File size: 4,058 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 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 | import OAI.Probability.FreeIsing.Ising.CopyObservables
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]
def cavityLogEdgeVector
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) (t : ℝ≥0) :
EdgeSpace d (differenceSpace beta μ) :=
WithLp.toLp 2 (fun c => observableDifference (beta := beta)
(memLp_cavityLogObservable hμ hbeta t TreeEscape.root (prefixFlip c TreeEscape.root)))
theorem norm_cavityLogEdgeVector_sq_le
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) (t : ℝ≥0)
(o : TreeVertex d) :
‖cavityLogEdgeVector hμ hbeta t‖ ^ 2 ≤ (d : ℝ) *
(cavityLogVarianceCoefficient (Real.tanh beta) *
∫ q, (IsingConditionalCopyBounds.fieldDifference beta o t q) ^ 2 ∂copies beta μ) := by
rw [PiLp.norm_sq_eq_of_L2]
calc
_ ≤ ∑ _c : Fin d, cavityLogVarianceCoefficient (Real.tanh beta) *
∫ q, (IsingConditionalCopyBounds.fieldDifference beta o t q) ^ 2 ∂copies beta μ := by
apply Finset.sum_le_sum
intro c hc
exact norm_cavityLogDifference_sq_le hμ hbeta t o TreeEscape.root (prefixFlip c TreeEscape.root)
_ = _ := by simp
theorem cavityLogEdgeVector_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) (o : TreeVertex d) (n : ℕ),
|inner ℝ (cavityLogEdgeVector hμ hbeta t)
((forwardOperator (by omega : 0 < d) (differenceReflection hμ)
((d : ℝ) - 1)⁻¹ ^ n) (cavityLogEdgeVector hμ hbeta t))| ≤
(K * ((n : ℝ) + 1) * (Real.sqrt ((d : ℝ) - 1))⁻¹ ^ n) *
((d : ℝ) * (cavityLogVarianceCoefficient (Real.tanh beta) *
∫ q, (IsingConditionalCopyBounds.fieldDifference beta o t q) ^ 2 ∂copies beta μ)) := by
obtain ⟨K, hK, hpow⟩ := forwardOperator_pow_bound hμ hd hbeta hthreshold
refine ⟨K, hK, ?_⟩
intro t o n
let f := cavityLogEdgeVector hμ hbeta t
let N := forwardOperator (by omega : 0 < d) (differenceReflection hμ) ((d : ℝ) - 1)⁻¹
have hinner : |inner ℝ f ((N ^ n) f)| ≤ ‖N ^ n‖ * ‖f‖ ^ 2 := by
calc
|inner ℝ f ((N ^ n) f)| ≤ ‖f‖ * ‖(N ^ n) f‖ :=
by simpa only [Real.norm_eq_abs] using norm_inner_le_norm (𝕜 := ℝ) f ((N ^ n) f)
_ ≤ ‖f‖ * (‖N ^ n‖ * ‖f‖) :=
mul_le_mul_of_nonneg_left ((N ^ n).le_opNorm f) (norm_nonneg f)
_ = _ := by ring
exact hinner.trans (mul_le_mul (hpow n)
(norm_cavityLogEdgeVector_sq_le hμ hbeta t o) (sq_nonneg ‖f‖)
(by positivity))
theorem summable_cavityLogEdgeVector_correlations
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hd : 3 ≤ d) (hbeta : 0 ≤ beta)
(hthreshold : Real.tanh beta ≤ 1 / Real.sqrt ((d : ℝ) - 1)) (t : ℝ≥0) :
Summable (fun n : ℕ => |inner ℝ (cavityLogEdgeVector hμ hbeta t)
((forwardOperator (by omega : 0 < d) (differenceReflection hμ)
((d : ℝ) - 1)⁻¹ ^ n) (cavityLogEdgeVector hμ hbeta t))|) := by
let f := cavityLogEdgeVector hμ hbeta t
let N := forwardOperator (by omega : 0 < d) (differenceReflection hμ) ((d : ℝ) - 1)⁻¹
apply Summable.of_nonneg_of_le (fun n => abs_nonneg _)
(f := fun n => ‖N ^ n‖ * ‖f‖ ^ 2)
· intro n
calc
|inner ℝ f ((N ^ n) f)| ≤ ‖f‖ * ‖(N ^ n) f‖ :=
by simpa only [Real.norm_eq_abs] using norm_inner_le_norm (𝕜 := ℝ) f ((N ^ n) f)
_ ≤ ‖f‖ * (‖N ^ n‖ * ‖f‖) :=
mul_le_mul_of_nonneg_left ((N ^ n).le_opNorm f) (norm_nonneg f)
_ = _ := by ring
· exact (summable_forwardOperator_pow hμ hd hbeta hthreshold).mul_right _
end Problem367.IsingCopySpectral
end
end OAI
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