta : ℝ} |
| variable {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ] |
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| 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 |
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| 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 |
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| 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' |
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| 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) |
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| 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] |
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| 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) |
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| 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) |
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| 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] |
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| 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 |
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| 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 |
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| 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)) |
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| end Problem367.IsingCopySpectral |
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| end |
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| end OAI |
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