| import OAI.Probability.InvariantIsing.Cavity.OffsetFieldPopulation |
| import OAI.Probability.InvariantIsing.Fields.PhysicalFieldPopulationLimit |
| import OAI.Probability.InvariantIsing.Cavity.CavityPrefixAffineFamily |
|
|
| namespace OAI |
|
|
| /-! An affine block template and any physical model with the same limiting |
| finite populations have asymptotically identical mean pressure. -/ |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators Topology |
| namespace InvariantIsing |
|
|
| theorem affine_template_pressure_comparison {m n : ℕ} (hm : 0 < m) (hn : 0 < n) |
| (q : ℕ) (hq : 0<q) (spec c : Fin m → ℕ) (hspec : ∑ a, spec a=n) |
| (ρ lam : Fin m → ℝ) (hρ : ∀ a, 0<ρ a) (hcount : ∀ a, (spec a : ℝ)=n*ρ a) |
| (μ : (M : ℕ) → Measure (Orthogonal M)) [∀ M, IsProbabilityMeasure (μ M)] |
| [∀ M, (μ M).IsMulRightInvariant] |
| (e : (M : ℕ) → Fin M → Fin m) |
| (he : Tendsto (fun M a => (spinGroupSize (e M) a : ℝ)/M) atTop (𝓝 ρ)) |
| {A : Type*} [Fintype A] [DecidableEq A] |
| (g : (M : ℕ) → Fin M → A) (γ b : A → ℝ) |
| (hg : Tendsto (fun M a => (spinGroupSize (g M) a : ℝ)/M) atTop (𝓝 γ)) |
| (group : Fin n → A) (hgroup : ∀ a, (spinGroupSize group a : ℝ)=n*γ a) |
| {K B : ℝ} (hK : 0≤K) (hlam : ∀ a, |lam a|≤K) (hB : 0≤B) (hb : ∀ a, |b a|≤B) : |
| let N := fun j => (∑ a, c a)+(q+j)*n |
| Tendsto (fun j => (∫ U, rotatedPressure (fun i => lam (e (N j) i)) (matrixRotation U⁻¹) |
| (fun i => b (g (N j) i)) ∂μ (N j))- |
| ∫ U, rotatedPressure (fun i => lam (cavityAffineLabel hm spec c hspec (N j) i)) |
| (matrixRotation U⁻¹) (offsetBlockField (∑ a, c a) (q+j) (fun i => b (group i))) ∂μ (N j)) |
| atTop (𝓝 0) := by |
| intro N |
| have hN j : 0<N j := Nat.add_pos_right _ (Nat.mul_pos (by omega) hn) |
| have hNl : Tendsto N atTop atTop := tendsto_atTop_mono (fun _ => Nat.le_add_left _ _) |
| (by simpa only [Nat.add_comm] using cavityRationalDimension_tendsto n q hn) |
| let f := fun j => cavityAffineLabel hm spec c hspec (N j) |
| have hfcard j a : spinGroupSize (f j) a=cavityAffineCount spec c q j a := by |
| dsimp only [f,N] |
| rw [show (∑ a, c a)+(q+j)*n=cavityPrefixAffineSize n q c j by |
| simp only [cavityPrefixAffineSize,Nat.add_comm]] |
| rw [cavityPrefixAffineLabel_progression hm spec c hspec hn] |
| exact cavityOrderedGroup_card _ _ a |
| have hf : Tendsto (fun j a => (spinGroupSize (f j) a : ℝ)/N j) atTop (𝓝 ρ) := by |
| apply tendsto_pi_nhds.mpr |
| intro a |
| have hratio : (spec a : ℝ)/n=ρ a := by |
| rw [hcount, mul_div_cancel_left₀ _ (Nat.cast_ne_zero.mpr hn.ne')] |
| simpa only [hfcard,N,cavityAffineSize,Nat.add_comm,hratio] using |
| cavityAffineCount_ratio spec c hn hq a |
| let gamma : Option A → ℝ := fun a => Option.elim a 0 γ |
| let field : Option A → ℝ := fun a => Option.elim a 0 b |
| have hfield a : |field a|≤B := by |
| cases a with |
| | none => simpa only [field,Option.elim_none,abs_zero] using hB |
| | some a => exact hb a |
| have hpop := offsetSiteGroup_population_tendsto hn (∑ a, c a) q hq group γ hgroup |
| have htarget := someSiteGroup_population_tendsto N (fun j => g (N j)) γ (hg.comp hNl) |
| have hh := physical_mean_joint_populations_tendsto N hN (fun j => μ (N j)) |
| f (fun j => e (N j)) ρ hρ hf (he.comp hNl) lam hK hlam |
| (fun j => offsetSiteGroup (∑ a, c a) (q+j) group) |
| (fun j i => some (g (N j) i)) gamma field |
| (by simpa only [N,gamma,Nat.cast_add,Nat.cast_mul] using hpop) htarget hB hfield |
| have hfield_eq j : (fun i => field (offsetSiteGroup (∑ a, c a) (q+j) group i))= |
| offsetBlockField (∑ a, c a) (q+j) (fun i => b (group i)) := |
| offsetSiteGroup_field group b |
| convert hh using 1 |
| funext j |
| congr 1 |
| exact congrArg (fun z => ∫ U, rotatedPressure (fun i => lam (f j i)) |
| (matrixRotation U⁻¹) z ∂μ (N j)) (hfield_eq j).symm |
| |
| end InvariantIsing |
| |
| end |
| |
| end OAI |
| |
