File size: 3,477 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 | import OAI.Probability.FreeIsing.Ising.CopyCavityBounds
import OAI.Probability.FreeIsing.Trees.OrbitAverage
namespace OAI
noncomputable section
open MeasureTheory
open scoped NNReal
namespace Problem367.IsingCopyOrbitAverage
open IsingObservationInnovation IsingObservationFactor IsingConditionalCopyBounds
open TreeOrbitAverage
variable {d : ℕ} {beta : ℝ}
variable {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ]
def copyAction (g : TreeAut d) : CopySpace d ≃ᵐ CopySpace d :=
let e := ObservationCopyStationarity.arrayRelabelEquiv (A := WienerSampler.Path) g.symm
e.prodCongr (e.prodCongr e)
theorem measurePreserving_copyAction
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (g : TreeAut d) :
MeasurePreserving (copyAction g) (copies beta μ) (copies beta μ) :=
IsingCopyStationarity.measurePreserving_copies_relabel hμ g.symm
theorem fieldDifference_copyAction_ae
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (g : TreeAut d)
(t : ℝ≥0) (v : TreeVertex d) :
(fun q => fieldDifference beta v t (copyAction g q)) =ᵐ[copies beta μ]
fieldDifference beta (g.toEquiv v) t := by
obtain ⟨hl, hr⟩ := IsingCopyStationarity.measurePreserving_observations
(beta := beta) (μ := μ)
filter_upwards [hl.quasiMeasurePreserving.ae
(IsingCopyStationarity.field_array_relabel_ae hμ g.symm t),
hr.quasiMeasurePreserving.ae
(IsingCopyStationarity.field_array_relabel_ae hμ g.symm t)] with q hq hq'
change field beta v t (relabel g.symm q.2.1) -
field beta v t (relabel g.symm q.2.2) =
field beta (g.toEquiv v) t q.2.1 - field beta (g.toEquiv v) t q.2.2
rw [congrFun hq v, congrFun hq' v]
rfl
theorem fieldDifference_sq_invariant
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (t : ℝ≥0)
{stem : List (Fin d)} (o : TreeVertex d) (ho : o.1.length ≤ stem.length)
(g : TreeAut d) (hg : FixesStemBall stem g) :
∀ᵐ q ∂copies beta μ,
(fieldDifference beta o t (copyAction g q)) ^ 2 =
(fieldDifference beta o t q) ^ 2 := by
filter_upwards [fieldDifference_copyAction_ae hμ g t o] with q hq
rw [hq, hg o ho]
theorem integral_fieldDifference_sq_mul_extension_average
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(t : ℝ≥0) {stem : List (Fin d)} {n : ℕ}
(o : TreeVertex d) (ho : o.1.length ≤ stem.length) (s₀ : Extension stem n)
(F : TreeVertex d → CopySpace d → ℝ)
(hmF : ∀ v, Measurable (F v))
(hF01 : ∀ v q, 0 ≤ F v q ∧ F v q ≤ 1)
(hcov : ∀ g, FixesStemBall stem g → ∀ v, ∀ᵐ q ∂copies beta μ,
F v (copyAction g q) = F (g.toEquiv v) q) :
(∫ q, (fieldDifference beta o t q) ^ 2 * F (endpoint s₀) q ∂copies beta μ) =
∫ q, (fieldDifference beta o t q) ^ 2 * WeightedOrbitAverage.average
(fun s : Extension stem n => F (endpoint s)) q ∂copies beta μ := by
apply TreeOrbitAverage.integral_weight_mul_extension_average s₀ copyAction
(measurePreserving_copyAction hμ) (fun q => (fieldDifference beta o t q) ^ 2) F
(fieldDifference_sq_invariant hμ t o ho) hcov
apply (IsingCopyStationarity.integrable_fieldDifference_sq hμ hbeta t o).mul_bdd
(hmF (endpoint s₀)).aestronglyMeasurable
exact Filter.Eventually.of_forall (fun q => by
simpa only [Real.norm_eq_abs, abs_of_nonneg (hF01 (endpoint s₀) q).1]
using (hF01 (endpoint s₀) q).2)
end Problem367.IsingCopyOrbitAverage
end
end OAI
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