| import OAI.Probability.InvariantIsing.Spectral.FiniteSpectralMeasure |
|
|
| namespace OAI |
|
|
| /-! Atom masses of a finitely supported external-field law, allowing |
| several labels to describe the same field value. -/ |
| noncomputable section |
| open MeasureTheory |
| open scoped BigOperators |
| namespace InvariantIsing |
|
|
| def fieldAtomMass {A : Type*} [Fintype A] (γ c : A → ℝ) (x : ℝ) : ℝ := |
| ∑ a, if c a=x then γ a else 0 |
|
|
| lemma finiteField_atom_mass {A : Type*} [Fintype A] (γ c : A → ℝ) |
| (hγ : ∀ a, 0 ≤ γ a) (x : ℝ) : |
| (finiteSpectralMeasure γ c).real {x}=fieldAtomMass γ c x := by |
| have hi := integral_finiteSpectralMeasure γ c hγ |
| (({x} : Set ℝ).indicator (fun _ => (1:ℝ))) |
| change (∫ y, ({x} : Set ℝ).indicator (1 : ℝ → ℝ) y ∂finiteSpectralMeasure γ c) = _ at hi |
| rw [integral_indicator_one (measurableSet_singleton x)] at hi |
| simpa only [fieldAtomMass,Set.indicator_apply,Set.mem_singleton_iff,mul_ite, |
| mul_one,mul_zero] using hi |
|
|
| lemma fieldAtomMass_nonneg {A : Type*} [Fintype A] (γ c : A → ℝ) |
| (hγ : ∀ a, 0 ≤ γ a) (x : ℝ) : 0 ≤ fieldAtomMass γ c x := by |
| apply Finset.sum_nonneg |
| intro a _ |
| split_ifs |
| · exact hγ a |
| · rfl |
|
|
| lemma fieldAtomMass_pos_at {A : Type*} [Fintype A] (γ c : A → ℝ) |
| (hγ : ∀ a, 0 < γ a) (a : A) : 0 < fieldAtomMass γ c (c a) := by |
| have hh := Finset.single_le_sum (f := fun i => if c i=c a then γ i else 0) |
| (fun i _ => by split_ifs; exact (hγ i).le; rfl) (Finset.mem_univ a) |
| simp only [ite_true] at hh |
| exact (hγ a).trans_le hh |
|
|
| lemma fieldAtomMass_eq_of_law_eq {A B : Type*} [Fintype A] [Fintype B] |
| (γ c : A → ℝ) (δ d : B → ℝ) (hγ : ∀ a, 0 ≤ γ a) (hδ : ∀ j, 0 ≤ δ j) |
| (he : finiteSpectralMeasure γ c=finiteSpectralMeasure δ d) (x : ℝ) : |
| fieldAtomMass γ c x=fieldAtomMass δ d x := by |
| rw [← finiteField_atom_mass γ c hγ,← finiteField_atom_mass δ d hδ,he] |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
|
|
