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a89d282 | 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 | import OAI.Probability.InvariantIsing.Magnetic.MagneticWassersteinMain
import Mathlib.MeasureTheory.Measure.Prod
namespace OAI
/-! The joint conditional-Haar hypothesis for random spectra and fields.
Only the one-dimensional marginal orbit law is prescribed at each dimension. -/
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped BigOperators
namespace InvariantIsing
abbrev FieldSpectralData (N : ℕ) := (Fin N → ℝ) × (Fin N → ℝ)
def dataPhysicalPressure {N : ℕ} (d : FieldSpectralData N) (U : Orthogonal N) : ℝ :=
rotatedPressure d.1 (matrixRotation U⁻¹) d.2
lemma measurable_random_specialPressure {Ω : Type*} [MeasurableSpace Ω] {N : ℕ}
(eig field : Ω → Fin N → ℝ) (heig : Measurable eig) (hfield : Measurable field)
(U : Ω → SpecialOrthogonal N) (hU : Measurable U) :
Measurable (fun ω => rotatedPressure (eig ω) (specialRotation (U ω)) (field ω)) := by
have hE (σ : Spin N) : Measurable (fun ω =>
rotatedEnergy (eig ω) (specialRotation (U ω)) σ+fieldEnergy (field ω) σ) := by
apply Measurable.add
· exact ((Finset.measurable_sum _ fun i _ =>
((measurable_pi_apply i).comp heig).mul
(((measurable_specialRotation_eval (spinVector σ) i).comp hU).pow_const 2))).const_mul (1/2 : ℝ)
· exact Finset.measurable_sum _ fun i _ => ((measurable_pi_apply i).comp hfield).mul_const _
unfold rotatedPressure logPartition
exact ((Finset.measurable_sum _ fun σ _ => (hE σ).exp).const_mul
(Fintype.card (Spin N) : ℝ)⁻¹).log.const_mul (N : ℝ)⁻¹
lemma measurable_dataPhysicalPressure {N : ℕ} (hN : 0 < N) :
Measurable (fun z : FieldSpectralData N × Orthogonal N => dataPhysicalPressure z.1 z.2) := by
have hh := measurable_random_specialPressure
(fun z : FieldSpectralData N × Orthogonal N => z.1.1)
(fun z : FieldSpectralData N × Orthogonal N => z.1.2)
(measurable_fst.comp measurable_fst) (measurable_snd.comp measurable_fst)
(fun z => cavityOrientationLift hN z.2⁻¹)
((measurable_cavityOrientationLift hN).comp measurable_snd.inv)
convert hh using 1
funext z
simp only [dataPhysicalPressure,rotatedPressure,cavityOrientationLift_energy]
lemma measurable_dataPressureMean {N : ℕ} (hN : 0 < N)
(H : Measure (Orthogonal N)) [SFinite H] :
Measurable (fun d : FieldSpectralData N => ∫ U, dataPhysicalPressure d U ∂H) :=
(measurable_dataPhysicalPressure hN).stronglyMeasurable.integral_prod_right'.measurable
/-- Conditional jointly on the spectrum and field, the pressure has its Haar orbit law.
This formulation allows arbitrary dependence across dimensions and between the two data sets. -/
def ConditionalFieldOrbitLaw {Ω : Type*} [MeasurableSpace Ω] {N : ℕ}
(P : Measure Ω) (d : Ω → FieldSpectralData N) (pressure : Ω → ℝ)
(H : Measure (Orthogonal N)) : Prop :=
P.map (fun ω => (d ω,pressure ω))=
((P.map d).prod H).map (fun z => (z.1,dataPhysicalPressure z.1 z.2))
end InvariantIsing
end
end OAI
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