File size: 1,962 Bytes
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 | import OAI.Probability.InvariantIsing.Spectral.MeasureMagneticTransport
import OAI.Probability.InvariantIsing.Core.FiniteFieldLawCoupling
namespace OAI
/-! The finite-field magnetic value depends on the field law, not on its
choice of positive atomic labels. -/
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped BigOperators
namespace InvariantIsing
theorem measureMagnetic_eq_of_field_law_eq {A B : Type*} [Fintype A] [Fintype B]
(ν : ProbabilityMeasure ℝ) (a b : ℝ)
(hcompact : IsCompact (ν : Measure ℝ).support)
(hbound : (ν : Measure ℝ).support ⊆ Icc a b)
(ha : a∈(ν : Measure ℝ).support) (hb : b∈(ν : Measure ℝ).support)
(γ c : A → ℝ) (δ d : B → ℝ) (hγ : ∀ i, 0 < γ i) (hδ : ∀ j, 0 < δ j)
(hγsum : ∑ i, γ i=1) (hδsum : ∑ j, δ j=1)
(he : finiteSpectralMeasure γ c=finiteSpectralMeasure δ d) :
(finiteMagneticFunctional (measureR (ν : Measure ℝ) b) γ c).toReal=
(finiteMagneticFunctional (measureR (ν : Measure ℝ) b) δ d).toReal := by
have hmass := fieldAtomMass_eq_of_law_eq γ c δ d (fun i => (hγ i).le)
(fun j => (hδ j).le) he
have hh := measureMagnetic_transport_le ν a b hcompact hbound ha hb
(sameFieldCoupling γ c δ d) γ c δ d
(sameFieldCoupling_nonneg γ c δ d hγ (fun j => (hδ j).le))
(fun i => (hγ i).le) hδ hγsum hδsum
(sameFieldCoupling_row γ c δ d hγ hmass) (sameFieldCoupling_col γ c δ d hδ hmass)
have hk := measureMagnetic_transport_le ν a b hcompact hbound ha hb
(sameFieldCoupling δ d γ c) δ d γ c
(sameFieldCoupling_nonneg δ d γ c hδ (fun i => (hγ i).le))
(fun j => (hδ j).le) hγ hδsum hγsum
(sameFieldCoupling_row δ d γ c hδ (fun x => (hmass x).symm))
(sameFieldCoupling_col δ d γ c hγ (fun x => (hmass x).symm))
rw [sameFieldCoupling_cost,add_zero] at hh hk
exact le_antisymm hh hk
end InvariantIsing
end
end OAI
|