| import OAI.Probability.FreeIsing.Ising.CopyEdgeVariance |
| import OAI.Probability.FreeIsing.Ising.CopyPathVector |
| import OAI.Probability.FreeIsing.Trees.ColoredPathCoefficients |
| import OAI.Probability.FreeIsing.Trees.PathPrefix |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory |
| open scoped NNReal |
|
|
| namespace Problem367.IsingCopySpectral |
| open IsingObservationInnovation IsingObservationFactor |
| open ColoredEdgeLifts ColoredPathAverage ConnectedHull |
|
|
| variable {d : ℕ} {beta : ℝ} |
| variable {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ] |
|
|
| theorem inner_rooted_cavityLogVector_eq_of_length_eq |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| (t : ℝ≥0) (a b : TreeVertex d) (hlen : a.1.length = b.1.length) (n : ℕ) : |
| inner ℝ (cavityLogVector hμ hbeta t (root d) (takeVertex a 1)) |
| (cavityLogVector hμ hbeta t (takeVertex a n) a) = |
| inner ℝ (cavityLogVector hμ hbeta t (root d) (takeVertex b 1)) |
| (cavityLogVector hμ hbeta t (takeVertex b n) b) := by |
| obtain ⟨g, hga, hgroot⟩ := exists_rootFixing_treeAut_map a b hlen |
| have hgroot' : g.toEquiv (root d) = root d := hgroot |
| have hfirst := g.map_takeVertex_of_root_eq hgroot' a 1 |
| have hlast := g.map_takeVertex_of_root_eq hgroot' a n |
| rw [hga] at hfirst hlast |
| have h := inner_cavityLogVector_map hμ hbeta g t (root d) (takeVertex a 1) |
| (takeVertex a n) a |
| rw [hgroot', hfirst, hlast, hga] at h |
| exact h.symm |
|
|
| theorem inner_pathValue_eq_rooted_cavityLogVector |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| (t : ℝ≥0) {n : ℕ} {i : Fin d} (p : Path d n i) |
| (v : TreeVertex d) (hlen : v.1.length = n + 1) : |
| inner ℝ (cavityLogEdgeVector hμ hbeta t i) |
| (pathValue (differenceReflection hμ) (cavityLogEdgeVector hμ hbeta t) p) = |
| inner ℝ (cavityLogVector hμ hbeta t (root d) (takeVertex v 1)) |
| (cavityLogVector hμ hbeta t (takeVertex v n) v) := by |
| rw [pathValue_cavityLogEdgeVector] |
| change inner ℝ (cavityLogVector hμ hbeta t (root d) (prefixFlip i TreeEscape.root)) |
| (cavityLogVector hμ hbeta t (tailVertex p) (headVertex p)) = _ |
| have hfirst : takeVertex (headVertex p) 1 = prefixFlip i TreeEscape.root := by |
| cases n <;> apply Subtype.ext <;> rfl |
| have htail : tailVertex p = takeVertex (headVertex p) n := by |
| apply Subtype.ext |
| change (colors p).dropLast = (colors p).take n |
| rw [List.dropLast_eq_take, colors_length] |
| simp |
| rw [← hfirst, htail] |
| apply inner_rooted_cavityLogVector_eq_of_length_eq hμ hbeta t _ v _ n |
| simpa only [headVertex, colors_length] using hlen.symm |
|
|
| theorem inner_forwardOperator_eq_rooted_cavityLogVector |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hd : 2 ≤ d) (hbeta : 0 ≤ beta) |
| (t : ℝ≥0) (n : ℕ) (v : TreeVertex d) (hlen : v.1.length = n + 1) : |
| inner ℝ (cavityLogEdgeVector hμ hbeta t) |
| ((forwardOperator (by omega : 0 < d) (differenceReflection hμ) |
