| import OAI.Probability.FreeIsing.Ising.CopyGeodesicCorrelations | |
| import OAI.Probability.FreeIsing.Trees.RootedTransitivity | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory | |
| open scoped NNReal | |
| namespace Problem367.IsingCopySpectral | |
| open IsingObservationInnovation IsingObservationFactor ColoredEdgeLifts | |
| variable {d : ℕ} {beta : ℝ} | |
| variable {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ] | |
| theorem norm_cavityLogVector_sq_eq | |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) | |
| (t : ℝ≥0) (v w : TreeVertex d) : | |
| ‖cavityLogVector hμ hbeta t v w‖ ^ 2 = | |
| ∫ q, (cavityLogDifference hbeta t v w q) ^ 2 ∂copies beta μ := by | |
| rw [← real_inner_self_eq_norm_sq, inner_cavityLogVector_eq_integral] | |
| simp only [pow_two] | |
| theorem norm_cavityLogVector_sq_eq_of_adj | |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) | |
| (t : ℝ≥0) {u v x y : TreeVertex d} (huv : treeAdj u v) (hxy : treeAdj x y) : | |
| ‖cavityLogVector hμ hbeta t u v‖ ^ 2 = | |
| ‖cavityLogVector hμ hbeta t x y‖ ^ 2 := by | |
| obtain ⟨g, hgu, hgv⟩ := exists_treeAut_map_adjacent_pair huv hxy | |
| have h := inner_cavityLogVector_map hμ hbeta g t u v u v | |
| rw [hgu, hgv] at h | |
| simpa only [real_inner_self_eq_norm_sq] using h.symm | |
| theorem integral_cavityLogDifference_sq_eq_of_adj | |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) | |
| (t : ℝ≥0) {u v x y : TreeVertex d} (huv : treeAdj u v) (hxy : treeAdj x y) : | |
| (∫ q, (cavityLogDifference hbeta t u v q) ^ 2 ∂copies beta μ) = | |
| ∫ q, (cavityLogDifference hbeta t x y q) ^ 2 ∂copies beta μ := by | |
| rw [← norm_cavityLogVector_sq_eq hμ hbeta t u v, | |
| ← norm_cavityLogVector_sq_eq hμ hbeta t x y] | |
| exact norm_cavityLogVector_sq_eq_of_adj hμ hbeta t huv hxy | |
| theorem norm_cavityLogEdgeVector_sq_eq | |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) | |
| (t : ℝ≥0) {u v : TreeVertex d} (huv : treeAdj u v) : | |
| ‖cavityLogEdgeVector hμ hbeta t‖ ^ 2 = (d : ℝ) * | |
| ∫ q, (cavityLogDifference hbeta t u v q) ^ 2 ∂copies beta μ := by | |
| rw [PiLp.norm_sq_eq_of_L2] | |
| calc | |
| _ = ∑ _c : Fin d, ‖cavityLogVector hμ hbeta t u v‖ ^ 2 := by | |
| apply Finset.sum_congr rfl | |
| intro c hc | |
| change ‖cavityLogVector hμ hbeta t TreeEscape.root | |
| (prefixFlip c TreeEscape.root)‖ ^ 2 = _ | |
| exact norm_cavityLogVector_sq_eq_of_adj hμ hbeta t (Or.inl ⟨c, rfl⟩) huv | |
| _ = (d : ℝ) * ‖cavityLogVector hμ hbeta t u v‖ ^ 2 := by simp | |
| _ = _ := by rw [norm_cavityLogVector_sq_eq] | |
| end Problem367.IsingCopySpectral | |
| end | |
| end OAI | |