| 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 |
| |