| ((d : ℝ) - 1)⁻¹ ^ n) (cavityLogEdgeVector hμ hbeta t)) = |
| (d : ℝ) * inner ℝ (cavityLogVector hμ hbeta t (root d) (takeVertex v 1)) |
| (cavityLogVector hμ hbeta t (takeVertex v n) v) := by |
| apply inner_forwardOperator_pow_eq_card_mul hd |
| intro i p |
| exact inner_pathValue_eq_rooted_cavityLogVector hμ hbeta t p v hlen |
|
|
| theorem abs_inner_rooted_cavityLogVector_le_norm_pow |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hd : 2 ≤ d) (hbeta : 0 ≤ beta) |
| (t : ℝ≥0) (n : ℕ) (v : TreeVertex d) (hlen : v.1.length = n + 1) |
| {u w : TreeVertex d} (huw : treeAdj u w) : |
| |inner ℝ (cavityLogVector hμ hbeta t (root d) (takeVertex v 1)) |
| (cavityLogVector hμ hbeta t (takeVertex v n) v)| ≤ |
| ‖forwardOperator (by omega : 0 < d) (differenceReflection hμ) |
| ((d : ℝ) - 1)⁻¹ ^ n‖ * |
| ∫ q, (cavityLogDifference hbeta t u w q) ^ 2 ∂copies beta μ := by |
| apply abs_pathCoefficient_le_norm_pow hd (differenceReflection hμ) n |
| (cavityLogEdgeVector hμ hbeta t) |
| · intro i p |
| exact inner_pathValue_eq_rooted_cavityLogVector hμ hbeta t p v hlen |
| · exact (norm_cavityLogEdgeVector_sq_eq hμ hbeta t huw).le |
|
|
| theorem exists_rooted_cavityLogVector_correlation_bound |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hd : 3 ≤ d) (hbeta : 0 ≤ beta) |
| (hthreshold : Real.tanh beta ≤ 1 / Real.sqrt ((d : ℝ) - 1)) : |
| ∃ K : ℝ, 0 ≤ K ∧ ∀ (t : ℝ≥0) (n : ℕ) (v : TreeVertex d), |
| v.1.length = n + 1 → ∀ (u w : TreeVertex d), treeAdj u w → |
| |inner ℝ (cavityLogVector hμ hbeta t (root d) (takeVertex v 1)) |
| (cavityLogVector hμ hbeta t (takeVertex v n) v)| ≤ |
| (K * ((n : ℝ) + 1) * (Real.sqrt ((d : ℝ) - 1))⁻¹ ^ n) * |
| ∫ q, (cavityLogDifference hbeta t u w q) ^ 2 ∂copies beta μ := by |
| obtain ⟨K, hK, hpow⟩ := forwardOperator_pow_bound hμ hd hbeta hthreshold |
| refine ⟨K, hK, ?_⟩ |
| intro t n v hlen u w huw |
| exact (abs_inner_rooted_cavityLogVector_le_norm_pow hμ (by omega) hbeta t n v |
| hlen huw).trans (mul_le_mul_of_nonneg_right (hpow n) |
| (integral_nonneg (fun q => sq_nonneg _))) |
|
|
| theorem exists_rooted_cavityLogDifference_pair_bound |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hd : 3 ≤ d) (hbeta : 0 ≤ beta) |
| (hthreshold : Real.tanh beta ≤ 1 / Real.sqrt ((d : ℝ) - 1)) : |
| ∃ K : ℝ, 0 ≤ K ∧ ∀ (t : ℝ≥0) (n : ℕ) (v : TreeVertex d), |
| v.1.length = n + 1 → ∀ (u w : TreeVertex d), treeAdj u w → |
| |∫ q, cavityLogDifference hbeta t (root d) (takeVertex v 1) q * |
| cavityLogDifference hbeta t (takeVertex v n) v q ∂copies beta μ| ≤ |
| (K * ((n : ℝ) + 1) * (Real.sqrt ((d : ℝ) - 1))⁻¹ ^ n) * |
| ∫ q, (cavityLogDifference hbeta t u w q) ^ 2 ∂copies beta μ := by |
| obtain ⟨K, hK, h⟩ := exists_rooted_cavityLogVector_correlation_bound hμ hd hbeta hthreshold |
| refine ⟨K, hK, ?_⟩ |
| intro t n v hlen u w huw |
| simpa only [inner_cavityLogVector_eq_integral] using h t n v hlen u w huw |
|
|
| end Problem367.IsingCopySpectral |
|
|
| end |
|
|
| end OAI |
|
|
