| import OAI.Probability.InvariantIsing.Cavity.CavityCanonicalMass |
| import OAI.Probability.InvariantIsing.Cavity.ConsecutiveGroupProfile |
|
|
| namespace OAI |
|
|
| /-! Finite field populations admit exact block templates and canonical |
| all-dimensional labels with the corresponding limiting proportions. -/ |
| noncomputable section |
| open Filter |
| open scoped BigOperators Topology |
| namespace InvariantIsing |
|
|
| lemma spinGroupSize_equiv {A B : Type*} [DecidableEq A] [DecidableEq B] |
| {N : ℕ} (E : A ≃ B) (g : Fin N → B) (a : A) : |
| spinGroupSize (fun i => E.symm (g i)) a=spinGroupSize g (E a) := by |
| simp only [spinGroupSize,Equiv.symm_apply_eq] |
|
|
| def countedFieldGroup {A : Type*} [Fintype A] {N : ℕ} (s : A → ℕ) |
| (hs : ∑ a, s a=N) : Fin N → A := |
| let E := Fintype.equivFin A |
| fun i => E.symm (cavityOrderedGroup (fun j => s (E.symm j)) |
| ((Equiv.sum_comp E.symm s).trans hs) i) |
|
|
| lemma countedFieldGroup_size {A : Type*} [Fintype A] [DecidableEq A] |
| {N : ℕ} (s : A → ℕ) (hs : ∑ a, s a=N) (a : A) : |
| spinGroupSize (countedFieldGroup s hs) a=s a := by |
| rw [countedFieldGroup,spinGroupSize_equiv] |
| exact (cavityOrderedGroup_card _ _ _).trans (by simp) |
|
|
| def canonicalFieldLabel {A : Type*} [Fintype A] (a0 : A) (s : A → ℕ) |
| (N : ℕ) : Fin N → A := |
| let E := Fintype.equivFin A |
| fun i => E.symm (cavityResidueLabel |
| (Fintype.card_pos_iff.mpr ⟨a0⟩) (fun j => s (E.symm j)) (Equiv.sum_comp E.symm s) N i) |
|
|
| lemma canonicalFieldLabel_population {A : Type*} [Fintype A] [DecidableEq A] |
| (a0 : A) (s : A → ℕ) (hs : 0 < ∑ a, s a) : |
| Tendsto (fun N a => (spinGroupSize (canonicalFieldLabel a0 s N) a : ℝ)/N) |
| atTop (𝓝 (fun a => (s a : ℝ)/(∑ a, s a))) := by |
| let E := Fintype.equivFin A |
| have hh := cavity_canonical_mass_tendsto (Fintype.card_pos_iff.mpr ⟨a0⟩) hs |
| (fun j => s (E.symm j)) (Equiv.sum_comp E.symm s) |
| apply tendsto_pi_nhds.mpr |
| intro a |
| have ha := (tendsto_pi_nhds.mp hh) (E a) |
| dsimp only [canonicalFieldLabel] |
| simp only [spinGroupSize_equiv] |
| simpa only [E,Equiv.symm_apply_apply,spinGroupSize,cavitySpectralGroup] using ha |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
|
|
