File size: 2,663 Bytes
f61e547 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 | 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
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