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c4cdbbd | 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 | 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
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