| import OAI.Probability.InvariantIsing.Cavity.CavityBaseGroups |
| import Mathlib.Analysis.SpecificLimits.Basic |
|
|
| namespace OAI |
|
|
| /-! Adding finitely many exceptional axes preserves group growth and |
| limiting spectral masses. The grouped base axes still partition all sites. -/ |
|
|
| noncomputable section |
| open Filter |
| open scoped Topology BigOperators |
|
|
| namespace InvariantIsing |
|
|
| lemma cavityBaseGroupDimension_tendsto {m d : ℕ} |
| (k : ℕ → Fin m → ℕ) (g : Fin d → Fin m) (a : Fin m) |
| (hk : Tendsto (fun j => k j a) atTop atTop) : |
| Tendsto (fun j => cavityBaseGroupDimension (k j) g a) atTop atTop := |
| (tendsto_add_atTop_nat (Fintype.card {i : Fin d // g i=a})).comp hk |
|
|
| lemma cavityBaseGroupDimension_sum {N m d : ℕ} (k : Fin m → ℕ) |
| (e : (((a : Fin m) × Fin (k a)) ⊕ Fin d) ≃ Fin N) (g : Fin d → Fin m) : |
| ∑ a, cavityBaseGroupDimension k g a=N := by |
| simpa only [Fintype.card_sigma, Fintype.card_fin] using |
| Fintype.card_congr (cavityBaseGroupEquiv k e g) |
|
|
| lemma cavityBaseGroupDimension_mass_sum {N m d : ℕ} (hN : 0 < N) |
| (k : Fin m → ℕ) (e : (((a : Fin m) × Fin (k a)) ⊕ Fin d) ≃ Fin N) |
| (g : Fin d → Fin m) : |
| ∑ a, (cavityBaseGroupDimension k g a : ℝ)/N=1 := by |
| rw [← Finset.sum_div, ← Nat.cast_sum, cavityBaseGroupDimension_sum k e g, |
| div_self (Nat.cast_ne_zero.mpr hN.ne')] |
|
|
| lemma cavityBaseGroupDimension_mass_tendsto {m d : ℕ} (N : ℕ → ℕ) |
| (hN : Tendsto N atTop atTop) (k : ℕ → Fin m → ℕ) (g : Fin d → Fin m) |
| (a : Fin m) {ρ : ℝ} |
| (hk : Tendsto (fun j => (k j a : ℝ)/N j) atTop (𝓝 ρ)) : |
| Tendsto (fun j => (cavityBaseGroupDimension (k j) g a : ℝ)/N j) atTop (𝓝 ρ) := by |
| have hi : Tendsto (fun j => ((N j : ℝ))⁻¹) atTop (𝓝 0) := |
| tendsto_inv_atTop_zero.comp (tendsto_natCast_atTop_atTop.comp hN) |
| have hc := hi.const_mul (Fintype.card {i : Fin d // g i=a} : ℝ) |
| have hh := hk.add hc |
| simpa only [cavityBaseGroupDimension, Nat.cast_add, div_eq_mul_inv, add_mul, |
| mul_zero, add_zero] using hh |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
|
|
