File size: 3,924 Bytes
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 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 | 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
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